prr.hec.gov.pkprr.hec.gov.pk/jspui/bitstream/123456789/7723/1/muhammad afzal f… · abstract this...

Analysis of Acoustic Scattering

Problems in Flexible Duct Involving

Step Discontinuity

Muhammad Afzal

Department of Mathematics

Quaid-i-Azam University

Islamabad,

Pakistan 2015


Problems in

Flexible Duct Involving Step

Discontinuity

By

Muhammad Afzal

Supervised By

Prof. Dr. Muhammad Ayub



Islamabad,

Pakistan 2015


Problems in Flexible Duct Involving

Step Discontinuity

By

Muhammad Afzal

A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT OF THE

REQUIREMENT FOR

THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

Supervised By

Prof. Dr. Muhammad Ayub



Islamabad,

Pakistan 2015

ABSTRACT

This dissertation addresses a class of boundary value problems arising in the modelling

of scattering of fluid-structure coupled waves in ducts or channels with abrupt

geometric changes and discontinuous material properties. The main aim is to invoke

the semi-analytic methods such as mode-matching and Galerkin techniques to render

the solutions to these archetypal boundary value problems. The envisaged scattering

problems are governed by Laplace or Helmholtz equations together with high-order

boundary conditions describing flexible duct or channel boundaries. It is substantiated

that the separation of variables reduces these boundary value problems to non-Strum-

Liouville eigen-systems wherein the eigen-functions are neither orthogonal nor

linearly independent. In this effect, generalized orthogonal relations are established

which, together with a mode-matching technique, render a solution by matching

modes across the interfaces of the geometric and material discontinuities. The

emphasis in the dissertation is on membrane and elastic plate bounded ducts involving

step discontinuities at interfaces with varying properties. Precisely, soft, rigid and

flexible step discontinuities are taken into account. The viability of the modematching

technique from the perspectives of the generalized orthogonal relations to address

aforementioned acoustic scattering problems in ducts or channels with soft, rigid or

flexible vertical strips is mathematically analysed and is supported with apposite

numerical experiments. Due to the slow convergence rate of mode-matching solutions

for scattering problems involving flexible strips, Galerkin method is employed as an

attractive and potential alternative. The expressions for energy fluxes in different

regimes are obtained and the analysis of power balance is performed.

i

CONTENTS

ABSTRACT I

CONTENTS III

LIST OF FIGURES VII

1 INTRODUCTION 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Avant Garde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Dissertation Catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 PRELIMINARIES 7

2.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3 Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . 12

2.1.4 Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Canonical Problems in Structural Acoustics . . . . . . . . . . . . . . 16

2.2.1 Problem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1.1 Mode-Matching Solution . . . . . . . . . . . . . . . 17

2.2.1.2 Expressions for Energy Flux and Power Balance . 19

2.2.1.3 Numerical Results and Discussion . . . . . . . . . 20

2.2.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2.1 Expression for Energy Flux and Power Balance . 28

2.2.2.2 Numerical results and discussion . . . . . . . . . . 30

2.3 Low Frequency Approximation . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Galerkin/Modal Approach . . . . . . . . . . . . . . . . . . . . . . . . . 34

iii

iv CHAPTER 0

3 ACOUSTIC PROPAGATION AND SCATTERING IN A TWO-DIMENSIONAL

WAVEGUIDE INVOLVING STEP DISCONTINUITY AND FLANGED JUNC-

TION 35

3.1 Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Mode-Matching Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Zero Displacement . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Zero Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Low Frequency Approximation Solution . . . . . . . . . . . . . . . . 41

3.4 Expressions for Power Distribution . . . . . . . . . . . . . . . . . . . 43

3.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . 44

3.6 Validation of Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 ACOUSTIC SCATTERING IN PLATE-MEMBRANE BOUNDED WAVEG-

UIDE INVOLVING STEP DISCONTINUITY 61

4.1 Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Mode-Matching Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Edge Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.1 Clamped Edge Condition . . . . . . . . . . . . . . . . . . . . . 67

4.3.2 Pin-jointed Edge Condition . . . . . . . . . . . . . . . . . . . . 67


4.4.1 Power Distribution Versus Height of Outlet Duct . . . . . . 69

4.4.1.1 Fundamental mode incidence . . . . . . . . . . . . 69

4.4.1.2 Secondary mode incidence . . . . . . . . . . . . . . 69

4.4.2 Power Distribution Versus Frequency . . . . . . . . . . . . . 72

4.4.2.1 Fundamental Mode Incidence . . . . . . . . . . . . 72

4.4.2.2 Secondary Mode Incidence . . . . . . . . . . . . . . 72

4.5 Validation of Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 ACOUSTIC SCATTERING IN A MEMBRANE BOUNDED DUCT WITH

ABRUPT HEIGHT CHANGE 77

5.1 Canonical Problem with Rigid Vertical Strip . . . . . . . . . . . . . . 78

5.1.1 Mode-Matching Solution . . . . . . . . . . . . . . . . . . . . . 80

5.1.2 Expresion for Power Balance . . . . . . . . . . . . . . . . . . . 82

5.1.3 Numerical Discussion and Results . . . . . . . . . . . . . . . 83

5.2 Canonical Problem With Vertical Membrane . . . . . . . . . . . . . . 87

5.2.1 Mode Matching Solution . . . . . . . . . . . . . . . . . . . . . 89

5.2.2 Power Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.3 Numerical Results and Discussion . . . . . . . . . . . . . . . 93

5.3 Galerkin Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.1 Zero Displacement at y =b . . . . . . . . . . . . . . . . . . . . 99

5.3.2 Zero Gradient at y =b . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Modal Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

CONTENTS v

5.4.1 Zero Displacement at y = a and y

=b . . . . . . . . . . . . . . 102

5.4.2 Zero Displacement at y = a and Zero Gradient at y

=b . . . 102


6 ACOUSTIC SCATTERING IN ELASTIC PLATE BOUNDED DUCT WITH ABRUPT HEIGHT CHANGE 107

6.1 Problem With Rigid Vertical Strip . . . . . . . . . . . . . . . . . . . . . 108

6.1.1 Mode Matching Solution . . . . . . . . . . . . . . . . . . . . . 110

6.2 Problem With Vertical Elastic Plate . . . . . . . . . . . . . . . . . . . . 112

6.2.1 Galerkin Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 113


7 CONCLUSION 121

BIBLIOGRAPHY 125

LIST OF FIGURES

2.1 Duct geometry for a general problem. . . . . . . . . . . . . . . . . . . . . 9

2.2 The geometric configuration of the duct for Problem 1 . . . . . . . . . .

2.3 Real part of normal velocity field versus y for the fundamental mode

16

and secondary mode incidences. . . . . . . . . . . . . . . . . . . . . . . .

2.4 Imaginary part of normal velocity field versus y for the fundamental

21

mode and secondary mode incidence. . . . . . . . . . . . . . . . . . . . .

2.5 Real part of normal velocity field versus y for the fundamental mode

21

and secondary mode incidences. . . . . . . . . . . . . . . . . . . . . . . .

2.6 Imaginary part of normal velocity field versus y for the fundamental

22

mode and secondary mode incidences. . . . . . . . . . . . . . . . . . . .

2.7 Real part of pressure versus y for the fundamental mode and sec-

22

ondary mode incidences. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8 Imaginary part of pressure versus y for the fundamental mode and

23

secondary mode incidences. . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9 The reflected power versus frequency (Hz) for the fundamental mode

23

incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.10 The reflected power versus frequency (Hz)for the secondary mode

24

incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.11 The power reflected via−membrane, fluid and both versus frequency

24

(Hz) for the fundamental mode incidence. . . . . . . . . . . . . . . . . .

2.12 The power reflected via−membrane, fluid and both versus frequency

25

(Hz) for the secondary mode incidence. . . . . . . . . . . . . . . . . . . . 25

2.13 The duct configuration for prototype problem II. . . . . . . . . . . . . .

2.14 The real and imaginary parts of vertical membrane displacement ver-

26

sus y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.15 The real and imaginary parts of normal velocity field versus y. . . . . 31

2.16 The real and imaginary parts of pressure versus y. . . . . . . . . . . . . 32

vii viii CHAPTER 0

2.17 The reflected power plotted against the number of terms. . . . . . . . . 32

2.18 The error term plotted against the number of terms. . . . . . . . . . . .

2.19 The reflected power, error and their combination versus the number

33

of terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Non-dimensional geometry of the waveguide . . . . . . . . . . . . . . .

3.2 Zero displacement at edges and incident forcing via fundamental

36

mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Zero displacement at edges and the incident forcing via secondary

45

mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Zero gradients at edges and the incident forcing via fundamental mode. 46

3.5 Zero gradients at edges and the incident forcing via secondary mode. 46

3.6 Zero displacement at edges and the incident forcing via fundamental

mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7 Zero gradients at edges and incident forcing via fundamental mode. . 47

3.8 Zero displacement at edges and incident forcing via fundamental mode. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.10 Zero gradient at edges and the incident forcing via fundamental mode. 49

3.11 Zero gradient at edges and the incident forcing via secondary mode. 49


mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50



mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


3.17 Zero gradient at edges and the incident forcing via secondary mode. 52


mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53



mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


3.23 Zero gradient at edges and the incident forcing via secondaryl mode. 55


mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56


LIST OF FIGURES

3.26 The real part of pressures in two duct regions are plotted against the

ix

duct height at matching interface. . . . . . . . . . . . . . . . . . . . . . . .

3.27 The imaginary part of pressures in two duct regions are plotted against

57

the duct height at matching interface. . . . . . . . . . . . . . . . . . . . .

3.28 The real part of normal velocities in two duct regions are plotted

57

against the duct height at matching interface. . . . . . . . . . . . . . . .

3.29 The imaginart part of normal velocities in two duct regions are plot-

58

ted against the duct height at matching interface. . . . . . . . . . . . . .

3.30 The real part of normal velocities in two duct regions are shown

58

against the duct height after using Lanczos filters. . . . . . . . . . . . .

3.31 The imaginary part of normal velocities in two duct regions are shown

60

against the duct height after using Lanczos filters. . . . . . . . . . . . . 60

4.1 Non-dimensional geometry of the waveguide. . . . . . . . . . . . . . . . 62

4.2 For fundamental mode incident and clamped edge conditions. . . . 70

4.3 For fundamental mode incident and pin-jointed edge conditions. . . . 70

4.4 For secondary mode incident and clamped edge conditions. . . . . . . 71

4.5 For secondary mode incident and pin-jointed edge conditions. . . . . 71

4.6 For fundamental mode incident and clamped edge conditions. . . . 72

4.7 For fundamental mode incident and pinjointed edge conditions. . . . 73

4.8 For secondary mode incident and clamped edge conditions. . . . . . . 74

4.9 For secondary mode incident and pinjointed edge conditions. . . . . .

4.10 At matching interface the real part of pressures are plotted against

74

the duct height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.11 At matching interface the imaginary part of pressures are plotted

75

against the duct height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.12 At matching interface the real part of normal velocities are plotted

75

against the duct height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.13 At matching interface the imaginary part of normal velocities are

76

plotted against the duct height. . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1 Non-dimensional geometry of the waveguide . . . . . . . . . . . . . . . 78

5.2 The real part of pressures against y at matching interface. . . . . . . . . 84

5.3 The imaginary part of pressures against y at matching interface. . . . 84

5.4 The real part of normal velocities against y at matching interface. . . . 85

5.5 The real part of normal velocities against y at matching interface. . . . 85

5.6 The real and imaginary parts of the normal velocity versus y. . . . . . 86

5.7 The real and imaginary parts of the normal velocity versus y. . . . . .

5.8 The power components versus frequency (Hz) for fundamental mode

86

incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.9 The power components versus frequency (Hz) for secondary mode

87

incident (which cuts-on at 553 Hz). . . . . . . . . . . . . . . . . . . . . . . 87 x CHAPTER 0

5.10 Non-dimentional geometery of the duct with verticle membrane strip. 88

5.11 The real part of pressures at the matching interface. . . . . . . . . . . . 93

5.12 The imaginary part of pressures at the matching interface. . . . . . . . 94

5.13 The real part of normal velocities at the matching interface. . . . . . . 94

5.14 The imaginary part of normal velocities at the matching interface. . .

5.15 For the fundamental mode incident the power components are plot-

95

ted against frequency (Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.16 For the secondary mode incident the power components are plotted

95

against frequency (Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.17 The power balance, error term and the power balance identity are

96

plotted against truncated terms (N). . . . . . . . . . . . . . . . . . . . . . 96

5.18 The real part of normal velocities using Galerkin solution at matching

interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.19 The imaginary part of normal velocities using Galerkin solution at matching

interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.20 The real part of normal velocities using modal solution at matching interface. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.21 The imaginary part of normal velocities using modal solution at matching

interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.22 A comparison of the power components versus frequency for Galerkin solution

(solid and dashed line) and Modal solution (dotted on solid and dashed lines)

subject to the fundamental mode incidence and zero displacement condition at y

= a and y =b. . . . . . . . . . . . . . . 105



subject to the secondary mode incidence (which cuts-on at 553Hz and zero

displacement condition at y = a and y

=b. 105



subject to the fundamental mode incidence, zero displacement condition at y = a

and zero gradient condition at y =b. . 106



subject to the secondary mode incidence (which cuts-on at 553Hz), zero

displacement condition at y = a and zero gradient condition at y =b. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . 106

6.1 The physical configuration of the elastic plate bound waveduide with rigid vertical

strip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 The geometry of the elastic plate bounded waveguide with abrupt height change.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 The real component of pressures versus y at the matching interface. . 116

LIST OF FIGURES xi

6.4 The imaginary component of pressures versus y at the matching interface. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.5 The real component of normal velocities versus y at the matching interface. . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.6 The imaginary component of normal velocities versus y at the matching interface.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.7 The real part of W(y) (dashed line) and ψ2x (0,y) (solid line). . . . . . . 118

6.8 The imaginary part of W(y) (dashed line) and ψ2x(0,y) (solid line). . . 118

6.9 The comparison of the power components versus frequency (Hz) for the

fundamental mode incidence (which cuts-on at 191Hz). The dotted lines show the

results of Galerkin approach. . . . . . . . . . . . . . . 119

6.10 The comparison of the power components versus frequency (Hz) for the

secondary mode incidence (which cuts-on at 191Hz). The dotted lines show the

results of Galerkin approach. . . . . . . . . . . . . . . . . 119

1 INTRODUCTION

1.1 MOTIVATION

The problems involving wave propagation in ducts or channels with flexible walls have

been of great interest to scientists and engineers. The curiosity of modeling and

demystifying the physical situations involving the propagation of electromagnetic,

water or sound waves in ducts or channels is excited due to diverse applications, for

instance, in structural mechanics and hydrodynamics. Refer, for example, to wherein

Dalrymple and Martin [1] studied the propagation of water waves in an infinitely

stretched rectangular duct and wherein Peat [2] studied the acoustic exhaust system.

Moreover, a spectrum of problems of other classes can also be reduced to those in

ducts or channels by exploiting the periodicity or symmetry of the underlying

structure. Erabs [3] used the diffraction grating of equal spacing to arrive at a simple

waveguide problem that can be solved using the Weiner-Hopf technique. Another

example can be found in Dalrymple and Martin [4]. They used the periodicity of

breakwater in the interaction of water waves to arrive at a simple waveguide problem.

However, according to Biggs and Porter [5] the simplification can only be made if the

array of breakwaters or diffraction grating is equally spaced.

This thesis concerns with the acoustic wave propagation in waveguides or ducts.

The study is important mainly because of envisaged applications of scattering

phenomena in structural design and ducting system of buildings and aircrafts. Duct like

structures are also widely used in heating, ventilation and air conditioning (HVAC)

systems and the acoustic scattering in these ducts is a common feature. The later

becomes more intriguing when there occurs an abrupt change in height or underlying

material properties posing challenging mathematical riddles. For instance, welds,

rivets and small physical variations in the properties of adjacent panels in an aircraft

wing give rise to scattering of fluid-structure coupled waves. It is of vital importance

for design engineers to fathom the qualitative features of sudden variation in panel

depth or the traceof a weld. The presence of two or more of such phenomena gives

rise to the possibility of resonance, which in turn can lead to a structural fatigue. The

engineering noise localization and reduction is another important issue present in the

industrial and mechanical processes. Typically, the noise is generated by the

mechanical devices such as combustion engines and fans that propagates through the

2

networks of ducts to the outside world. The unwanted sound travels significant

distances due to reflection and transmission through the internal walls of the duct.

1.2 STATE OF THE ART

The problem of noise reduction has been extensively studied over the last few

decades. Huang [6, 7, 8] and Huang and Choy [9] considered the reactive silencer used

in HVAC system for reducing ducted tonal fan noise. The duct, considered parallel to

the axis of the expansion chamber, was taken to be bounded by a membrane with

varying height. Due to the variation in height of the membrane, the device was tuned

thereby providing stop-band at specified frequency. The flexible channels have been

considered by Dowed and Voss [10] to analyze the cavity-backed panel at the low

frequency range in the presence of fluid flow. Afterwards, Kang and Fuchs [11]

examined their proficiency in the case of cavity-backed micro perforated membranes.

Recently, Lawrie and Guled [12, 13] investigated the performance of two dimensional

modified reactive silencers due to their potential use as hybrid devices in HVAC ducting

systems. The investigation proposed that the stop-band by the silencer can be

broadened and/or shifted with height of the membrane.

It is well known that the problems involving structures having planar boundaries

with abrupt change in material properties may lead to the solution by the classical

Wiener-Hopf (WH) technique; see, for instance, Noble [14]. Refer also to Brazier-Smith

[15], Norris and Wickham [16], and Cannell [17, 18]. Koch [19, 20] considered an

acoustically lined duct of rectangular configuration and used the WH technique to

debate the attenuation of sound. Rawlin [21] applied WH technique to discuss the

radiation of sound from an unflanged cylindrical duct with absorbing boundaries.

Hassan and Rawlin [22, 23] and then Ayub et al. [24, 25, 26, 27, 28] used this technique

to discuss the acoustic scattering in bifurcated and trifurcated waveguide problem.

Demir and Buykoksy [29] predicted the transmission loss in locally lined reactive

silencer using WH technique. A brief historical perspectives of WH technique can be

found in [30].

This standard WH approach fails for more complicated geometries such as ducts

or channels involving discontinuities and/or ducts or channels bounded by wave

bearing boundaries (corresponding to high order boundary conditions). For non-planar

boundaries, there are no standard solution methods. The prob- 1.2. STATE OF THE ART

lems with structures comprising of two planar surfaces, joined together to form a

wedge, may be solved by recourse to the Kontorovich-Lebedev transform [31, 32], or

the Sommerfeld integral [33, 34]. The problems with geometric discontinuities that are

tractable analytically include those involving wave propagation in waveguides having

abrupt change in height [35].

In the recent years, mode-matching (MM) techniques have been devised to deal

with more complicated geometries and the problems involving propagation in

ducts/channels with high order boundary conditions. Such methods were originally

developed to solve canonical problems governed by Laplace or Helmholtz equation

and the duct/channel boundaries described by Dirichlet, Neumann or Robin

conditions. The solution of these problems can be expressed as eigenfunction

expansion form by using the separation of variable technique. The underlying eigen-

sub-systems is Sturm-Liouville (SL) possessing well defined orthogonality principles.

Consequently, in the process of mode-matching across the interface between two

regions, the orthogonality relation renders a well-behaved infinite system of linear

algebraic equations. Therefore, numerous problems involving complicated geometric

structures and material discontinuities in a wide range of applications associated with

water waves, acoustics and electromagnetic theory have been solved using mode-

matching technique wherein orthogonality relations prove to be extremely useful. We

refer the reader, for instance, to the works by Lebedev et al. [36], Evans and Linton

[37], Meiver [38], Evans and Porter [39, 40] and Dalrymple and Martin [1].

The solution to the analogous problems with soft, rigid or impedance boundary

conditions are usually solvable by means of MM technique along with standard OR.

However, the envisaged model problems involve second or higher order derivatives in

boundary conditions and hence the use of a standard OR becomes inappropriate. For

instance, Folk and Herczynski [41, 42] dealt with an elastic system for which separation

of variables leads to a non-SL eigensystem. Albeit, they are able to derive an

orthogonality relation rendering a solution to the problem. Fama [43] formally derived

an OR to study the elastic response of circular cylinder. Several other scientists

exploited ORs to study a spectrum of problems, for example, Drazin and Rein [44] for

Sommerfeld equation, Murphy et al. [45] and, Zheng-Dong and Hagiwara [46] for fluid

loaded elastic structures, Seligson [47] in electromagnetism, Scandrett and Frenzen

[48] to discuss the porous media effect and, Rao and Rao [49] to solve the fourth order

derivative containing equations.

The presence of higher order fields equations leads to non-SL system wherein the

use of classical orthogonality principles is inappropriate. Even a simple second-order

field equation in conjunction with second or higher order boundary conditions, for

example associated with the fluid-coupled motion of membrane or elastic plate

(Junger and Feit [50], the eigen-sub-systems are non-SL.

The eigenvalues are usually defined as the roots of complicated dispersion relation

that give rise to non-orthogonal eigen-functions. The complexity involved in dispersion

relation and OR, and the imposition of edge conditions at the end of the boundaries

specified with higher order derivative might be the reason behind this. However,

Lawrie and Abraham [51] were pioneers who considered a generalized form of

boundary value problem and demonstrated related orthogonality principles. Since

then similar type of ORs have been exploited in literature to deal with assorted physical

situations. We refer, for instance, to Warren et al. [35] for acoustic scattering in a

waveguide involving discontinuity in structure and material properties, Kaplunov [52]

for the dispersion of wave in elastic layered media, Kirby and Lawrie [53] for the

dissipative silencer, Guled and Lawrie [13] for the tuning of reactive silencer and Teng

4

et al. [54], Evans and Porter [55], Linton and Chung [56], Chung and Linton [57], and

Porter and Evans [58] for problems related to water waves.

Recently Lawrie [59] proved analytic properties of the non-SL systems and

discussed the convergence issue. Refer also to [51] wherein some of these properties

are also discussed. She proved that the eigenfunctions belonging to this class are

linearly dependent and substantiated, by means of Green’s function, that the

corresponding insatz suitably converges point-wise. Moreover, the completeness

property was proved only for wave bearing boundaries (membrane and elastic plate).

Manam [60] adopted similar approach to prove the properties associated to the

propagation of water waves. To mention a few examples, we refer to the studies in

[61, 62, 63, 64, 65, 66, 67].

In contrast, Huang [6, 68] used low frequency approximation (LFA) approach where

the field potential relies on the infinite sums of Fourier integrals and length bounding

surfaces. The Fourier coefficients can be determined by substituting the velocity

potential into membrane condition together with the imposed condition. The

approach works well for the case when a membrane is attached at the mouth of the

expansion chamber of course in low frequency regime. Unfortunately, the extension

of this method to general problem, for example those with upper/below membrane

bounded ducts or channels, is no more appropriate. In these situations the membrane

is used as an acoustic absorbent to reduce the low frequency noise. But the recent

development by Lawrie and Guled [13] to study reactive silence and then Nawaz and

Lawrie [67] to discuss the acoustic scattering at flanged junction in elastic plate

bounded duct proved that the LFA renders very accurate solution in the low frequency

regime.

A Galerkin approach is also exploited to study a wide range of related problems.

For example, Norris and Wickhman [16] discussed the diffraction of acoustic waves at

the mutual interface of two plates, Brazier-Smith [15] explained the acoustic

properties of two plates in half-plan, Gorman [69] used superposition- 1.3. AVANT GARDE

Galerkin approach to analyze the plate vibration, Huang [7, 9] analyzed the drumlike silencer and

the reflection of broadband sound from plate covering the duct cavities, and more recently Liu et

al. [70] conferred a silencer of plates along with generalized form of boundary conditions. Some

relevant recent developments on hybrid silencer wherein the fluid-structure coupled system

arised can also be found in [71, 72, 73].

1.3 AVANT GARDE

The main aim of this dissertation is to analyze the acoustic scattering problems in

flexible duct involving step discontinuity. This work can be seen as a continuation of

the investigation made by Warren et al. [35] in the following perspective:

(1) The inclusion of flanged junction at matching interface and the comparative

study of MM and LFA solution methods.

(2) The elastic plate and membrane bounded duct problems together with step

discontinuity using MM technique.

The replacement of acoustically rigid vertical strip with membrane strip and a comparative analysis of the Galerkin approach and the Modal approach 5

(3) blended with MM technique.

(4) The shifting of membrane bounded duct problems to elastic plate problems and

their Galerkin solutions.

(5) Consideration of fundamental mode incident as well as secondary mode

incidence for all the problems.

1.4 DISSERTATION CATALOG

The thesis is organized in the order precised below.

CHAPTER 2 commences with the review of generalized form of boundary value

problems that belong to the class of non-SL problems. Since, the thesis is concerned

particularly with the flexible bounded ducts, the properties of non-SL systems and

components of energy flux are discussed. These results are crucial in proving the

convergence of MM solution. Note that the properties of a nonSL system can be found

in [74], whereas, components of energy are utilized by Warren et al. [35] for a

membrane bounded duct. These expressions incorporate both the fluid-borne and the

structure-borne components of energy flux that can also be derived using the

approach taken by Crighton and Oswell [75] together with the appropriate OR. The

power expressions, MM scheme, solution validation and related terms are explained

by solving two canonical problems.

CHAPTER 3 investigates the MM and LFA solutions of a two dimensional waveguide

problem with a flanged junction. The merits and demerits of the approaches are

compared for the scattering of fluid-structure coupled waves. The boundary value

problem involving high-order boundary conditions turns out to be non-SL problem in

nature where the use of standard OR enables the MM solution. The low frequency

approximation is derived which proves to be surprisingly accurate for the structure-

borne mode incidence. In order to validate the truncated model expansion, the

distribution of power in duct regions is discussed and Gibbs oscillations are

incorporated by reconstruction of the normal velocity field using Lanczos filter.

In CHAPTER 4, the propagation and scattering of acoustic waves in a flexible wave-

guide involving step discontinuity at an interface is considered. The emerging

boundary value problem is solved by employing a MM technique. The physical

scattering process and attenuation of duct modes versus frequency regime and change

of height is studied. Moreover, the MM solution is validated through a series of

numerical experiments by testifying the power conservation identity and matching

interface conditions.

6

CHAPTER 5 deals with two variations of a model problem involving the scattering of

fluid-structure coupled waves in a waveguide with abrupt change in height. The first

problem is relevant to that considered by Warren et al [35]. which is discussed by using

MM technique. The replacement of rigid vertical strip with a vertical membrane gives

rise to another interesting problem wherein the MM solution is poorly convergent.

Galerkin and Modal approaches have been applied as an alternative and their results

are compared.

In CHAPTER 6, both the membrane bounded duct problems studied in Chapter 5,

have been shifted to elastic plate bounded duct problems. As in Chapter 5, the MM

technique has been utilized to solve problem with the rigid vertical strip whereas the

problem with vertical elastic plate has been solved by using the Galerkin approach.

The contents of CHAPTER 5 and 6 have been completed in collaboration with Dr Jane

B. Lawrie at Brunel University London.

2 PRELIMINARIES

A plethora of real world problems in engineering design and structural mechanics

involves propagation and scattering of acoustic, elastic or electromagnetic waves in

pipes and ducts having abrupt changes in material properties or geometry. A typical

example is the silencer design for vehicles with an abrupt change in cross-sectional

area and a shielded bounding wall. The discrete nature of the wavenumber spectrum

in such problems allows the total wave field representation by a superposition of

traveling wave modes in each region of constant duct properties. The analysis of

reflection and transmission of waves in pipes and ducts is therefore performed mostly

by matching modes across the interface at discontinuities in pipe or duct properties.

If the eigenfunctions in each region form a complete orthogonal basis, the

orthogonality relations allow the eigenfunction coefficients to be determined by

solving a simple system of linear algebraic equations. The complexity of an

orthogonality relation depends not only on the type of boundary that forms the surface

of waveguide but also on the order of field equation. For structures involving soft, hard

or impedance boundaries and at most a second order field equation, the solution can

be computed in terms of an eigenfunction expansion by virtue of separation of

variables. The emerging orthogonality relation is found to be very simple and the

resulting eigen-sub-system turns out to be Sturm-Liouville (SL). Consequently, in the

process of mode-matching across the interface between two regions, the

orthogonality relation renders a well-behaved infinite system of linear algebraic

equations. Therefore, numerous problems involving complicated geometric structures

and material discontinuities in a wide range of applications associated with water

waves, acoustics and electromagnetic theory have been solved using mode-matching

technique wherein orthogonality relations prove to be extremely useful. We refer the

reader, for instance, to the works by Lebedev et al. [36], Evans and Linton [37], Peat

[2], Evans and Porter [39] and Dalrymple and Martin [1, 4].

In contrast, for high-order field equations, separation of variables leads to

eigenfunction expansions for which the resulting eigen-systems are no more SL, even

with simple boundary conditions. For instance, Folk and Herczynski [41, 42] dealt with

an elastic system for which separation of variables leads to a non-SL eigensystem.

Albeit, they are able to derive an orthogonality relation rendering a solution to the

problem. Moreover, a separable second-order field equation together with high-order

boundary conditions, for example those describing the fluid-coupled motion of a

8 CHAPTER 2

membrane or elastic plate, give rise to a non-SL problem. We refer, for instance, to the

works by Nawaz and Lawrie [67] and, Junger and Feit [50]. The eigenvalues are defined

as the roots of a complicated dispersion relation and the associated eigenfunctions are

not usually orthogonal even with respect to a weight function.

It is worthwhile precising that the derivation of an appropriate orthogonality

relation is not sufficient enough to completely determine a solution to the

aforementioned problems. An additional difficulty is the choice of appropriate edge

conditions at the junction of discontinuity. A practical and convenient mean of

imposing the edge conditions is critical.

The aim of this chapter is to discuss a class of boundary value problems and the

analysis of associated non-SL systems from the perspectives of orthogonality relations.

In particular, we discuss two prototype problems in order to describe the process of

mode matching for analyzing reflection and transmission in waveguides having

material discontinuities and abrupt geometrical changes. Precisely, the problems

undertaken are concerned with the scattering of acoustic waves in a two-dimensional

waveguide consisting of two semi-infinite duct sections wherein the effects of rigid and

flexible walls are taken into account. The expressions for power and energy flux for

each of the flexible wall is discussed and are substantiated through apposite numerical

simulations.

The rest of the chapter is arranged in the following order. Section 2.1 is dedicated

to the mathematical formulation of the problems. In particular, dispersion relations

and related properties are discussed in Section 2.1.1, the boundary conditions relevant

to rigid walls, membrane and elastic plate are presented in Section 2.1.2 and the

orthogonality relations and energy fluxes are discussed in Sections 2.1.3 and 2.1.4

respectively. In Section 2.2, two canonical problems are studied and the solution

framework is explained. The unknown scattering coefficients are obtained for both

problems and the expressions of energy fluxes are derived. The chapter concludes with

Sections 2.3 and 2.4 discussing low frequency approximation, Galerkin and modal

approaches.

2.1 MATHEMATICAL FORMULATION

In this section, we present a brief review of the basic notions and results available in

the literature. Most of the discussion follows that in the works [51, 59, 74]. 9

Consider a two dimensional semi-infinite domain bounded above by a membrane

or an elastic plate and below by a rigid wall; see Figure 2.1. Assume that the domain is

filled with a compressible fluid. Without loss of generality, assume that the surfaces

are aligned with y = a and y

= 0 respectively, where a is a positive real number.

2.1. MATHEMATICAL FORMULATION

FIGURE 2.1. Duct geometry for a general problem.

Let ψtot(x,y,t) be the transient fluid velocity potential in the waveguide satisfying the

wave equation

, (2.1)

where (x,y) are the Cartesian coordinates, t is the time variable and c is the sound speed.

Assuming an incident forcing with time harmonic dependence, transient velocity

potential ψtot can be expressed as

ψtot(x,y,t) =ℜenΨ(x,y)e−iωto, (2.2)

where ω is the frequency pulsation. The dimensional time harmonic fluid velocity

potential Ψ(x,y), on suppressing the time dependence in (2.1) by virtue of (2.2), satisfies

the Helmholtz equation

Ã

(2.3)

10 CHAPTER 2

Here and throughout this text k =ω/c is coined as wavenumber.

For the sake of convenience, the problem (2.3) can be non-dimensionalized with

respect to the length scale 1/k and the time scale 1/ω using transforma-

tions x =kx, y =ky and t =ωt. The non-dimensional velocity potential ψ(x,y) = ψ(x,y) then

satisfies the equation

¡∇2 +1¢ψ(x,y) = 0. (2.4)

The boundaries of the waveguide can be described, in general, in a physically

admissible form as

µ ∂ ¶∂ψ µ ∂ ¶

Lp +Mp ψ= 0, p ∈0,a, y ∈R, (2.5)

∂x ∂y ∂x

³ ∂ ´

where a = ka; refer, for instance, to [51, 59, 74, 76]. The operators Lp ∂x and

³ ∂ ´ p ∈0,a are the differential operators of the form

Mp ∂x for

µ ∂ ¶ XHp p ∂2h µ ∂ ¶ XJp p ∂2j Lp ∂ x = ch ∂ x2h and Mp ∂ x = j=0dj ∂

x2j , (2.6)

h=0

where the coefficients chp and dj

p are constants and, Hp and Jp are non-negative integers.

2.1.1 DISPERSION RELATIONS

It is well known that the velocity potential ψ admits an eigenfunction expansion

= X

∞ ±isnx for x 6= 0, (2.7) ψ(x,y) AnYn(y)e

n=0

2.1. MATHEMATICAL FORMULATION 11

where An are the mode amplitudes, sn are the wavenumbers and Yn(y) are the

eigenfunctions. Therefore, the expression (2.7), by virtue of (2.4)–(2.5), render

Yn00(y)−γ2nYn(y) = 0, (2.8)

Pp(sn)Yn0(p)+Qp(sn)Yn0(p) = 0, p ∈0,a, (2.9)

where prime indicates differentiation with respect to y and

γ2n = (sn

2 −1).

The coefficients Pp(sn) and Qp(sn) are the characteristic polynomials to be de-

³ ∂ ´ ∂ Mp ³

∂∂

x ´ on eigentermined

by the action of differential operators Lp ∂x ∂y and function expansion (2.7) subject to the

condition (2.5). It is important to note that Pp(sn) ≡Lp(isn) and Qp(sn) ≡Mp(isn) [74].

The eigenfunctions Yn(y) can be obtained explicitly by solving (2.8)–(2.9) in terms of

Pp and Qp as

1

Yn(y) =P0(sn)cosh(γny)− Q0(sn)sinh(γny), γn

subject to the condition K (γn)

= 0, where

n 2 osinh(γna)

K (γn) := γnP0(sn)Pa(sn)−Q0(sn)Qa(sn)

γn n o

P0(sn)Qa(sn)−Pa(sn)Q0(sn) cosh(γna),

is termed as dispersion relation. The above dispersion relation can be solved

numerically for γn. It is worthwhile precising that the roots γn of the dispersion relation

satisfy the following properties [74]

i. For each root γn, there is another root −γn.

ii. There is a finite number of real roots. iii. There is

an infinite number of imaginary roots.

iv. The complex roots ±γc and their complex conjugates occur for some

frequency ranges.

12 CHAPTER 2

The real and imaginary roots are taken by employing a convention that the positive

roots, +γn, are either positive real or have a positive imaginary part. They are sorted

sequentially by placing real root first and then by increasing imaginary part, so that γ0

is the largest real root. For any complex root γc lying in the upper half of the complex

γ−plane, the root also lies in same half plane. The sequence of such pairs is taken

according to the magnitude of their imaginary part, and in the order γc is followed by

. Furthermore, it is assumed that all roots have multiplicity one.

2.1.2 BOUNDARY CONDITIONS

The canonical problems governed by Laplace or Helmholtz equations in a channel or a

duct with boundaries described by rigid, soft or impedance conditions lead to an SL

system with well defined orthogonality principles thereby yielding an infinite system

of linear algebraic equations with unknown scattering amplitudes. The resulting

systems are usually diagonally dominated and the off-diagonal elements decay very

rapidly so that an accurate solution can be obtained by truncation and inversion.

In contrast, the problems involving higher order boundary conditions have non-

orthogonal eigenfunctions and therefore the associated eigen-sub-systems are non-SL.

In this work, we consider a duct or a channel with a rigid lower surface and a membrane

or an elastic plate as the upper boundary. The relevant condition for a rigid horizontal

lower boundary can be described by

0, y = 0, x ∈R. (2.10)

If the upper boundary is a membrane then the appropriate condition is given by

µ

0, y = a, x

∈R, (2.11)

wherein the non-dimensional parameters µ and α are the (in vacuo) membrane

wavenumber and the (in vacuo) fluid loading parameter respectively, defined as

c ω2ρ

µ= c m and α= Tk 3 . (2.12)

In above, T denotes the membrane tension per unit length (in the normal direction)

and cm = T/ρm the speed of waves (in vacuo) on the membrane where ρm is the

membrane mass per unit area, and ρ is the compressible fluid density [35]. Moreover,

if the upper surface represents an elastic plate then the nondimensional form of the

boundary condition is given by (see, for instance, [67])


µ ∂4 ¶

0, (2.13)

where the non-dimensional parameter µ1 is the in vacuo plate wavenumber and α1 is a

fluid loading parameter defined by

12(1 υ2)c2ρ

4 = − p = 12(1−υ2)c2ρ

µ1 k2h2E and α1 k3h3E , (2.14)

in terms of Young’s modulus E, plate density ρp and the Poisson’s ratio υ. Note that the

conditions (2.12), (2.13) and (2.14) can be deduced from the generalized from (2.5) by

respectively substituting (P0(sn),Q0(sn))

= (1,0), (Pa(sn),Qa(sn))

= (−sn2 +µ2,α) and (Pa(sn),Qa(sn))

= (sn4 −µ4

1,−α1) into (2.9).

The subtlety occurs when dealing with the second or fourth-order boundary

conditions. In these cases, the edge condition must be applied at the junction between

any two boundaries when at least one of them is described by a highorder condition.

Clearly, the SL orthogonality relation leaves no choice to apply those edge conditions.

From this perspective, new orthogonality conditions are discussed in the subsequent

section.

2.1.3 ORTHOGONALITY RELATIONS

In this section, we discuss the orthogonality relations(ORs) satisfied by eigenfunctions

Yn, for n = 0

,1,2,··· , for membrane and elastic plate bounded ducts or channels.

For a membrane bounded duct, the eigenfunction expansion (2.7) must satisfy the

boundary value problem

¡∂ψ∇2 +=10

¢,ψ(x,y) = 0 ((xx,,yy)) ∈∈R

R××0(0,,a),

∂y

µ R×a.

This leads to the following system of equations

Yn00(y)−(sn2 −1)Yn(y) = 0, (2.15)

Yn0(0) = 0, (2.16)

14 CHAPTER 2

(sn2 −µ2)Yn0(a)−αYn(a) = 0, (2.17)

for the eigenfunctions Yn for all n = 0,1,

···. Recall that the eigenvalues are the roots of

equation (2.17). On multiplying Equation (2.17) with Ym0 (a), for some m = 0,1,··· , and

after simple manipulations, we arrive at

h i

a ³ ´

Yn00(y)Ym(y)−Ym00(y)Yn(y) dy = 0. (2.19)

Consequently, by (2.15) it is found that

a o

Ym(y)Yn(y)dy = 0. (2.20)

Now, when m 6=n, we have

a

Yn0(a)Ym0 (a)+α Ym(y)Yn(y)dy = 0. (2.21) 0

However, if m =n we get sm2 −sn

2 = 0. We set

h i2 a

Cn := Yn0(a) +αˆ Y . (2.22) 0

Note that Cn 6= 0. Hence on combining (2.21) and (2.22) it is obvious to write

a

α Ym(y)Yn(y)dy +Yn0(a)Ym0 (a) =δmnCn, (2.23) 0

(sm2 −sn

2)Yn0(a)Ym0 (a)−α Yn0(a)Ym(a)−Ym0 (a)Yn(a) = 0,

Together with (2.16), Equation (2.18) therefore provides

(2.18)


which is the generalized form of OR for a membrane bounded duct. Here δmn is the

Kronecker’s Delta function.

Similarly, for elastic plate bounded ducts, the eigenfunction expansion must satisfy

¡∂ψ∇2 +=10

¢,ψ(x,y) = 0 ((xx,,yy)) ∈∈R

R××0(0,,a),

∂y

µ 0, (x,y) ∈R×a,

leading to the system of equations

Yn00(y)−(sn2 −1)Yn(y) = 0,

Yn0(0) = 0, (2.24)

(sn4 −µ41)Yn0(a)−α1Yn(a) = 0.

Therefore, the generalized OR for elastic plate bounded duct is given by

a

α1YmmnDn, (2.25) 0

where non-zero coefficients Dn are defined by

h i2 a

Dn := 2 snYn0(a) +α1 ˆ Y . (2.26) 0

It is important to note that the eigenfunctions Yn(y) for n = 0,1,2,

··· are linearly

dependent for flexible bounded ducts. For membrane bounded ducts,

Yn(y) satisfy

a, (2.27) n=0

16 CHAPTER 2

along with identity

X∞

Yn

a, (2.29) n=0 n=0

together with

and sn −= 1. (2.30)

n=0 n=0 Dn

The analytic proves of generalized forms of these ORs can be found in [74]. It is well

known that the number of such linearly dependent sums is always equal to half of the

order of highest derivative involved in the boundary conditions; refer, for instance, to

[35, 74]. That is, the number of edge conditions imposed at the corners of the

boundary are half of the order of boundary conditions. Since for the case of membrane

bounded duct the highest derivative involved is of second order, therefore one edge

condition is required at the membrane edge. However, for the case of elastic plate

bounded duct, two edge conditions will be required on the elastic plate edges. In

addition, a Green’s function can be constructed in terms of the eigenfunctions in the

following form

α =δ(y −υ)+δ(y +υ)+δ(y +υ−2a), 0 ≤υ,y ≤ p, (2.31) n=0 En

where δ(y) is the Dirac mass at 0. This relation is useful in the evaluation of the sums in

order to have a point-wise convergent eigenfunction expansion.

2.1.4 ENERGY FLUX

The understanding of the energy flux is important to measure the accuracy of the

approximate solution. Moreover, it provides a physical insight of the boundary value

problem in terms of reflected and transmitted powers. In this section, we briefly recall

the expressions of energy fluxes for flexible waveguides. It is worthwhile precising that

the presented solution should obey the power conservation law, that is, the power fed

into the system must be equal to the sum of reflected and transmitted power under

adiabatic conditions.

1.

n=0 Cn =

For elastic plate bounded ducts we have

(2.28)


In the problem under consideration, the power fed into the system will transfer

through compressible fluid in the duct and through the walls of the duct. The energy

flux in the fluid present inside the flexible duct of height a in terms of non-dimensional

time harmonic fluid velocity potential is defined by

∂ε ½ a µ∂ψ¶∗ ¾

1¯fluid =ℜe i ˆ0 ψ ∂x dy , (2.32)

∂t

where superposed asterisk (∗) denotes the complex conjugate; see [50, 35]. If the duct

has a flexible surface taken as a membrane at y = a then the energy flux per unit length

in z−direction along the surface is given by

½ i µ∂ψ¶µ ∂2ψ ¶∗¾

e . (2.33)

∂t ¯

memb =ℜ α ∂y ∂x∂y

Furthermore, if the flexible boundary at y = a is replaced by a thin elastic plate then the

energy flux per unit length in z−direction along the plate is given by

½ i ·µ ∂2ψ ¶µ ∂3ψ ¶∗ µ∂ψ¶µ ∂4ψ ¶∗¸¾

2.2 CANONICAL PROBLEMS IN STRUCTURAL ACOUSTICS

Here we cater to two problems arising in structural acoustics concerned with the

scattering of acoustic waves in two-dimensional waveguides consisting of two semi-

infinite duct sections. The material properties of each duct section is assumed to be

different from the other one. The rigid and flexible walls are taken into account. The

rigid wall is defined in terms of Neumann boundary condition whereas the flexible

walls are defined in terms of higher order conditions. The application of the

orthogonality relation is established and the mode-matching technique is invoked to

1

¯plate =ℜe α ∂x∂y ∂x2∂y − ∂y ∂3x∂y . (2.34)

∂t

In order to further elucidate the solution framework presented in this section, two

archetypal problems in structural acoustics involving higher order boundary conditions

are studied in the next section.

18 CHAPTER 2

obtain approximate solutions. Later on, the expressions for energy fluxes are discussed

and numerical illustrations are provided.

2.2.1 PROBLEM I

Consider a two-dimensional semi-infinite duct comprising the non-dimensional region

[0,∞)×[0,a]. The duct is bounded below by an acoustically rigid wall at y = 0 and bounded

above by a membrane at y = a in a non-dimensional setting. It is assumed that the duct

is closed by a rigid vertical wall at x = 0; see Figure 2.2. The rest of the physical

configuration is taken to be the same as in Section

2.1.

FIGURE 2.2. The geometric configuration of the duct for Problem 1

A time harmonic acoustic wave is incident from positive x−axis on x = 0. On striking

with rigid vertical wall of the duct at x = 0, it is scattered. As the rigid vertical wall is

perfectly reflecting, there is no transmission, and consequently only the incident and

the reflected field will propagate in the duct.

Membrane

Rigid

Rigid Compressible fluid

2.2. CANONICAL PROBLEMS IN STRUCTURAL ACOUSTICS 19

Since the vertical wall at x = 0 is assumed to be rigid, the appropriate boundary

condition on non-dimensional fluid velocity potential ψ(x,y) is given by

(2.35)

In addition, an edge condition will be required at the corner where the membrane is

connected with rigid vertical wall. The edge condition not only ensures the uniqueness

of the solution but also describes how the membrane and rigid vertical surface are

connected. The choice of edge conditions can significantly alter the scattered field. We

refer to the articles [15, 16, 77] for a comprehensive list of appropriate edge

conditions. In the sequel, we choose a zero displacement edge condition, that is,

0, x = 0, y = a. (2.36)

Therefore, fluid velocity potential ψ(x,y) satisfies the boundary value problem

0, (x,y)

(0, ) 0,

µ∂y∂2= 2¶ ∈ ∞ ×

2 +µ ψy +αψ= 0, (x,y) ∈(0,∞)×a, (2.37)

∂x

∂ψ∂x = 0, (x,y) ∈0×[0,a),

∂ψ

∂y = 0, (x,y) = (0,a),

where the parameters µ and α are defined in (2.12).

The aim here is to obtain the solution to the boundary value problem (2.37).

This will be done using the mode-matching technique in the next subsection.

¡∇2 +1

¢ψ(x,y) = 0,

∂ψ

(x,y) ∈(0,∞)×(0,a),

20 CHAPTER 2

2.2.1.1 MODE-MATCHING SOLUTION

Consider an incident wave of arbitrary duct mode. Then by separation of variables,

the eigenfunction expansion of fluid velocity potential is given by

Rn cosh(γny)eisnx, `∈0,1, (2.38)

q α

The first term in expansion (2.38) involves the forcing F` = C`s` which is chosen for the

algebraic convenience and to ensure that the incident power is unity. The counter ` is

considered to incorporate two distinct incident duct modes. It assumes the values 0 or

1 according to the fundamental mode or the first higher mode incidence respectively.

The dispersion relation (2.39) can be solved numerically for γn which, in turn, satisfy

the properties listed in Section 2.1.1. Using the eigenfunctions cosh(γny) for n = 0,1,2,

···

in (2.23), the OR is revealed to be a

mn, (2.40)

The unknown reflected mode coefficients Rn for n = 0,1,2,

··· in the eigenfunction

expansion (2.38) can be found by exploiting the rigid vertical wall condition. Indeed,

using (2.38) in the rigid vertical wall condition renders X∞

Rnsn cosh(γny) =F`s`cosh(γ`y), n=0

which on multiplication by αcosh(γmy) and integration over y from 0 to a yields a a ∞

X

RnsnF`s`. n=0

Subsequently, the use of orthogonality relation (2.40) on both sides of the above

relation implies

q where sn = γ2n +1 and γn satisfies the following dispersion

relation

(γ2n +1−µ2)γn sinh(γna)−αcosh(γna) = 0. (2.39)

where = αa µ

3γ2m +1−µ2 ¶

2

Cm : 2 + 2γ2m [γm sinh(γma)] . (2.41)


X∞ RnsnCmδmn −γmγn sinh(γma)sinh(γna) n=0

=F`s`Cmδm`−γmγ`sinh(γma)sinh(γà). (2.42) Now,

rearranging and simplifying (2.42), the explicit form of the reflected field

amplitudes is obtained as

γm sinh(γma)E Rm

=F`δm`+ , (2.43) smCm

where the constant E is defined by ∞

E :=−F`s`γ`sinh(γà)+ X

Rnsnγn sinh(γna) =−iψxy(0,a), (2.44) n=0

and can be evaluated by virtue of the edge condition (2.36). In order to do so, we multiply

(2.43) by γm sinh(γma) and sum over index m, to obtain

∞ ∞ γ2 sinh2(γma)E

S m=0 smCm

2.2.1.2 EXPRESSIONS FOR ENERGY FLUX AND POWER BALANCE

In order to calculate the incident power, we first substitute the incident field

ψinc(x,y) =F`cosh(γ`y)e−is`x, (2.47)

in (2.32) to obtain the power traveling through fluid by

½a ¾

Pinc|fluid

=ℜeF`F`∗s`∗ei(s`−scosh(γ`y)cosh(γ`y)dy .

Using OR (2.40) for m =n =`, Pinc|fluid becomes

X X m Rmγm sinh(γma) =F`γ`sinh(γà)+ . m=0 m=0 smCm

By edge condition (2.36) the value of E is found to be

(2.45)

2F γ sinh(γ a) ∞ 2 2

E = − ` ` ` where S = X γm sinh (γma). (2.46)

22 CHAPTER 2

Pinc|fluid Cè

i

Now, recall that s` is either real or pure imaginary depending upon the value

= q 2 −1 for a non-SL system, therefore (2.48) will survive only for real of γ` s`

values of s` and reduces to

½ 1 2 1 2 2 2 ¾

Pinc|fluid =ℜe F`s`C`− α F`s`γ`sinh (γà) . (2.49) α

Since, F` =p

α/s`C`, we have

½ 1 2 2 2 ¾

Pinc|fluid =ℜe 1− F`s`γ`sinh (γà) . (2.50) α

Similarly, using the incident field (2.47) in (2.33), the power traveling along the

membrane is given by

½ 1 2 2 2 ¾

Pinc|memb =ℜe α F`s`γ`sinh (γà) . (2.51)

Hence, the total incident power is

Pinc =Pinc|fluid +Pinc|memb = 1. (2.52)

Likewise, the expression for reflected power can be calculated by considering the

reflected field

X∞ isnx

ψref (x,y) = Rn cosh(γny)e . (2.53) n=0

Substituting the expression for reflected field (2.53) into (2.32), the power reflected

through fluid appears to be

| ½

1 ∞ ∞sm∗ )xαa

cosh(γmy)cosh(γny)dy¾

,

Pref fluidRnR 0


which by virtue of (2.40) simplifies to

∞∞ ¾ X 2X 2 2 2

Pref |fluid|Rn| snCn|Rn| snγn sinh (γna) . (2.54) n=0n=0

On the other hand, the reflected field (2.53) together with (2.33) provides the power

reflected through membrane by

½ 1 X∞ 2 2 2 ¾

Pref |memb =ℜe |Rn| snγn sinh (γna) . (2.55)

α n=0

Since the total reflected power is

Pref =Pref |fluid +Pref |memb,

therefore,

½¾ 2

Pref =ℜe|Rn| snCn . (2.56)

α n=0

Thus, for x > 0, the energy flux in negative x−direction (2.52) will be equal to the energy

flux in positive x−direction (2.56), that is,

½¾ 2

ℜe|Rn| snCn = 1. (2.57) α n=0

This is known as the power balance or the conserved power identity.

2.2.1.3 NUMERICAL RESULTS AND DISCUSSION

In this section, the solution of the model problem is discussed numerically. The solution

of non-SL problem is suitably convergent [59]. Thus, the equations (2.38) and (2.43) are

truncated upto N terms for some positive integer N in order to check the accuracy of

presented algebra and distribution of energy flux. This not only validates the proposed

solution technique but also provides a useful physical information about the boundary

value problem. While carrying the parametric investigation, the speed of sound in air c =

343ms−1 and density of air ρ = 1.2043kgm−3 are taken from Kaye and Laby [78]. The other

parameters chosen for analysis may vary from case to case and will be stated explicitly.

24 CHAPTER 2

For the configuration considered herein the density and the tension of membrane are

taken to be ρm = 0.1715kgm−2 and T = 350Nm whereas the height of the duct is fixed at a =

0.085m. These parameters are consistent with that of Warren et al.

[35].

In Figures 2.3-2.4, the real and imaginary parts of non-dimensional normal velocity

condition (2.35) stated for rigid vertical wall are plotted in the region 0×[0,a]. It can be

seen that ℜe©

ψx(0,y)ª and ℑm

©ψx(0,y)

ª are zero in y ∈ [0,a). However, near (x,y) = (0,a), (that

is, the corner where the membrane is attached

ÂHΨxL

FIGURE 2.3. Real part of normal velocity field versus y for the fundamental mode and secondary

mode incidences.

ÁHΨxL

F 0

F 1

0.2 0.4 .6 0 0.8 y

0.0002

0.0002

0.0004

0.0006

F 0

F 1

0.2 0.4 0.6 0.8 y

0.015

0.010

0.005

0.005


FIGURE 2.4. Imaginary part of normal velocity field versus y for the fundamental mode and secondary mode incidence.

to the rigid vertical wall) there are periodic fluctuations which show the singular behavior

of normal velocity field at the corner.

Similarly, in Figures 2.5-2.6 the non-dimensional normal velocity in y−direction ψy(0,y)

is plotted in region 0×[0,a]. Clearly, it is observed that ℜeψy(0,y) and ℑmψy(0,y) are zero

at y = 0 and y = a, and are non-zero for y ∈ (0,a). This is exactly the situation which is being

stated algebraically in normal velocity condition (2.37) and edge condition (2.36).

The variation of non-dimensional pressure in region 0×[0,a] are delineated ÂIΨyM

FIGURE 2.5. Real part of normal velocity field versus y for the fundamental mode and secondary

mode incidences.

ÁIΨyM

F 0

F 1

0.2 0.4 0.6 0.8 y

0.8

0.6

0.4

0.2

26 CHAPTER 2

FIGURE 2.6. Imaginary part of normal velocity field versus y for the fundamental mode and secondary mode incidences.

in Figures 2.7-2.8.

Note that in Figures 2.3-2.8 both the fundamental mode incidence and secondary

mode incidence are taken into account while the frequency is fixed at f = 550Hz.

Recall that the incident power is scaled at unity (2.52) and one end of the duct is

closed with rigid vertical wall. Therefore, all the incident power goes ÂHΨL

FIGURE 2.7. Real part of pressure versus y for the fundamental mode and secondary mode incidences.

ÁHΨL

F 0

F 1

0.2 0.4 0.6 0.8 y

1.5

1.0

0.5

0.5

F 0

F 1

0.2 0.4 0.6 0.8 y

1.0

1.5

2.0

2.5


FIGURE 2.8. Imaginary part of pressure versus y for the fundamental mode and secondary mode incidences.

on reflection and no power is transmitted. Here two different duct mode incidences are

considered, that is, the fundamental mode incidence (` = 0) and secondary higher mode

incidence (` = 1). In Figure 2.9, the non-dimensional reflected power is plotted against

frequency 1Hz ≤ f ≤ 1200Hz while the fundamental mode incidence is considered. It can

be seen that the power balance (2.57) is achieved successfully in the whole frequency

regime.

FIGURE 2.9. The reflected power versus frequency (Hz) for the fundamental mode incidence.

F 0

F 1

0.2 0.4 0.6 0.8 y

0.4

0.3

0.2

0.1

0.1

0.2

P ref

0 200 400 600 800 1000 Frequency

0.2

0.4

0.6

0.8

1.0 Power

28 CHAPTER 2

Figure 2.10 depicts the reflected power plotted against frequency 1Hz ≤ f ≤ 1200Hz

while the secondary mode incidence is taken into account. From Figure 2.8, it can be

seen that the power balance (2.57) is achieved successfully in regime f ≥ 452Hz but it is

not satisfied for f < 452Hz. The accuracy is still not improved on increasing the number N

for the later case. However, increasing frequency upto f = 452Hz the reflected power

becomes unity. This is the point where the membrane bounded duct starts propagating

and which is known as the cut-on point for the secondary mode incidence.

FIGURE 2.10. The reflected power versus frequency (Hz)for the secondary mode incidence.

Figures 2.11-2.12 are plotted against frequency to see the distribution of power

transferring through fluid and along the membrane for both the fundamental mode

incidence (`= 0) and the secondary mode incidence (`= 1).

P ref

0 200 400 600 800 1000 Frequency

0.5

1.0

1.5

2.0

Power

P ref fluid

P ref memb

P ref

0 200 400 600 800 1000 Frequency

0.2

0.4

0.6

0.8

1.0 Power


FIGURE 2.11. The power reflected via−membrane, fluid and both versus frequency (Hz) for the

fundamental mode incidence.

FIGURE 2.12. The power reflected via−membrane, fluid and both versus frequency (Hz) for the

secondary mode incidence.

Figure 2.11 substantiates that for the fundamental mode incidence (` = 0), most of

the power travels along the membrane and therefore, it can be regarded as structure-

born mode. In contrast, as the secondary mode incidence (` = 0) is cut-on, the maximum

of the power travels through fluid and so it is fluid-born mode; refer to Figure 2.12.

2.2.2 PROBLEM 2

In the second canonical problem studied in this section, the same physical configuration

is considered as in Problem 1 except that the rigid vertical strip at x = 0 is now replaced

by membrane. This will alter the vertical wall condition prescribed in (2.35) and edge

condition (2.36), whereas the rest of the equations in (2.37) remain unchanged. The

physical configuration of the problem is illustrated in Figure 2.13.

P ref memb

P ref fluid

P ref

600 800 1000 1200 Frequency

0.2

0.4

0.6

0.8

1.0 Power

30 CHAPTER 2

FIGURE 2.13. The duct configuration for prototype problem II.

The vertical membrane wall condition is given by

µ ∂2 2¶∂ψ µ αψ 2E

∂y2 + ∂x − = 1δ(y)+2E2δ(y −a), (x,y) ∈0×(0,a), (2.58)

where δ is used to impose two extra edge conditions on vertical membrane edges and,

E1 and E2 are material dependent constants . Thus, there are now three edge conditions

corresponding to zero displacement at each membrane edge, that is

ψx(0,y) = 0, y = 0,a, (2.59)

and

ψy(0,a) = 0. (2.60)

The expressions for the scattered fields and the dispersion relation defined in (2.38)-

(2.39) remain valid for this case, however, the expression for scattering coefficients Rn is

changed.

The values of Rn can be found by using the vertical membrane condition (2.58) and

the edge conditions (2.59). The eigenfunction expansion (2.38) together with membrane

condition (2.58) yields

nno F`Rn isnα cosh(γny)

=


Let us define

= 2E1δ(y)+2E2δ(y −b), (x,y) ∈0×(0,a). (2.61)

and ∆−n =isn(γ2n +µ2)+α, (2.62)

and also note that (∆+n)∗ = −∆−

n. Now multiplying (2.61) by αcosh(γmy) and integrating over

y from 0 to a, we obtain

a ∞ X

Rndy n=0

a

=F`

Finally, using orthogonality relation (2.40) on both sides, (2.63) gives

X∞ Rn∆+nCmδmn −γmγn sinh(γma)sinh(γna) n=0

=F`∆−`Cmδm`−γmγ`sinh(γma)sinh(γ`a)+αE1 +αE2 cosh(γma), which on simplification

leads to

F`∆−`δm Rm = +,

(2.64)

∆+m ∆+m ∆+m

where the constant E3 is defined by

E Rn∆+nγn sinh(γna). (2.65)

The constants Ej (j = 1,2,3) can be evaluated by applying the edge conditions

(2.59)-(2.60). In order to apply, the edge conditions (2.59), we multiply (2.64) by sm

cosh(γmy) and sum over index m, that is,

X∞ ∞ sm cosh(γmy)δm

Rmsm

m=0∆

+m

my)

32 CHAPTER 2

+ 1

m=0 ∆

+mCm

ma) +

2 m=0 ∆

+mCm

+m=0 ∆+mCm , (2.66)

Thus, for y = 0,a:

my)

αE1+ 2

m=0 ∆

+mCm m=0 ∆

+mCm

+E3 X∞ γm sinh(γma)sm cosh(γmy) =F`s`cosh(γ`y)(1− ∆`− ). (2.67) m=0 ∆+mCm ∆+`

Similarly, the edge condition (2.60) can be imposed to get

ma)

1 + 2

m=0 ∆

+mCm m=0 ∆

+mCm

∆+m ∆+` . (2.68)

Finally, the constants E1 −E3 can be obtained by solving (2.65), (2.67) and (2.68)

simultaneously.

2.2.2.1 EXPRESSION FOR ENERGY FLUX AND POWER BALANCE

Since, the eigenfunction expansion remains same for both rigid vertical wall and vertical

membrane and only the expression of Rn is different, the expression of reflected power

determined for rigid vertical wall (2.56) can be adopted here as well. However, in case of

membrane-like vertical strip, the power balance (2.57) can only be achieved successfully

after the truncation of large number of terms, that is, N > 250. Thus it is useful to derive


a power balance identity which not only holds for any number of terms but also confirms

those terms which converge at large number of terms. For this we rewrite (2.64) as:

Rm∆+mCm =F`∆−`C`δm`+αE1 +αE2 cosh(γma)+γm sinh(γma)E3. (2.69)

On multiplying (2.69) with −1/α2Rm∗ sm

∗ and summing over m, it is found that

RmRm∗ sm∗ ∆+mCmRm∗ sm∗

α m=0

R

R(2.70)

Note that

¾½¾

RmRm∗ smCmeRmRCm .

(2.71) α m=0α n=0

Now, on collecting the real parts of (2.70) and making use of (2.71), it is obvious to write

½ 1

X∞ 2 ¾

X

ℜe |Rm| smCm Rmsm α

m=0m=0

R

R (2.72)

Note also that

F`∗s`

∗, (2.73) m=0

∞ X

F), (2.74) m=0

F ), (2.75) m=0

34 CHAPTER 2

where the first terms in (2.73) and (2.74) are zero due to the chosen edge conditions

(2.59). Hence, from (2.72), we obtain

ℜe½α 1 mX∞=0|Rm|2 smCm¾ enF` F`∗s`∗

+αE2F`∗s`∗cosh(γ`a)−iψ∗xy(0,a)E3

o

∗

R` = (∆+`)∗ + (∆+`)∗C` + (∆+`)∗C` + (∆`+)∗C` . (2.77)

On using (2.77) into (2.76), it is obtained after simplifications that

½¾ 2

ℜe|Rm| smCmF`s α m=0

+αF`s`

+F`s` . (2.78)

The three middle terms of the right hand side of (2.78) are imaginary. Hence,

½ 1 ∞ ¾ ( i ∆+ )

On using (2.80), (2.79) implies

n 1 X∞ 2 o iµ2

ℜe |Rm| smCm

xy(0,a) α m=0

o

. (2.81)

On the other hand, from (2.64)

. (2.76)

ℜe α mX

=0|Rm|2 smCm =ℜe α2 ψ∗xy(0,a)E3 − α`

Now, with the aid of (2.60) and (2.65) it is useful to write

. (2.79)

E3 =µ2ψxy(0,a)+ψxyy(0,a). (2.80)


α

Hence, from (2.82) it is clear that when N →∞, Υ→ 0 and Pref → 1.

2.2.2.2 NUMERICAL RESULTS AND DISCUSSION

The model problem for membrane vertical wall does not preserve the power balance

with an arbitrary number of truncated terms N. However, it can be recovered by

increasing N. In order to see the validation of other imposed conditions on the model

problem, Equations (2.67)-(2.68) and (2.64) are solved simultaneously after truncation

at N terms. The values of the physical parameters in this section are consistent with

Problem 1.

In Figures 2.14, the real and imaginary parts of non-dimensional vertical

membrane displacement are plotted at y ∈ [0,a]. It is depicted that ℜe©

ψx(0,y)ª and

ℑm©

ψx(0,y)ª are zero when y

= 0 or y = a, which is consistent with the edge conditions

(2.59).

Similarly, Figure 2.15 shows the non-dimensional normal velocity in y− direction

plotted in the region 0×[0,a]. It is shown that ℜe©

ψy(0,y)ª and ℑm

©ψy(0,y)

ª are zero at y =

0 and y = a, and are non-zero for y

∈ (0,a). This is in agreement with the normal velocity

condition (2.58) and edge condition (2.60).

In Figure 2.16, the non-dimensional pressure is delineated in region 0×[0,a].

It important to see the convergence of reflected power as N varies. Figures 2.17-

2.18 show the reflected power Pref and the error term Υ versus N. It is observed that on

increasing N, the reflected power Pref approaches to unity while the error term Υ tends

to zero.

For the vertical membrane case the power balance (2.57) holds only when

N is large although the other conditions of model problem are being satisfied at small

N. However, the power balance identity (2.82) remains valid at any arbitrary number

N > 4; refer to Figure 2.19. Therefore, the power balance is not only useful to describe

the problem physically but also proves a useful check for the accuracy of obtained

solution.

While considering only real parts

Pref −Υ= 1,

where

(2.82)

½ i¾

Υ=ℜexyy(0,a) . (2.83)

36 CHAPTER 2 2.3. LOW FREQUENCY APPROXIMATION

FIGURE 2.14. The real and imaginary parts of vertical membrane displacement versus y.

FIGURE 2.15. The real and imaginary parts of normal velocity field versus y.

2.3 LOW FREQUENCY APPROXIMATION

The low frequency approximation (LFA) is often used to discuss the low frequency

range of noise for example ‘buzz-saw’ noise in aero-engines [79] and tonal fan noise

[80] in chimney stacks of power station. The dissipative devices works well at mid and

high frequency range of noise but not suitable at the low frequency range of noise.

x 0 , y

x 0 , y

0.2 0.4 0.6 0.8 y

10

5

5

10

x 0 , y

y 0 , y

y 0 , y

0.2 0.4 0.6 0.8 y

10

5

5

10

15

y 0 , y

37

Some passive devices like Helmholtz resonators are effective and contain a range of

industrial applications [81, 82]. But these have

FIGURE 2.16. The real and imaginary parts of pressure versus y.

FIGURE 2.17. The reflected power plotted against the number of terms.

not extensively used in HVAC. However, the study of active devices, such as expansion

chamber and reactive silencer, to cancel out noise within the duct is the fast growing

area of research [83, 84]. The LFA technique is very potential approach for

understanding the scattering of waves in low frequency regime. The approach is based

on the propagation of limited number of duct modes usually equal to the imposed

, 0 y

0 , y

0.2 0.4 0.6 0.8 y

5

4

3

2

1

1

2

, 0

0 100 200 300 400 500 N

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10 P

38 CHAPTER 2

condition. It also relies on matching integral quantities such as mean pressure and

mean velocity instead of using OR. For example in 2.3. LOW FREQUENCY APPROXIMATION

FIGURE 2.18. The error term plotted against the number of terms.

FIGURE 2.19. The reflected power, error and their combination versus the number of terms.

prototypes the eigen-expansion of field potential in duct takes the form

N

ψ(x,y) =F`cosh(γ`y)e−is`x + X

Rn cosh(γny)eisnx. n=0

0 100 200 300 400 500 N

0.02

0.04

0.06

0.08

0.10

P ref

P ref -

0 100 200 300 400 500 N

0.2

0.4

0.6

0.8

1.0

Power

39

Here N is the truncation parameter which allows the number of modes to propagate.

Note that in Problem 1, the equations (2.35-2.36) have not been still utilized, therefore

N = 1, permits two reflected modes to propagate. The solution process is explained in

detail in Chapter 3.

2.4 GALERKIN/MODAL APPROACH

In contrast to Problem 1, if the vertical rigid strip at matching interface is replaced with

higher order derivative containing boundary condition the MM method converges only

after the collection the large number of truncated modes (see Problem 2). However, a

small variation in MM approach gives another technique of finding solutions coined as

Galerkin approach. This method has been extensively studied and used to solve a wide

range of related problems (see [15, 16, 69, 70, 85]). The method is constructs solution

on basis functions chosen so that the edge conditions are satisfied at the boundaries

(that is, at either end of the membrane/plate). To address the boundary value

problems in the thesis the Galerkin method has been developed in following two

different ways: First, the displacement of the vertical membrane is represented as a

generalized Fourier sine series the basis functions of which are chosen à priori to satisfy

atleast the conditions at the edges of the membrane. The second case (Modal

approach) is a variation of the Galerkin method in which displacement of the vertical

membrane is represented as a modal expansion in which the basis functions are the

eigenfunctions for the duct section of greater height. These eigenfunctions are linearly

dependent and are not orthogonal in the usual sense but have well known properties

[59]. The method is further explained in Chapter 5 and 6.

ACOUSTIC PROPAGATION AND

SCATTERING IN A TWO-DIMENSIONAL

WAVEGUIDE INVOLVING STEP DISCONTINUITY AND FLANGED

JUNCTION

In this chapter, acoustic wave scattering in two-dimensional waveguide involving

flanged junction and step discontinuity is discoursed. The study is important in view to

its use as a component in a modified silencer for heating ventilation and air-

conditioning (HVAC) ducting systems. It is well studied phenomenon [6, 7, 13] that a

membrane attached across the mouth of an expansion chamber can effectively reduce

the transmission of low-frequency noise along a ducting system. The work aims to

investigate the MM and LFA solutions of a two dimensional waveguide problem with

flanged junction. The relative merits of each approach are compared for the scattering

of fluid-coupled wave. The boundary value problem with higher order derivatives

develops a non-SL problem where the use of generalized OR enables the MM solution.

The contents of this chapter are arranged in the subsequent order. The

twodimensional problem which incorporates vertical discontinuity is formulated in

Section 3.1, whereas, Section 3.2 includes a MM solution with the implication of

different physical edge conditions. Section 3.3 is dedicated to produce the low-

frequency approximation while the expressions for power distribution are detailed in

Section 3.4. Few numerical illustrations related to power distribution and modes

coefficients are provided in Section 3.5. The solution to the model problem is validated

in Section 3.6.

3.1 BOUNDARY VALUE PROBLEM

Consider a two-dimensional infinite rectangular waveguide containing two semiinfinite

duct sections of different heights a and b, where a < b. In dimensional

Cartesian frame of reference these sections occupy the regions (−∞,0]×[0,a] and

35

[0,∞)×[0,b], respectively. At 0×[d,b] where d < a, a vertical flange or strip is placed which

joins both the duct sections at (x,y) = (0,a) and (x,y) = (0,b). The lower surface of the

waveguide lies along R×0 is acoustically rigid whilst the upper boundaries are

membranes which lie on (−∞,0)×a and (∞,0)×b. The side of the flange is assumed to

41

be rigid at x = 0−, y

∈ (d,a) and soft at x = 0+, y ∈ (d,b). The interior region of the waveguide

is filled with a compressible fluid of density ρ and sound speed c. The harmonic time

dependence, e−iωt in which ω = ck is the radians frequency and k is a fluid wave number

is taken into account. The problem is non-dimensionalized with respect to length scale

k−1 and time scale ω−1 under the transformation x = kx and y = ky etc. The non-dimensional

geometry of the problem is shown in Figure 3.1.

FIGURE 3.1. Non-dimensional geometry of the waveguide

The non-dimenstional velocity potential ψ(x,y) in waveguide can be defined

as

ψ1(x,y), (−∞,0)×[0,a],

ψ(x,y) = (3.1)

ψ2(x,y), (0,∞)×[0,b],

where ψ1(x,y) and ψ2(x,y) are the reduced velocity potentials in the regions

(−∞,0]×[0,a] and [0,∞)×[0,b], respectively and satisfy the Helmholtz equation ¡ 2 ¢

∇ +1 ψ= 0, (3.2)

with unit non-dimensional wave number. The rigid horizontal surface in both duct

sections are given by

∂ψj

0, y 0, x R, j 1,2. (3.3)

42 CHAPTER 3

∂y = = ∈ =

3.1. BOUNDARY VALUE PROBLEM

For the upper bounds of duct, the non-dimensional membrane boundary conditions are

µ

1,2, (3.4)

where for j = 1 and j = 2 the condition is applied at y = a, x ∈ (−∞,0) and y = b, x ∈ (0,∞),

respectively. At x = 0−, y ∈ (d,a) the rigid side of vertical flange is given by

0, (3.5)

wheareas, at x = 0+, y ∈(d,b) the soft face of the flange is defined by

ψ2 = 0. (3.6)

Therefore at x = 0−, (d,a) the normal component of velocity vanishes and at x = 0+, y ∈ (d,b)

the fluid pressure vanishes while both are continuous at x = 0, y ∈(0,d). That is

(3.7)

∂x 0, 0×(d,a)

and

=(

ψ1, 0×(0,a)

ψ2 . (3.8)

0, 0×(d,b)

In additions the edge conditions are applied at the points where membranes are

connected with the vertical surface. These conditions describe how the membranes are

connected to the vertical surface. The choice of edge conditions can significantly affect

the scattered field. The zero displacement and zero gradient choices of edge conditions

are considered here. The choice of zero displacement condition at both edges contain

ψ1y(0,a) =ψ2y(0,b) = 0, (3.9)

whereas, the zero gradient edge conditions are

43

ψ1xy(0,a) =ψ2xy(0,b) = 0. (3.10)

The above formulated problem can solved by using the MM and LFA techniques.

Subject to zero displacement and zero gradient edge conditions the MM and LFA

solutions are presented in next sections.

3.2 MODE-MATCHING SOLUTION

In this section, we obtain the mode-matching solution of the boundary

value problem

∂x = ∂x χ0×(0,d), 0×(0,b),

∂ψj ∂y2 = 0, ´

³³∂∂∂∂xx222

++µµ22´ψψ21yy ++αψαψ21 == 00,,

R×0,

(−∞,0)×a,

(0,∞)×b,

ψ

ψ11xyy(0(0,a,a))==00==ψψ2y2xy(0,(0b,)b, ),

for the model problem presented in the previous section, where χ is the characteristic

function, that is, for any domain D ⊂Rd

1, x ∈D,

χD(x) =

0, x ∈Rd \D.

¡∇2 +1

¢ψ= 0,

ψψ∂ψ∂ψ∂22x12===ψ00∂ψ,1,χ10×(0

,a),

(−∞,0)×[0,a]∪(0,∞)×[0,b

], x = 0−, y ∈(d,a), x = 0+,

y ∈(d,b)

0×(d,b),

44 CHAPTER 3

Let an incident wave of an arbitrary duct mode is propagating in waveguide from

negative x direction towards x = 0. It will scatter at interface into infinite number of

reflected and transmitted modes. By using the separation of variable method the

Equations (3.2)-(3.4) yield the eigenfunction expansion form of scattered fields as

An cosh(τny)e−iηnx (3.11)

and

X∞ isnx

ψ2(x,y) = Bn cosh(γny)e . (3.12)

n=0

Note that the first term in (3.11) is an incident wave in which the indices ` is either

0 or 1 corresponds, respectively, to the fundamental mode and the secondary mode.

The forcing term F` = p

α/C`η` is chosen such that to ensure

q q the incident power to be

unity. The quantities ηn = τ2n +1 and sn = γ2

n +1 are the non-dimensional wave numbers

which are either positive real or positive imaginary. The τn and γn, n

= 0,1,2,... are the

roots of the following dispersion relations

K(ζ,p) = (ζ2 +1−µ2)ζsinh(ζp)−αcosh(ζp) = 0, (3.13) 3.2. MODE-MATCHING SOLUTION

where ζ = τn, p = a in left duct region (−∞,0)×[0,a] and ζ = γn, p = b in right duct region

(0,∞)×[0,b]. The roots of (3.13) can be found numerically and their properties have been

stated in Chapter 2. The scattered field coefficients (An,Bn), n = 0,1,2,... be the unknowns

to determine. These can be found by following the MM process established in Chapter

2 for non-SL systems. Thus for two different membrane bounded duct sections of

waveguide, (2.23) yields the ORs as

a

), (3.14)

when x,y ∈(−∞,0)×(0,a), wheras, when x,y ∈(−∞,0)×[0,a] the OR is

b

), (3.15)

where

= αa µ

3τ2m +1−µ2 ¶

2

Cm : + 2 [τm sinh(τma)] (3.16) 2 2τm

45

and Dm can be obtained from (3.16) by replacing a with b, τm with γm and Cm with Dm. In

order to obtain the system of equations that containing An and Bn as unknowns, on

substituting (3.11) and (3.12) into continuity condition of normal velocity (3.7), it is

obvious to find

∞

X X∞ Bnsn cosh(γny), 0×(0,d)

F`η`cosh(τ`y)−n=0 Anηn cosh(τny) = n0=,0 0×(d,a) . (3.17)

On multiplying (3.17) by αcosh(τmy) and integrating from 0 to a, we get

a a

F`η`αˆ cosh(τmy)cosh(τ`y)dy

− X

Anηnαˆ

cosh(τmy)cosh(τny)dy

d

Bnsn . (3.18)

Define d

Rmn ), (3.19)

and using the OR (3.14), (3.18) gives

F`η`Cmδm`−τmτ`sinh(τma)sinh(τ`a)

X∞

BnsnRmn. (3.20) n=0

The simplification and rearrangement of equation (3.20) yields

m m

Am =F`δm`− E1 − X BnsnRmn, Cmηm Cmηm n=0

where

(3.21)

E1 =−iψ1xy(0,a). Similarly for Bn

put (3.11) and (3.12) into (3.8) to get

(3.22)

∞

X∞ F`cosh(τ`y)+ X

An cosh(τny), 0×(0,d)

Bn cosh(γny) = n=0 . (3.23)

46 CHAPTER 3

τ sinh(τ a) α ∞

n=0 0, 0×(d,b)

On multiplying by αcosh(γmy) and integrating from 0 to b, the pressure condition (3.23)

implies b

∞∞

XX

F`R`m + AnRnm. (3.24) n=0n=0

On using the OR for right hand duct (3.15) into (3.24) and simplifying the resulting

expression, it is obtained that

γ sinh(γ b) α µ

∞ ¶

Note that equations (3.21) and (3.25) contain the constants Ei (i = 1,2) to be unknowns.

In order to calculate these unknowns the choice of edge conditions lead to the following

possible cases.

3.2.1 ZERO DISPLACEMENT

In this case the zero displacement edge conditions (3.9) has been used to determine

the constants Ei (i

= 1,2). The comparison of (3.26) and (3.9) yields E2 = 0. But for E1, on

multiplying (3.21) by τm sinh(τma) and summing over m it is found

that

X∞ X∞ Amτm sinh(τma) =F` τm sinh(τma)δm` m=0 m=0

m m

Bm = E2 + F`R`m + X AnRnm ,

Dm Dm n=0

where

(3.25)

E2 =ψ2y(0,b). (3.26)

47

∞ ∞ Bnsnτm sinh(τma)Rmn

− E . (3.27) m=0 CmηmCmηm

S . (3.29)

On using (3.9) and rearranging one may get

2F

E1 =− `τ`sinh(τ`a) + α X∞ X∞ Bnsnτm sinh(τma)Rmn ,

S1 S1 n=0m=0 Cmηm

where

(3.28)

48 3.3. LOW FREQUENCY APPROXIMATION SOLUTION

3.2.2 ZERO GRADIENT

In this case the constants Ei (i = 1,2) has been found by using the choice of zero gradient

at the both edges, that is (3.10). The comparison of (3.22) and (3.10) gives E1 = 0,

however, the value of E2 can be found by multiplying (3.25) by smγm sinh(γmb), summing

over m and using (3.10). Thus

mb) µ X∞ ¶

m=0 Dm

The appropriate choice of Ei (i = 1,2) together with (3.21) and (3.25) form an infinite

system of linear algebraic equations that are truncated and then solved numerically.

3.3 LOW FREQUENCY APPROXIMATION SOLUTION

In this section LFA solution is formulated and the results are compared with MM

solution. The LFA is based on matching integral quantities such as mean pressure and

mean velocity instead of using OR. So only the fundamental forcing incident is

considered herein. It is pertinent to mention here that the LFA is not expected to valid

for higher mode incident. The expressions for velocity potentials (3.11) and (3.12)

remain valid in this case with truncation of n =M for left hand duct and n =N for right

hand duct. That are

M

where

E2 =− F`R`m + AnRnm ,

S2 m=0 Dm n=0

(3.30)

∞ s

£γ sinh(γ b)

¤2

S2 = X m m m . (3.31)

49

n=0

Thus, M +1 and N +1 modes in reflected and transmitted fields, respectively will be

taken into consideration. But the total number of modes, M +N +2, are consistent with

number the physical conditions applied at x = 0. The numeric values of M and N depend

upon the geometrical configuration of duct. Four cases are discussed here for zero

displacement and zero gradient edge conditions. However, the continuity of mean

pressure and velocity flux is used in all cases. At x = 0, y ∈(0,d), the continuity conditions

of mean pressure and normal velocity are

d d

ψ1dy =ˆ ψ2dy, (3.34)

0 0 and d d

ψ1xdy =ˆ ψ2xdy. (3.35)

0 0

γn n=0 n=0

τn τ0

and XM Anηn sinh(τnd) XN Bnsn sinh(γnd) F0η0 sinh(τ0d) + = . (3.37)

n=0 τn n=0 γn τ0

On substituting (3.32) and (3.33), the zero displacement condition (3.9) gives

M

ψ1(x,y) =F0 cosh(τ0y)eiη0x + X

An cosh(τny)e−iηnx, n=0

and

(3.32)

N ψ2(x,y) = X

Bn

cosh(γny)eisnx. (3.33)

On using (3.32) and (3.33), we have

N Bn sinh(γnd) XM An sinh(τnd) F0 sinh(τ0d) X

− = , (3.36)

50 CHAPTER 3

n=0

In subsequent cases, different situations for the underlying structure are discussed.

CASE I: d<a<b

This is most general case, at x = 0, it includes both vertical flange and discontinuity in

height. But at x = 0+, y ∈(d,b) mean pressure is zero. That is

b

ψ2dy = 0, (3.42) d

which implies

XN (sinh(γnb)−sinh(γnd)) =

Bn 0. (3.43) n=0 γn

And at x = 0−, y ∈(d,a), the zero velocity flux is given by

a

ψ1xdy = 0, (3.44) d

3.4. EXPRESSIONS FOR POWER DISTRIBUTION

which gives

XM ηn(sinh(τna)−sinh(τnd)) = F0η0(sinh(τ0a)−sinh(τ0d))

An . (3.45) n=0 τn τ0

X Anτn sinh(τny) =−F0τ0 sinh(τ0a), n=0

and N

(3.38)

X Bnγn sinh(γny) = 0, n=0

whilst the zero gradient condition (3.10) leads to

M

(3.39)

X Anηnτn sinh(τny) =F0τ0η0 sinh(τ0a), n=0

and N

(3.40)

X Bnsnγn sinh(γny) = 0. (3.41)

51

Thus for zero displacement we have six equations (3.36)-(3.39), (3.43) and (3.45) and

for zero gradient we have (3.36)-(3.37),(3.40)-(3.41), (3.43) and (3.45). On considering

M = N = 2 for each, these sets of equations can be solved simultaneously for model

coefficients.

CASE II: a=b, d<a

In this case a vertical flange or strip is assumed in a continuous duct. The duct modes

and system of equations of Case-1 remain valid for this case with a =b.

CASE III: d=a, a<b.

In this case no vertical flange but discontinuity in height is considered. On taking M =

1, two duct modes for left hand duct and N = 2, three duct modes for right hand duct,

the equations (3.36)-(3.39) and (3.43) for zero displacement condition and the

equations (3.36)-(3.37),(3.40)-(3.41) and (3.43) for zero gradient conditions are solved

simultaneously.

CASE IV: d=a=b.

In this case neither vertical flange nor discontinuity in height is assumed. For zero

displacement condition the equations (3.36)-(3.39) and for zero gradient (3.36)-(3.37),

(3.40)-(3.41) are simultaneously solved whilst M =N = 1.

3.4 EXPRESSIONS FOR POWER DISTRIBUTION

In this section the expressions for scattered power distribution are presented.

As discussed in chapter 2 that the choice of F` = q

C`α

η` enables to take the incident power

to be unity. However, the expressions for reflected and transmitted powers are found

to be

=

and

½ 1 X∞ 2 ¾

Ptrans =ℜ |Bn| snDn ,

α n=0

whilst the conserve power identity is

x > 0, (3.47)

52 CHAPTER 3

X∞ 2 ¾

Pref|An| ηnCn , x < 0 (3.46) n 0

3.5 NUMERICAL RESULTS AND DISCUSSION

In this section the physical problem is solved numerically after truncation of (3.21) and

(3.25) upto m = 0,1,...N terms. The reduced system contains N

+1 linear algebraic

equations which are then solved simultaneously. The power distribution in duct

regions and mode coefficients are plotted against frequency for different duct mode

incident (` = 0,1) . The consideration of power not only validates the MM and LFA

solutions to be physically more suitable but also to discuss the relative merits of each

approaches. In Figures 3.2-3.31 each graph comprises membrane mass density ρm =

0.1715kgm−2 and tension T = 7500Nm which are consistent with Huange [7]. The figure

caption in each case is stated subject to the incident duct modes whichever are taken,

the type of edge condition which is used and the number of terms N taken.

CASE I: d<a<b

Figures 3.2-3.7 has been computed using dimensional duct heights: a = 0.06m,

b = 0.085m and flange d = 0.045m. The reflected power (P-ref), transmitted power (P-

trans) ,and their sum (PB) are plotted against frequency for different edge conditions

(see Figures 3.2-3.5). The dashed curves show the results of low frequency

approximation solution whereas solid lines depict the results obtained due to MM. In

Figures 3.6-3.7 the modulus of first three reflected and transmitted modes are plotted

against frequency. Figure 3.2 shows that in the interval of frequency between 1 to 450

Hz almost all the power is reflected. In this frequency range the MM and LFA solutions

curves coincide. But as frequency increases above 450 Hz, we can see that about 90

percent of incident power goes on reflection. Also in this frequency range the power

balance identity (3.48) is not achieved by LFA solution. On the other hand for zero

gradient conditions, almost all the energy is reflected depicting a good agreement

between MM and LFA solutions (see Figure 3.4). Figures 3.3 and 3.5 show the

component of powers when secondary mode forcing (` = 1) is incident (propagating at

553 Hz). The incident power is now fluid-borne while propagation is totally reflected

which decreases steadily by increasing frequency. The comparison of Figures 3.3 and

3.5 suggests that the zero gradient condition at edges increases the rate of

transmission with increasing frequency.

Figures 3.6-3.7 show the results of modulus of reflected and transmitted mode

coefficients for zero displacement and zero gradient edge conditions, respectively. The

Pinc =Pref +Ptran. Note

that these expression can also be found in [74].

(3.48)

53

frequency regime is taken from 1 to 1253 Hz. It is clear that the fundamental reflected

and transmitted modes curves obtained by MM and LFA solution coincide exactly.

However, it appears that the LFA curves do not seem to match exactly for secondary

and higher modes. Also there is a corner at 553 Hz which shows that after 553Hz the

secondary mode becomes propagating. 3.5. NUMERICAL RESULTS AND DISCUSSION

FIGURE 3.2. Zero displacement at edges and incident forcing via fundamental mode.

FIGURE 3.3. Zero displacement at edges and the incident forcing via secondary mode.

The results for absolute value of reflected and transmitted modes in Case-II and Case-

III are quite similar to that of Case-I.

54 CHAPTER 3

CASE II: a=b, d<a

In this case both duct sections comprise same dimensional height a = b = 0.06m

while d = 0.045m. Figure 3.8 shows that for the zero displacement edge condition the

power distribution in duct region is very similar to that of Figure 3.2 of Case 1. But

most of power is reflected when zero gradient condition is considered (see Figures 3.4

and 3.10). For the secondary mode incident power components behave similar to that

of Figures 3.3 and 3.5 but transmission rate is more rapid

FIGURE 3.4. Zero gradients at edges and the incident forcing via fundamental mode.

FIGURE 3.5. Zero gradients at edges and the incident forcing via secondary mode.

in this situation.

55

CASE III: d=a, a<b

In Figures 3.14-3.19, we let a = d = 0.06m and b = 0.085m. As for the previous case, when

the fundamental forcing is incident maximum power goes on reflection which

decreases slowly by increasing frequency. It drops down to 25 percent of incident

energy at the frequency value 553 Hz and then steadily increases by increasing

frequency. While not quite overlying the results of LFA and MM however PB seems to

be agreed for both approaches. 3.5. NUMERICAL RESULTS AND DISCUSSION

FIGURE 3.6. Zero displacement at edges and the incident forcing via fundamental mode.

FIGURE 3.7. Zero gradients at edges and incident forcing via fundamental mode.

56 CHAPTER 3

CASE IV: d=a=b

In this case the duct comprises simplest geometry in which d = a =b = 0.06. Figures 3.20-

3.23 correspond to the graphs of power components for zero displacement and zero

gradient edge conditions, respectively. figure11(a) shows that the transmitted

(reflected) power increases (decreases) sharply up to maximum value at 553Hz and

then steeply decreases (increases) by increasing frequency. For the secondary mode

incident the reflected and transmitted powers demonstrate a sharp inversion on

propagation (see Figure 3.21). The oveall trend for fundamental mode incident is

power reflection after 553Hz whilst for secondary mode incident it behave inversely.

However MM and LFA solutions match ex-

FIGURE 3.8. Zero displacement at edges and incident forcing via fundamental mode.


57

actly by considering either the power distribution or mode coefficients against

frequency regime

3.6 VALIDATION OF TECHNIQUE

In order to see the validation of MM solution the continuity conditions of normal

velocity and pressure at matching interface (3.7)-(3.8), are assessed. In Figures 3.26-

3.27 the real and imaginary parts of non-dimensional pressures ψj(0,y), j = 1,2 are

plotted in region 0 ≤ y ≤ a where f = 700Hz. Both the curves overlie in the region 0 ≤ y ≤

d while ψ2(0,y) goes on zero in the interval d ≤ y ≤ a which exactly satisfies the continuity

condition (3.8).

58 CHAPTER 3

FIGURE 3.10. Zero gradient at edges and the incident forcing via fundamental mode.

FIGURE 3.11. Zero gradient at edges and the incident forcing via secondary mode.

3.6. VALIDATION OF TECHNIQUE 59



60 CHAPTER 3





FIGURE 3.17. Zero gradient at edges and the incident forcing via secondary mode.

62 CHAPTER 3



64 CHAPTER 3


FIGURE 3.23. Zero gradient at edges and the incident forcing via secondaryl mode.




ÂIΨjM

66 CHAPTER 3

FIGURE 3.26. The real part of pressures in two duct regions are plotted against the duct height at matching interface.

ÁIΨjM

FIGURE 3.27. The imaginary part of pressures in two duct regions are plotted against the duct height at matching interface.

Similarly the real and imaginary parts of non-dimensional normal velocities ψjx(0,y),

j = 1,2 are plotted in region 0 ≤ y ≤ b (see Figures 3.28-3.29). It can be seen that ψ1x(0,y)

is zero when d ≤

y ≤ a as taken in (4.4) however there are oscillations around the mean

position in two curves. The amplitude of oscillations increases as y → d (corner) and

decreases as y → 0. This situation is because of singular nature of flow at corner which

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.00

0.05

0.10

0.15

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.5

0.4

0.3

0.2

0.1

0.0


cannot be improved even by increasing the truncated terms N. Moreover the

oscillations are due to Gibbs phenomenon [86] which can be improved on using the

spectral filter [87]. The same issue is comprehensively reported and addressed by

Nawaz and Lawrie [67].

ÂIΨjxM

FIGURE 3.28. The real part of normal velocities in two duct regions are plotted against the duct height at matching interface.

ÁIΨjxM

FIGURE 3.29. The imaginart part of normal velocities in two duct regions are plotted against the

duct height at matching interface.

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

5

5

10

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

20

10

10

20

68 CHAPTER 3

Therefore in order to calculate the explicit expression for this filter assuming

the truncated model expansion as

N

fN p ≤ y ≤ p. (3.49)

Consider instead the quantity

N p/N

f = ˆ fN(y +v)dv, (3.50) 2p

which is known to be the Lanczos filter. For x < 0, the left hand duct p

= a and ζn = τn

whilst for x > 0, the right hand duct p = b and ζn = γn. Thus by using the Lanczos filter in

normal velocity expressions the oscillations in normal velocity can be removed. This

−p/N

which is being the convolution of fN(y) with

S(υ) =( 0N, ,

||υυ||<>pp 2p

On using (3.49) into (3.50) and integrating, we arrive at

(3.51

)

N fNσ(y) =

X Anσn

p

cosh(ζny), n=0

where

(3.52

)

p sinh(ζnp/N) σn = ,

ζnp/N

(3.53

)


term has been found by following the procedure outlined in [67]. On using the Lanczos

filter [88], in truncated normal velocity expression, the curves of normal velocities are

smoothened as can be seen from Figures 3.30-3.31.

The satisfactory level of agreement in normal velocity curves is observed in Figures

3.30–3.31 as considered in (3.7). The satisfaction of these continuity conditions along

with the conserve power identity (3.48) validate the MM solution. ÂIΨjxM

FIGURE 3.30. The real part of normal velocities in two duct regions are shown against the duct height after using Lanczos filters.

ÁIΨjxM

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

4

2

2

4

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

10

5

5

10

70 CHAPTER 3

FIGURE 3.31. The imaginary part of normal velocities in two duct regions are shown against the duct height after using Lanczos filters.

ACOUSTIC SCATTERING IN PLATE-MEMBRANE BOUNDED

WAVEGUIDE INVOLVING STEP

DISCONTINUITY

This chapter includes the consideration of acoustic wave propagation in flexible

waveguide involving step discontinuity at interface. The emerging boundary value

problem is non-SL the solution to which is presented by using the MM technique. Two

incident duct modes; structure-borne mode and fluid-borne mode are considered in

elastic plate bounded duct. At matching interface it scattered into the model spectrum

of reflected and transmitted modes. The scattered potentials in duct regions comprise

eigen expansion form in which the unknowns are scattered wave amplitudes. The

implications of the conditions of continuity in pressure and velocity at the junction

(discontinuities) of two ducts corresponds to infinite system of algebraic equations. It is

the diagonally dominant system and off diagonal elements decay quickly for an accurate

solution. The truncation and use of appropriate edge conditions leads to the solution of

the problem. The solution is discussed to insight the physical scattering process and

attenuation of duct modes in frequency regime and change of height. Moreover the MM

solution is validated through the power conservation identity and matching interface

conditions.

The rest of the investigation is arranged in the following order. Section 4.1 is

dedicated to formulate the boundary value problem governing the wave propagation in

the wave-guide. A mode-matching solution is constructed in the subsequent Section 4.2.

The edge conditions and their implications on the scattering pattern are then discussed

in Section 4.3. Graphical results are presented to discuss the distribution of power against

frequency and vertical discontinuity which are placed in Section 4.4. Finally, the validation

of technique and important contributions of the investigation are summarized in Section

4.5.

61

4.1 BOUNDARY VALUE PROBLEM

Consider a two dimensional infinite wave-guide consisting of two semi-infinite duct

sections with different heights. The lower wall of both duct sections is assumed to be

acoustically rigid. The upper surface of inlet duct section consists of an elastic plate

whereas that of the outlet duct section is a membrane. The upper surfaces of the inlet

and outlet duct sections are connected by means of a vertical rigid strip and respectively

72 CHAPTER 4

meet the strip at heights a and b where b > a. In a two-dimensional Cartesian frame of

reference (x,y) the duct sections occupy the regions

(−∞,0)×(0,a) and (0,∞)×(0,b),

respectively. The waveguide is filled with compressible fluid of density ρ and sound speed

c.

Throughout this work, a harmonic time dependence e−iωt is assumed and suppressed

where ω is the angular frequency in radians. The problem is nondimensionalized relative

to length and time scales 1/k and 1/ω respectively by virtue of the transformation x = kx

and y = ky etc. The non-dimensional geometry of the problem is depicted in Figure 4.1.

FIGURE 4.1. Non-dimensional geometry of the waveguide.

Let ψ1(x,y) and ψ2(x,y) be the potential fields in the inlet and outlet duct sections

respectively. The non-dimensional velocity potential ψ(x,y) in the waveguide can be

defined as

ψ1(x,y), (x,y) ( ,0] [0,a],

ψ(x,y) = ψ2(x,y), ∀∀(x,y) ∈∈[0−,∞+∞)××[0,b], ,

which satisfies the Helmholtz equation

(4.1)

¡∇2 +1

¢ψ= 0, ∀(x,y) ∈(−∞,0]×[0,a]∪[0,∞)×[0,b]. (4.2)

73

4.1. BOUNDARY VALUE PROBLEM

The natural conditions in non-dimensional form at lower acoustically rigid wall for both

duct regions are

∂ψj

0, x R, y 0, j 1,2. (4.3)

∂y = ∈ = =

Since the upper surface of the inlet duct section comprises of an elastic plate, the

boundary condition at surface (−∞,0)×a in non-dimensional form is given by

µ ∂4 4¶

∂ x4 −µ1 ψ1y −α1ψ1 = 0. (4.4)

where subscript y indicates a derivative with respect to y, µ1 is the non-dimensional

invacuo plate wave-number and α1 is a fluid loading parameter defined in Chapter 2.

On the other hand, since the upper surface of outlet duct section is assumed to be

a membrane, following non-dimensional membrane boundary condition is imposed

µ

0, (x,y) ∈(0,+∞)×b,

(4.5)

where µ and α are respectively non-dimensional membrane wave-number and fluid

loading parameter which are also stated in Chapter 2. At the matching interface,

0×[0,a] (coined as aperture), the fluid pressure and the normal component of velocity

are continuous whereas the normal component of velocity vanishes on x = 0+, y ∈ (a,b).

Therefore, the following continuity conditions hold:

ψ1 =ψ2, (x,y) ∈0×(0,a) (4.6)

and

∂ψ1

∂ψ2 = , (x,y) ∈0×(0,a),

∂x (4.7)

∂x 0, (x,y) ∈0×(a,b).

In addition, the edge conditions are applied at the points where elastic plate and

membrane are joined with rigid vertical strip. These conditions not only ensure a

unique solution of the boundary value problem but also describe how the elastic plate

or membrane are connected to the strip. The choice of edge conditions can

significantly alter the scattered field. The zero displacement (resp. zero gradient)

condition at membrane edge is

74 CHAPTER 4

ψ2y(0,b) = 0 (resp. ψ2xy(0,b) = 0). (4.8)

The above mentioned class of boundary value problems having wave-bearing

boundaries has been discussed in detail by many researchers, see for instance [7, 35,

67]. The boundary conditions involve only even order derivatives in x since odd order

derivatives do occur in systems which are damped, and the occurrence of such

derivatives significantly alters the nature of the underlying eigen-system. In particular,

the dispersion relation will not be an even function of the wave number. It is not,

therefore, immediately obvious that the results presented herein apply to such

systems. Note also, that higher order derivatives in y are easily removed by recourse

to the governing wave equation. The underlying structure with its mathematical model

is quite significant and physically admissible [11, 51, 68]. The solution to the above

stated problem is presented in the next section.

4.2 MODE-MATCHING SOLUTION

In this section, we invoke the mode-matching technique to solve the boundary value

problem

∂x = ∂x

χ0×(0,a),

0×(0,b),

∂ψj = R×0,

∂y 0,

³∂∂x44 ´ 0 (−∞,0)×a,

(4.9)



1, x ∈D,

χD(x) =

0, x ∈Rd \D.

¡∇2 +1

¢ψ= 0,

ψ∂ψ12=ψ∂ψ2,1

(−∞,0)×[0,a]∪(0,∞)×[0,b],

0×(0,a),

µ

³ψψ∂∂x2112xyy+(0(0,a2,a´)ψ)==20y0=+=ψαψψ2y22xy(0=,(0b0,,)b,

),

(0,∞)×b,

75

Let an incident wave of an arbitrary duct mode be propagating in inlet duct section

from the negative x−direction towards x

= 0. At the planar junction of ducts or

discontinuity, that is, at x = 0, it will scatter into potentially large number of reflected and

transmitted modes. The eigen-expansion form of scattered velocity potentials in duct

regions take the forms

X∞ 1

AnYn (y)exp(−iηnx), (4.10) n=0

and

X∞ 2

ψ2(x,y) = BnYn (y)exp(isnx), , (4.11) n=0

where

Y and Yn2(y

) = cosh(γny). (4.12)

4.2. MODE-MATCHING SOLUTION

The first term in equation (4.10) represents the incident wave with an arbitrary

q α1

forcing F` = C`η` so that the incident power is unity. The counter ` assumes values 0 or 1

according to fundamental or secondary mode incidence respec-

q q tively. The parameters ηn = τ2n +1 and sn = γ2

n +1

are the complex wavenumbers of nth reflected and transmitted modes respectively,

where τn and γn for n = 0,1,2,

··· are the eigen-values of the eigen-system. The eigen-values

τn and γn are the roots of the dispersion relations

K1( τn,a) = ((τ2n +1)2 −µ4

1)τn sinh( τna)−α1τn cosh(τna) = 0

and

(4.13)

K2(γn,b) = (γ2n +1−µ2)γn sinh(γnb)−αγn cosh(γnb) = 0. (4.14)

The dispersion relations (4.13)-(4.14) can be solved numerically for τn and γn which, in turn,

satisfy the properties stated in Chapter 2.

The above proposed eigen-system is non-SL system and the eigenfunctions Yn1(y) and

Yn2(y) are linearly dependent (see Chapter 2) however satisfy the special ORs. The

appropriate ORs for given eigen-system can be obtained from

(2.25) and (2.23), that are a

Ym mnCn ), (4.15)

and b

76 CHAPTER 4

Ym mnDn (4.16)

Note that δmn is the Kronecker’s delta function and the prime indicates a differentiation

with respect to y whereas

α a α1Y 1 (b)Y 01(b)

The complex amplitudes of nth reflected and transmitted modes, An and Bn, are the

unknowns to be determined. The substitution of model expansion of scattered fields

(4.10)-(4.11) into the continuity conditions (4.6)-(4.7) lead to an infinite system of

algebraic equations thereby providing the values of An and Bn. The resultant algebraic

system can be solved by neglecting higher order modes corresponding to n > N for some

N ∈ N. Using (4.10)-(4.11) into (4.6), the continuity condition of pressure yields

F` Bn cosh(γny). (4.19)

Finally, multiplying (4.19) with α1 cosh(τmy), integrating over (0,a) and subsequently

exploiting OR (4.15) it is found that

AmBnRmn, (4.20) Cm Cm n=0

where

E1 =ψ1yyy(0,a), (4.21)

E2 =ψ1y(0,a) (4.22)

and a

Rmn =ˆ cosh(τmy)cosh(γny)dy. (4.23)

0

Similarly on invoking (4.10)-(4.11) into (4.7), multiplying by αcosh(γmy), integrating from 0

to b and then using OR (4.16) the expression for Bm is found to be

Cm := 2(τ 1 + m2τ2m n + 2 +

10(a)]2.

2 m 1)[Y

m

(4.17)

and αb µ

3γ2 2

Dm := + m +12 −µ ¶[Ym10(b)]2.

2 2γm (4.18)

77

γm sinh(γmb)E3 α ∞

Bm = + F`η`R`mAnηnRnm, (4.24) Dmsm Dmsm

where

E3 =ψ2xy(0,b). (4.25)

Note that E1 −E3 are constants which are unknown and can be determined by using the

edge conditions. At the elastic plate edge two type of conditions are considered : clamped

edge condition and pin-jointed edge condition.

In equations (4.21)-(4.22) into (4.25), the constants Ei (i = 1,2,3) are to be precised to

ensure the uniqueness of the scattering pattern and the MM solution. This requires

appropriate conditions at the points connecting elastic plate and membrane with vertical

strip. The subsequent section is dedicated to invoke different edge conditions thereby

fixing the values of these constants.

4.3 EDGE CONDITIONS

A common assumption, when modeling wave-guide structures, is that the duct walls are

clamped at the joint. In practice, however, the duct sections may be simply supported

together. Therefore this section investigates different effects that arise when the edges

are (a) clamped and (b) pin-jointed at the junction. The former edge conditions are

characterized by zero membrane displacement and zero gradient while the latter by zero

plate displacement and zero plate bending moment. A comprehensive list of appropriate

edge conditions can be found, for example, in references [15, 16]. As described in Chapter

2, for structures involving elastic plates or membranes, the number of edge conditions

are half of the order of plate/membrane conditions. In fact, this imposes additional

constraints on the solution to the underlying boundary value problem which also ensures 4.3. EDGE CONDITIONS

the uniqueness of the solution. In the sequel, two different admissible conditions,

precisely clamped edge and pin-jointed edge conditions, are considered in order to cater

various industrial applications.

4.3.1 CLAMPED EDGE CONDITION

In this case the elastic plate is connected along vertical rigid strip edge in the clamped

connection. The appropriate edge conditions correspond to be the zero displacement and

zero gradient. That is

ψ1y(0,a) = 0 (4.26)

and

ψ1xy(0,a) = 0. (4.27)

78 CHAPTER 4

X∞

On multiplying (4.20) with ηmτm sinh(τma) and using edge condition (4.27), m=0

it is found that

2F`η`τ` sinh(τ`a) α1 X∞ X∞ Bnηmτm sinh(τma)Rmn

E1 = − , (4.28)

S1 S1 n=0m=0 Cm

where

S1 = . m=0 Cm

From (4.22) and (4.26), it is obvious to find E2 = 0. On using the zero displacement edge

condition, (4.8) results

αF`η` ∞ γm sinh(γmb)R`m α ∞ ∞ Anηnγm sinh(γmb)Rnm

E

, (4.29)

where

X∞ γ2m sinh2(γmb)

S2 = . m=0 smDm

4.3.2 PIN-JOINTED EDGE CONDITION

For the case in which the plate is pin-jointed (simply supported) along the edge x = 0, y =

a. The appropriate edge conditions are

ψ1y(0,a) = 0, and ψ1xxy(0,a) = 0 (4.30)

On imposing (4.30) in a similar fashion as for clamped edge condition, it is found that E2 =

0 and

α1 ∞ ∞ Bnηmτm sinh(τma)Rmn E

, (4.31)

where

X∞ η2mτ2m sinh2(τma)

S3 = . m=0 Cm

Moreover the zero gradient condition

ψ2xy(0,b) = 0, (4.32)

79

is considered at membrane edge and yielding E3 = 0.


For the given non-SL system, (4.20) and (4.24) constitute a system of infinite number of

linear algebraic equations which, together with the different values of Ei(i = 1,2,3) for

either clamped edge or pin-jointed edge situations, is truncated and solved numerically.

The numerical solution converges point-wise to the desired solution. The truncation of

(4.20) and (4.24) at N corresponds to N +1 equations, where N is the number of truncated

modes.

In order to discuss wave propagation in similar structures as considered herein often

requires the study of the power balance. There are two admissible means of energy

propagation: through the fluid and along the flexible boundary. The convenient

expressions for the (non-dimensional) energy flux across an arbitrary vertical strip in a

duct bounded above by an elastic plate and membrane, and below by a rigid wall are

given by

( N µ 1 22 ¶) X

ℜ |An| ηnCn + |Bn| snDn = 1, (4.33) n 0 α1

α n=0

shows the transmitted power to outlet duct. The power expressions (4.34) to (4.35) can

be found in on using (4.10)-(4.11) in (2.32)-(2.34) together with appropriate ORs which

incorporate both the fluid and the structure-borne components of energy flux.

The dynamic interaction between a fluid and a structure is a major apprehension in

many engineering problems. These problems include systems as diverse as offshore and

submerged structures, storage tanks, bio-mechanical systems, ink-jet printers, aircrafts,

and suspension bridges. The interaction can extremely change the dynamic properties of

the structure. Therefore, it is desired to accurately model these diverse systems with the

=

where ( N )

1 X 2 Pref =ℜ |An| ηnCn , α1 n=0

gives the reflected power in inlet duct and

x < 0, (4.34)

( N )

1 X 2 Ptrans =ℜ |Bn| snDn , x > 0 (4.35)

80 CHAPTER 4

inclusion of the fluid-structure interaction. In order to see the fluid structure interaction,

the fluid and structural equations need to be represented as energy equations for

reflected and 4.4. NUMERICAL RESULTS AND DISCUSSION

transmitted modes. This analysis presents a treatment of the interaction of an acoustic

fluid with a flexible structures. The numerical results presented in this section consist of

comparison between reflected and transmitted components of power against frequency

and change of height, for both the structural-borne fundamental and the fluid-borne

second mode incidence, and to validate the MM technique, conditions are verified for

the real and imaginary parts of pressure and velocity at the interface x = 0.

In the sequel we assume that the inlet duct contains elastic plate of aluminum

with thickness h = 0.0006m and density ρp = 2700kg m−3. The values of Poisson’s ratio and

Young’s modulus are taken to be E = 7.2×1010Nm−2 and ν= 0.34 respectively; while ρa = 1.2kg

m−3 and c = 344ms−1. The outlet duct comprises membrane of mass density ρm = 0.1715kg

m−2 and tension T = 350Nm.

Figures 4.2-4.13 are delineated for two different field incidences, that is, the

fundamental mode incidence and secondary mode incidence. The results show that for

the fundamental mode incidence (` = 0) maximum of energy (in excess of 99% of energy)

is carried in the plate whereas for secondary mode incident (`= 1) in excess of 99% of

energy is in the fluid.

4.4.1 POWER DISTRIBUTION VERSUS HEIGHT OF OUTLET DUCT

In Figures 4.2-4.5, the power components are plotted versus kb (non- dimensional height)

by fixing physical height of inlet duct at a = 0.04m and varying

the height of outlet duct from a = 0.04m to b

= 4m.

4.4.1.1 FUNDAMENTAL MODE INCIDENCE

It is observed that for the case of fundamental mode incidence (` = 0), when a = b the

maximum power goes on reflection for both clamped and pin-jointed conditions, where

f := ω/2π = 250Hz. The overall trend on increasing outlet duct height is reflection over

periodic fluctuation at the point where every new mode becomes propagating. It is worth

mentioning that we have used rigid and the flexible walls of different conditions in the

configuration of inlet and outlet ducts. Therefore the inlet duct modes and outlet duct

modes are coupled due to flexible walls. For the fundamental mode incidence the

maximum of the incident power goes on reflection which is consistent with available

81

results; see for example [35] for rigid inlet and flexible outlet duct walls. Note that Pref ,

Ptrans and PB represent the reflected power, transmitted power and their sum (Power

Balance) respectively.

4.4.1.2 SECONDARY MODE INCIDENCE

Unlike fundamental mode incidence, when the secondary mode is incident (`= 1), at a = b

the 20% of incident power is transmitted whereas the other goes

FIGURE 4.2. For fundamental mode incident and clamped edge conditions.

b 5 10 15

0.2

0.4

0.6

0.8

1

P-ref P-trans PB

b 5 10 15 0

0.2

0.4

0.6

0

82 CHAPTER 4

FIGURE 4.3. For fundamental mode incident and pin-jointed edge conditions.

on reflection. On varying the physical height b of the outlet duct section, the 4.4. NUMERICAL RESULTS AND DISCUSSION

transmission reaches upto 70% of incident power at the point where a new mode is cut-

on. But once a new mode becomes propagating, reflection increases upto 80% and

transmission decreases inversely. The overall trend is that the reflection and transmission

behave inversely for both clamped and pin-jointed edges.

FIGURE 4.4. For secondary mode incident and clamped edge conditions.

b 5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b 5 10 15 0

0.2

0.4

0.6

0.8

1 P-ref P-trans PB

83

FIGURE 4.5. For secondary mode incident and pin-jointed edge conditions.

4.4.2 POWER DISTRIBUTION VERSUS FREQUENCY

Figures 4.6-4.9 depict the distribution of power in elastic plate and membrane bounded

ducts against frequency values. The physical height of inlet duct is

fixed at a = 0.04m while the outlet duct achieves height b = 0.06m.

4.4.2.1 FUNDAMENTAL MODE INCIDENCE

It can be seen that for the case of fundamental mode incidence (` = 0), the maximum

power goes on reflection for both clamped and pin-jointed edge conditions. However

relatively more reflection for later edge condition along with zero gradient condition at

membrane edge is observed. But the power balance identity (4.33) is achieved

successfully in whole frequency regime for both edge conditions.

FIGURE 4.6. For fundamental mode incident and clamped edge conditions.

4.4.2.2 SECONDARY MODE INCIDENCE

Figures 4.8-4.9 elucidates the power balance versus frequency in the case of secondary

mode incidence (` = 1). The graph in Figures 4.8-4.9 is obtained by choosing the clamped

edge condition at elastic plate edge (x = 0, y

= a

= 0.04m) and zero displacement condition

at membrane edge (x = 0, y

=b

= 0.06m), and finally plotting power components verses

frequency. It can be seen that in the frequency range 1 ∼ 234Hz the power balance identity

Frequency(Hz) 200 400 600

0.2

0.4

0.6

0.8

1

P-ref P-trans PB

84 CHAPTER 4

(4.33) is not achieved (dotted line) due to the cut-off inlet duct mode. At frequency f =

235Hz, the inlet duct

4.5. VALIDATION OF TECHNIQUE

FIGURE 4.7. For fundamental mode incident and pinjointed edge conditions.

mode becomes propagating and the 90% of the incident power goes on reflection which

decreases steadily by increasing frequency. However at f = 553Hz the reflected and

transmitted power is distributed equally in duct regions. It is the point where the

membrane bounded duct mode (outlet duct mode) becomes propagating. Once outlet

duct mode is cut-on the maximum power goes on transmission whereas reflection is very

small.

On the other hand, the graph on the right in Figures 4.8-4.9 is obtained by assuming

the pin-jointed condition at elastic plate edge and zero gradient condition at membrane

edge. The graph shows that as inlet duct mode is cut-on at f = 235Hz, the entire incident

power is reflected and consequently there is no transmission. However, once outlet duct

mode is cut-on (f = 553Hz) it suddenly decreases and maximum of incident power goes on

transmission.

4.5 VALIDATION OF TECHNIQUE

Figures 4.10-4.13 show the continuity of pressure (4.6) and normal velocity (4.7) at the

matching interface for a = 0.55 and b

= 0.82, the non-dimensional heights of inlet and outlet

ducts respectively.

Frequency(Hz) 200 400 600 0

0.2

0.4

0.6

0.8

1

P-ref P-trans

85

It is clearly substantiated in Figures 4.10-4.11 that at matching interface, that is, 0 ≤ y

≤ a, the real parts of non-dimensional pressures ℜψ1(0,y)and ℜψ2(0,y) show a good

agreement (see left graph in figure 4.6), where f = 700Hz. The imaginary parts behave

similarly (see right graph in Figures 4.10-4.11).

FIGURE 4.8. For secondary mode incident and clamped edge conditions.

FIGURE 4.9. For secondary mode incident and pinjointed edge conditions.

In Figures 4.12-4.13, the real and imaginary parts of normal velocities ψ1x(0,y) and

ψ2x(0,y) are plotted which also elucidate a very close agreement when 0 ≤ y ≤ a.

By virtue of the aforementioned numerical results and discussion we have the

following pronouncements.

Frequency(Hz) 200 400 600 800 1000 0

0.5

1

1.5

2

P-ref P-trans PB

Frequency(Hz) 200 400 600 800 1000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

P-ref P-trans PB

86 CHAPTER 4

1. The numerical agreement of continuity conditions (4.6)-(4.7) at matching interface

and validation of power balance identity (4.33) substantiate the validity of the MM

solution. 4.5. VALIDATION OF TECHNIQUE

ÂIΨjM

FIGURE 4.10. At matching interface the real part of pressures are plotted against the duct height.

ÁIΨjM

FIGURE 4.11. At matching interface the imaginary part of pressures are plotted against the duct

height.

2. It is important to note that for the fundamental mode incidence the pinjointed

edges minimize the power transmission as compared to the clamped edges.

However for secondary mode incidence, it increases the rates of power distribution

in duct sections.

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.095

0.100

0.105

0.110

0.115

0.120

j=1 j=2

0.2 0.4 0.6 0.8 1.0 y

0.08

0.07

0.06

0.05

0.04

87

3. It is worth commenting that the choice of edge condition does not affect the

attenuation of flexible duct modes. In fact, the choice of edge conditions ÂIΨjxM

FIGURE 4.12. At matching interface the real part of normal velocities are plotted against the duct height.

ÁIΨjxM

FIGURE 4.13. At matching interface the imaginary part of normal velocities are plotted against the duct height.

imposed on the flexible boundaries at the junction significantly affects the

transmission of energy along the duct. However, it does not affect the attenuation

of flexible duct modes as can be visualized in Figures 4.2-4.13, wherein the

attenuation is consistent for any selection of edge conditions.

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.0

0.2

0.4

0.6

0.8

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.0

0.4

0.6

0.8

1.0

1.2

ACOUSTIC SCATTERING IN A

MEMBRANE BOUNDED DUCT WITH

ABRUPT HEIGHT CHANGE

In this chapter, the aim is to resolve two variants of a model problem emerging from

the propagation of fluid-structure coupled waves in a waveguide having an abrupt

height change using semi-analytic techniques. The first problem is handled with a

mode-matching (MM) technique by retaining a finite number (say N) of terms in the

eigenfunction expansion. The power balance is achieved even for a very small N. A

slight modification to the boundary value problem renders the second pertinent

problem discussed in this chapter, however, the MM solution achieves the power

balance only if N → ∞. It is worthwhile precising that the conservation laws provide a

mean to check the validity of solution techniques from the perspective of yielding a

physically admissible solution. The semi-analytic methods, in particular MM

technique, usually preserve conservation laws even when N is very small [89]-[90].

However, in the aforementioned case, the truncated eigenfunction expansion of the

MM solution upto N terms has very low convergence rate and does not preserve power

balance. It is demystified that the involvement of the Dirac delta function at the edges

of vertical discontinuity is responsible for slow convergence. In order to handle this

situation, the Galerkin and modal approaches are therefore adopted and the results

are compared.

The chapter is organized as follows. In Section 5.1, the problem by Warren et al.

[35] is revisited. Two duct mode incidences and the rigid vertical strip at matching

interface are considered. The associated boundary value problem is resolved using

MM technique in Subsection 5.1.1, the power balance identity is obtained in

Subsection 5.1.2 whereas the numerical results are presented in Subsection 5.1.3. In

Section 5.2, the problem with membrane vertical strip is assumed and is resolved using

MM technique in Subsection 5.2.1. The convergence analysis of MM solution using

power balance is discussed in Subsection

5.2.2 and further elucidated numerically in Subsection 5.2.3. Sections 5.3 and 5.4

77

deal with the Galerkin and Modal approaches respectively. The obtained results are

compared for two choices of edge conditions in Section 5.5.

5.1. CANONICAL PROBLEM WITH RIGID VERTICAL STRIP 89

5.1 CANONICAL PROBLEM WITH RIGID VERTICAL STRIP

Consider a two-dimensional waveguide consisting of two semi-infinite sections of

different heights a and b with b > a, linked together by a vertical interface at 0×[a,b] in

the Cartesian frame of reference. The lower wall of the waveguide is coincident with

the horizontal line x = 0. Furthermore, it is assumed that the lower wall is acoustically

hard whereas the upper boundary segments (−∞,0)×

a and (0,∞)×b are membranes. The material properties of the vertical strip may vary.

In particular, we restrict ourselves to the cases of 1) a rigid strip and 2) a membrane.

The waveguide is filled with a compressible fluid of density ρ and sound speed c. The

non-dimensional geometry of the problem is shown in Figure 5.1.

FIGURE 5.1. Non-dimensional geometry of the waveguide

Consider a time harmonic incident wave propagating in the positive x direction

towards origin. Let k = ω/c be the fluid wavenumber where ω is the frequency pulsation

in radians. Then, the non-dimensional problem with respect to the length scale k−1 and

time scale ω−1 is governed by the non-dimensional Helmholtz equation, that is,

¡∇2 +1

¢ψ= 0, (5.1)

where the non-dimensional velocity potential ψ is defined by

= ψ1(x,y), (x,y) ∈(−∞,0)×[0,a],

ψ(x,y) (5.2)

90 CHAPTER 5

ψ2(x,y), (x,y) ∈(0,∞)×[0,b],

with ψ1 and ψ2 being the velocity potentials respectively in each half and, a =ka

and b =kb.

Recall that the boundary condition for the horizontal rigid surface y = 0 is given by

0, (x,y) ∈R

×0, j ∈1,2. (5.3)

Since, the upper boundaries of the waveguide are assumed to be membranes, ψj

satisfies the condition

µ

1,2, (5.4)

with the convention that for j = 1 the condition is applied at (−∞,0)×a whereas for j =

2 it is applied at (0,∞)×b.

At the matching interface 0×(0,a), we impose the continuity condition

ψ1 =ψ2,, y ∈(0,a). (5.5)

Moreover, a second condition giving appropriate information about the normal

component of the fluid velocity is required. Note that at x = 0+, the vertical strip is rigid

while the outside region is in vacuo and the contribution across the duct is neglected,

thus

0, x = 0+, y ∈(a,b). (5.6)

Therefore, an appropriate velocity condition is

∂ψ1

∂ψ2 = ∂x , (x,y) ∈0×(0,a),

(5.7)

∂x

0, (x,y) ∈0×(a,b).

Finally, edge conditions are required at the points where the membranes are

connected with the vertical strip. The most relevant and physically admissible edge

conditions correspond to zero membrane displacement, however, zero gradient

conditions are also used occasionally. Here we restrict ourselves to the zero

displacement edge conditions at the horizontal membrane edges, that is,


ψ1y(0,a) = 0, (5.8)

and

ψ2y(0,b) = 0. (5.9)

5.1.1 MODE-MATCHING SOLUTION

In this section, we invoke the mode-matching technique to solve the boundary value

problem

∂x = ∂x χ0×(0,a),

0×(0,b),

∂ψj

0, R 0, (5.10)

∂y = ×

³³ 0, ( ,0)×a,

∂ 2 0, (0,∞)×b,

ψ1y(0,a) = 0 =ψ2y(0,b),



= 1, x ∈D,

χD(x) (5.11)

0, x ∈Rd \D.

Note that, by separation of variables the velocity potentials can be expressed in

terms of the eigenfunction expansion as

ψ1(x,y) = F` An cosh(τny)e−iηnx, (5.12)

X∞ isnx

ψ2(x,y) = Bn cosh(γny)e , (5.13) n=0

¡∇2 +1

¢ψ= 0,

ψ∂ψ12=ψ∂ψ2,1

(−∞,0)×[0,a]∪(0,∞)×[0,b],

0×(0,a),

92 CHAPTER 5

where An and Bn are the complex amplitudes of nth reflected and transmitted modes

respectively. The first term in (5.12) is the incident wave having am-

plitude F` = p

α/C`η` chosen to ensure unit incident power. The counter ` is placed to take

into account the fundamental mode as well as the first higher mode incidence. It

assumes the values either 0 or 1 corresponding respectively to fundamental or first

higher mode incidences. The non-dimensional wave

q q numbers ηn = τ2n +1 and sn = γ2

n +1 are either positive real or

have positive imaginary part whereas τn and γn, n = 0,1,2,

··· , are the roots of the

dispersion relation K(ζ,p) = 0 where

K(ζ,p) = (ζ2 +1−µ2)ζsinh(ζp)−αcosh(ζp). (5.14)

For the left hand section of duct ζ = τn and p = a, while for the right hand duct section ζ

= γn and p = b. These roots are computed numerically and their properties are discussed

in detail in previous chapters.

Note that the eigenfunctions satisfy a generalized orthogonality relation (OR) in

each duct section. For the left hand duct the OR is a

αcosh(τmy)cosh(τny)dy −τmτn sinh(τma)sinh(τna) =δmnCm, (5.15) 0

where

= αa +µ

3τ2m +1−µ2 ¶

2

Cm : 2 [τm sinh(τma)] . (5.16) 2 2τm

For the right hand duct the OR can be obtained from (5.15)-(5.16) by replacing a with

b, τm with γm and Cm with

= αb +µ

3γ2m +1−µ2 ¶

2

Dm : 2 [γm sinh(γmb)] . (5.17)

2 2γm

Precisely, we have

b

αcosh(γmy)cosh(γny)dy −γmγn sinh(γmb)sinh(γnb) =δmnDm. (5.18) 0


Moreover, the properties discussed above, a Green’s function can be consructed for the eigenfunctions (as discussed in Chapter 2). In terms of eigenfunction for the duct

lying in x > 0, this takes the form

),

(5.19) q=0

for all y,υ∈[0,b].

The continuity condition (5.5), by virtue of expansions (5.12) and (5.13), implies

that

F` Bn cosh(γny), (5.20)

Therefore, on multiplying (5.20) with αcosh(τmy), integrating from 0 to a and then using

the OR (5.15) for the left hand duct, we obtain

τm sinh(τma)E

0

Similarly, the velocity condition (5.7), used in conjuction with the OR for the right hand

duct section leads to

γm sinh(γmb)E2 αF`η`R`m

Bm = + −AnηnRnm, (5.24)

Dmsm Dmsm Dmsm n=0

where

E2 =−iψ2xy(0,b). (5.25)

On ther other hand, using the edge conditions (5.8) and (5.9), it is found that

E1 = 0 and

where

Am =−F`δm`+ + BnRmn,

Cm Cm n=0

(5.21)

E1 =ψ1y(0,a), (5.22)

and a

Rmn =ˆ cosh(τmy)cosh(γny)dy. (5.23)

94 CHAPTER 5

α ∞ γ sinh(γ b) ½

X∞ ¾

E

−F`η`R`m + AnηnRnm ,

(5.26) n=0

where

∞ £

γ sinh(γ b)¤2

S.

(5.27)

Finally, the equations (5.21)-(5.24) and (5.26) lead to an infinite system of linear

algebraic equations from which the unknowns An and Bn for n = 0,1,2,

··· , can be found.

5.1.2 EXPRESION FOR POWER BALANCE

Recall from Chapter 2 that the non-dimensional reflected power is expressed in terms

of An for n = 0,1,2,··· , as

1 XJ1 2

Pref = |Am| Cmηm, (5.28) α m=0

where J1 is the number of cut-on modes in the left hand duct. The transmitted power

Ptrans is obtained from (5.28) by replacing Am,Cm, ηm, J1 with the Bm, Dm, sm, J2 (where J2 is

the number of cut-on modes in the right hand duct). Since, the amplitudes F` are

chosen to achieve unit incident power, the conservation of energy states that the sum

of the reflected and the transmitted power must be unity, that is

Pref +Ptrans = 1. (5.29)

In rest of this section, we substantiate that the power balance (5.29) is achieved

for the equations (5.21) and (5.24) subject to a truncation of the terms.

Note that Rmn is always real, indeed, since the roots of dispersion relation defined

in (5.14) are either real or pure imaginary for a membrane bounded duct. Let us retain

first N terms in (5.24) where N is a positive integer. Multiplying the truncated form of

(5.24) by Bm∗ Dmsm, and summing over m it is found that

N

X Bm∗ BmDmsm

m=0

N

+αF`η` X Bm∗ R`m

m=0


N N

n X B

m∗ Rnm (5.30) m=0

where superposed (∗) indicates the complex conjugate and the superscript N, for

example on ), indicates that N +1 terms are retained in the eigenfunction

expansion. Since

N α X

Bm∗ Rpm =Cδp`+τ∗p sinh(τ∗pa)³ψ1

Ny(0,a)

´∗, (5.31) m=0

from (5.21), with index p = n or ` which may be appropriate, Equation (5.30) after

rearranging the terms becomes

N X ³ 2´

|Bm| Dmsm +|Am|m m=0

=−i(ψ2N

y(0,b))∗ψ2N

xy(0,b)−i(ψ1N

y(0,a))∗ψ1N

xy(0,a)

+|F`|2 η`C`∗ +F`η`C`∗A∗` −Cn∗F`∗A`η`. (5.32)

Note that η` are real for an incident mode and consequently C` are also real.

On applying the edge conditions (5.8)-(5.9) and taking the real part of (5.32) it is

found that

Pref +Ptrans = 1. (5.33)

This identity, which holds for any value of N > 0, confirms that the approximate mode-

matching solution to (5.10) preserves the energy conservative law [35]. It corresponds

to the physical power balance only when N +1

> J1,J2 and is sufficiently large to ensure

that Am, m = 0,1,2,··· ,J1 and Bm, m = 0,1,2,··· ,J2 have converged adequately. Thus

satisfaction of identity (5.33) does not ensure that the solution to the physical problem

has been attained, only that the algebra is most likely correct. The obtained solution

of the modal problem is discussed numerically in the next section.

5.1.3 NUMERICAL DISCUSSION AND RESULTS

Equations (5.21), (5.24) and (5.26) lead to a suitably convergent linear algebraic

system, which on truncation upto N terms provides a solution of the physical problem

numerically. Note that the reduced system retains N +1 equations. The approximate

solution can be used to validate the continuity and the edge conditions. Moreover, the

reflected and transmitted powers can also be analyzed for both fundamental and

secondary incident modes.

96 CHAPTER 5

In Figures 5.2–5.3, the real and imaginary components of non-dimensional

pressures ψ1(0,y) for y ∈ [0,a] and ψ2(0,y) for y ∈ [0,b] are plotted at matching interface

at 700Hz. A good agreement is observed between the two curves in fluid region y ∈ [0,a]

in view of the continuity condition (5.5) on pressure at the matching interface.

In Figure 5.3, the real parts of the normal velocities ℜeψ1x(0,y) for y ∈ [0,a] and

ℜeψ2x(0,y) for y ∈ [0,b] are delineated at the matching interface at 700Hz.

ÂIΨjM

FIGURE 5.2. The real part of pressures against y at matching interface.

ÁIΨjM

FIGURE 5.3. The imaginary part of pressures against y at matching interface.

In the region 0 ≤ y ≤ a, both the curves match exactly while ℜeψ2x(0,y) ≈ 0 for a ≤ y ≤b

showing that the real part of the normal velocity satisfies (5.7). Clearly, the imaginary

j=1

j=2

0.2 0.4 0.6 0.8 y

0.80

0.82

0.84

0.88

0.90

0.92

0.94

j=1

j=2

0.2 0.4 0.6 0.8 y

0.15

0.10

0.05

0.00

0.05


parts of non-dimensional normal velocities show the same level of accuracy; see Figure

5.4.

Figure 5.6 shows the real and imaginary parts of the non-dimensional normal

velocity at matching interface, ℜeψ1y(0,y) and ℑmψ1y(0,y) for y ∈ [0,a]. Clearly,

ℜeψ1y(0,y) = 0 and ℑmψ1y(0,y) = 0 at y = 0 and y = a in agreement with the edge

condition (5.8). Likewise, the validation of condition (5.9) is ob-

ÂIΨjxM

FIGURE 5.4. The real part of normal velocities against y at matching interface.

ÁIΨjxM

FIGURE 5.5. The real part of normal velocities against y at matching interface.

served in Figure 5.7.

j=1

j=2

0.2 0.4 0.6 0.8 y

0.6

0.4

0.2

j=1

j=2

0.2 0.4 0.6 0.8 y

0

4

6

8

10

12

98 CHAPTER 5

In Figure 5.8-5.9, the power components are plotted against frequency by

considering two different incident fields: The fundamental mode incident (i.e.,

structure-borne incident) and the secondary mode incident (i. e., fluid-borne incident).

Figure 5.8 shows the reflected power (P-ref), transmitted power (Ptrans) and their sum

(PB) when the incident field comprises structural-borne incident, and it is clear that

the maximum of the power goes on reflection at discontinuity in duct height. For

secondary mode incidence (which cuts-on at

FIGURE 5.6. The real and imaginary parts of the normal velocity versus y.

FIGURE 5.7. The real and imaginary parts of the normal velocity versus y.

1 y 0 , y

1 y , 0 y

0.2 0.4 0.6 y

1

2

3

4

ly 0 , y

2 y 0 , y

2 y 0 , y

0.2 0.4 0.6 0.8 1.0 y

1

2

3

4

5

6

2 y 0 , y


553Hz), the same power components against frequency are shown in Figure 5.8. It can

be seen that for the fluid borne mode incidence, while cuts on, the maximum of the

power is reflected which decreases by increasing frequency. However, for both

incident fields the power balance identity (5.33) is achieved successfully regardless of

the level of truncation.

100 CHAPTER 5

FIGURE 5.8. The power components versus frequency (Hz) for fundamental mode incidence.

FIGURE 5.9. The power components versus frequency (Hz) for secondary mode incident (which cuts-on at 553 Hz).

5.2 CANONICAL PROBLEM WITH VERTICAL MEMBRANE

In this section, the case when rigid vertical strip in region 0×[a,b] is replaced by

membrane is considered. Therefore, the appropriate vertical strip condition is now

defined by ψ2xyy(x,y)+µ2ψ2x(x,y)−αψ2(x,y) = 2E3δ(y −a)+2E4δ(y −b), (5.34) for all (x,y)

∈0×(a,b), where E3 and E4 are constants, and δ is the Dirac mass at 0 which is being

P-ref

P-trans

PB

0 100 200 300 400 500 600 700 Frequency

0.2

0.4

0.6

0.8

1.0

Power

P-ref

P-trans

PB

600 700 800 900 1000 Frequency

0.2

0.4

0.6

0.8

1.0

Power

5.2. CANONICAL PROBLEM WITH VERTICAL MEMBRANE 101

used to impose two extra edge conditions on vertical membrane edges. The two extra

edge conditions corresponding to zero displacement at either vertical membrane edge

are given by

ψ2x(0,a) = 0, (5.35)

ψ2x(0,b) = 0. (5.36)

Note that the governing equations (5.1)-(5.5) along with the edge conditions (5.8)-

(5.9) stated for the rigid vertical strip case are still valid for the case of vertical

membrane strip. However, the transmission condition for normal velocity is changed

and takes the form

∂ψ2 , y ∈(0,a),

µ

The non-dimensional geometry of the problem is illustrated in Figure 5.10.

∂x

= M(y), y ∈(a,b),

where

(5.37)

1 n o M(y) = 2 αψ2 −ψ2xyy +2E3δ(y −a)+2E4δ(y −b) . (5.38)

102 CHAPTER 5

FIGURE 5.10. Non-dimentional geometery of the duct with verticle membrane strip.

5.2.1 MODE MATCHING SOLUTION

In this section, we deploy the mode-matching (MM) technique to resolve the new

boundary value problem

¡∇2 +1

¢ψ= 0,

ψψψψ221jyxx====ψMψ02,,1x(y, ),

ψ³∂∂2x2xyy2 ++

µ2µ´2ψψ12yx+

−αψαψ12=

=02,E

3δa(y −a)+2E4δ(y

−b),

Ω,

0×(0,a),

0×(0,a),

0×(a,b),

R×0,

0×(a,b),

(−∞,0)×a

,

(0,∞)×b,

(5.39

)


³ψψ∂∂12x2xy2(0(0+,,µaa2))´==ψ002y==+ψψαψ22xy(0(

02 ,,=bb0)),,,

where Ω= (−∞,0)×[0,a]∪(0,∞)×[0,b].

The solution formulated in the previous section for rigid vertical strip (5.12)(5.23)

can be easily adopted here by calculating the expression for Bn using the velocity

condition (5.37) and appropriate edge conditions (5.9) (5.35)–(5.36).

On substituting (5.12) and (5.13) into (5.37) and using the OR for right hand duct,

it is found that

α n γm

Bm = cosh(γma)E3 +cosh(γmb)E4 + sinh(γmb)E2

Dm∆m α X∞ 2 2³ ∞ ´o

(5.40) n=0 + Bn(isnγn −α)Qmn +iµ F`η`R`mAnηnRnm ,

where E2, ∆m and Qmn are defined by

∞

E2 = X

Bn∆nγn sinh(γnb), n=0

(5.41)

∆m = ism (5.42) a

Qmn =cosh(γmy)cosh(γny)dy,

(5.43) 0

respectively.

In order to find the constants Ej for j ∈ 1,··· ,4 the edge conditions (5.8) –

(5.9) and (5.35) – (5.36) can be used. Note that

E1 =ψ1y(0,a) = 0.

104 CHAPTER 5

Multiplying (5.40) by sm cosh(γmy) and summing over m, we get

X∞ sm cosh(γmy)n

ψ2x(0,y) =α cosh(γma)E3 +cosh(γmb)E4 m=0

Dm∆m

Qmn

+iµ2F`η`R`m AnηnRnmo. (5.44)

Therefore, using (5.35)-(5.36), we get

n ψ2x(0,p) =α

cosh(γma)E3 +cosh(γmb)E4 m=0 Dm∆m

Qmn

+iµ2F`η`R`m AnηnRnmo

, p ∈a,b. (5.45)

Similarly on multiplying (5.40) by γm sinh(γmb), summing over m and using

(5.9), it is found that

n γm

0 =α cosh(γma)E1 +cosh(γmb)E2 + sinh(γmb)E4

Dm∆m α

Qmn +iµ2F`η`R`m AnηnRnmo. (5.46)

This expression together with (5.45) leads to system of four equations in Ej. Finally,

these equations together with (5.21) and (5.40) are truncated and solved numerically.

5.2.2 POWER BALANCE

In order to understand power balance, consider the truncated form of (5.40), that

is,

BmDm∆m =αcosh(γma)E1 +αcosh(γmb)E2 +γm sinh(γmb)E4


N

+α X Bn(isnγ2n −α)Qmn +iµ2αF`η`R`m n=0

N

−iµ2α X

AnηnRnm. (5.47) n=0

The substitution of ∆m from (5.42) and then further simplifications lead to

i n

BmDmsm = 2 BmDm

ma)E3

µ N

o Qmn) N

+αF`η`R`m AnηnRnm. (5.48)

Multiplying by Bm∗ and summing over m, it is found that

XN Bm∗ BmDmsm

i N BmDm m=0 =

i XN ∗ n

− µ 2 m=0Bm γm sinh(γmb)E2 +αcosh(γma)E3

N

o Qmn N N

`R`m −α X

AnηnBm∗ Rnm. (5.49) m,n=0

Again (5.21) can be used to eliminate the terms containing Bm∗ Rpm (p =` or n as

appropriate) appearing on right hand side of (5.49), that is

N

106 CHAPTER 5

α X

Rpm . (5.50) m=0

Note that by virtue of OR for right hand duct, Qmn can be expressed as

where

αQmn =Dmδmn −γmγn sinh(γmb)sinh(γnb)−αTmn, (5.51)

Tmn : . (5.52)

Therefore, together with (5.50) and (5.51), the Equation (5.49) leads to

N

X ³ 2 2 ´ |Bm| Dmsm +|Am| Cmηm

m=0

i2 ∗

iµα2 ∗ iµα2 ¢∗E4

µ

i o

y(0,b)

+ iα XN B∗ Bn(isnγ2 −α)Tmn.

(5.53)

µ2 m,n

Now, expressing E1 and E2 in terms of ψ1N and ψ2

N, we obtain

N

X ³ 2 2 ´ |Bm| Dmsm +|Am| Cmηm

m=0

³ ´∗

iα2 ¢∗E3

µ α

xy(0,a)

+ i α2 ˆ b ¡ ¢∗n o

. (5.54)


µ a

Finally, applying the edge conditions and retaining only the real parts, (5.54) yields

Pref +Ptrans +IN = 1 (5.55)

where

ˆ b ¡¢∗ψ2Nxyy(0,y)dy =ℜ n− i

IN e 2 µ a

i N ∗ i

E4o. (5.56)

Remark that the identity (5.55) holds true for all values of N > 0. It should be noted,

however, that for the moderate values of N, IN is non-zero. Therefore, the truncated

solution does not preserve the energy conservation law, yet it can be used to confirm

the algebraic integrity of the solution. Clearly, when N → ∞ the energy must be

conserved. Recall that the vertical membrane satisfies the condition

ψ2xyy(0,y)+µ2ψ2x(0,y)−αψ2(0,y) = 2E3δ(y −a)+2E4δ(y −b), (5.57) for all y ∈ [a,b]. Therefore, by

letting N → ∞ in (5.56), (5.57) can be used to eliminate ψ2xyy(0,y) from (5.56) to

substantiate that

b ˆx(0,y)dy . (5.58) no lim IN =−ℜe i

N→∞ a

The form of integral is reminiscent of the non-dimensional flux across the membrane;

which should be zero since the acoustic field exterior to the duct is neglected. This can

be confirmed by using (5.57) thereby eliminating from the integral (5.58).

Consequently,

lim IN =ℜ ˆ ψ2x(0,y)ψ2xyy(0,y)dy n i

N→∞ α a

. (5.59)

Integrating by parts, we obtain

x(0,b) N

108 CHAPTER 5

i ib i

b o ψ2xy(0,y)ψ∗

a − αˆ

a 2xy(0,y)dy . (5.60)

Finally, using the edge conditions it is clear that IN → 0 as N →∞ and thus (5.58):

Pref +Ptrans → 1 as N →∞. (5.61)

5.2.3 NUMERICAL RESULTS AND DISCUSSION

In this section, the above truncated system is solved numerically for unknown modal

coefficients An and Bn for n = 0,1,2,

··· ,N. The results are used to reconstruct the matching

conditions (5.5) and (5.37) and to ensure that the eigenfunction expansion of the

solutions have converged adequately.

Figures 5.11- 5.14, delineate the real and imaginary parts of matching conditions

at the interface. An excellent agreement between two curves is observed in the fluid

region 0 ≤ y

≤ a. However, the normal velocity fields oscillates for vertical membrane

strip case. Such oscillations appear due to the abrupt truncation of poorly convergent

series and the Gibb’s phenomenon obscure all useful information about these series.

Therefore, in order to smooth the normal velocity curves for necessary information

either one has to increasing N or use Lanczos filters developed in Chapter 3.

ÂIΨjM

FIGURE 5.11. The real part of pressures at the matching interface.

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.315

0.320

0.325

0.330

0.335

0.340

0.345


In the next graphs the power components are shown against frequency. Figures

5.18-5.19 show the graphs when fundamental and secondary mode incident ÁIΨjM

FIGURE 5.12. The imaginary part of pressures at the matching interface.

ÂIΨjxM

FIGURE 5.13. The real part of normal velocities at the matching interface.

fields are considered. It can be seen at the moderate level of truncation that the power

balance identity (5.61) is not achieved exactly (however it has 97-99% accuracy).

In Figure 5.20 the power balance (PB) (that is, the sum of reflected and transmitted

power), the error term IN and their sum, are plotted versus N. Clearly, on increasing N,

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.315

0.320

0.325

0.330

0.335

0.340

0.345

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.2

0.2

0.4

0.6

110 CHAPTER 5

IN decreases and PB approaches to unity, nevertheless PB+IN is unity for all N > 0 in

agreement with (5.55).

It should be noted that the truncated modal solution does not preserve the ÁIΨjxM

FIGURE 5.14. The imaginary part of normal velocities at the matching interface.

FIGURE 5.15. For the fundamental mode incident the power components are plotted against frequency (Hz)

power balance unless N → ∞. The later result can be explained by considering

equations (5.34) and (5.37). The two delta functions present in (5.37) are introduced

in order to enforce the edge conditions on the vertical membrane and this equation is

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

2

4

6

8

P-ref

P-trans

PB

0 100 200 300 400 500 600 700 Frequency

0.2

0.4

0.6

0.8

1.0

Power


used to generate (5.35)-(5.36). The system of equations (5.21), (5.40), (5.45) and (5.46)

is then truncated and solved.

Unfortunately, even retaining a large number of terms, for example with

N > 350, the accuracy improves very slowly whereas the computational time

FIGURE 5.16. For the secondary mode incident the power components are plotted against frequency (Hz)

FIGURE 5.17. The power balance, error term and the power balance identity are plotted against truncated terms (N).

P-ref

P-trans

PB

600 700 800 900 1000 Frequency

0.2

0.4

0.6

0.8

1.0

Power

I N

PB + I N

PB

0 100 200 300 400 N

0.2

0.4

0.6

0.8

1.0

112 CHAPTER 5

increases. Hence, we conclude that the solution converges poorly and thus, one

requires an alternative technique.

113 5.3. GALERKIN APPROACH

5.3 GALERKIN APPROACH

The basic theme of the Galerkin approach is to select an appropriate basis function to

describe the displacement of the vertical membrane. An appropriate basis is the one

in which each function satisfies the edge conditions imposed at both end points.

Consider the vertical membrane condition

Wyy +µ2W −αψ2 = 0, 0×[a,b], (5.62)

where W(y) is the membrane displacement. Note that at y = a and y

= b, the vertical

membrane (5.62) is connected with semi infinite horizontal membranes. Two different

sets of edge conditions are chosen at these ends. Precisely, the zero displacement at y

= a and y =b

W(a) =W(b) = 0, (5.63)

and zero displacement at y = a and zero gradient at y

=b

W(a) =Wy(b) = 0, (5.64)

are chosen. The normal velocity condition can be written in the form

∂ψ1

∂ψ2 = ∂x , 0×(0,a),

(5.65)

∂x

W(y), 0×(a,b).

It is expedient to express W(y) as a Fourier sine series, that is,

W a)¤

, (5.66)

where the Fourier coefficients Gn for n = 1

,2,3,··· , are unknowns. Note that each term of

the series automatically satisfies the zero displacement conditions at y = a and the

eigenvalues λn for n = 0

,1,2,··· , are chosen to satisfy the remaining conditions at y =b.

More details are furnished in the subsequent subsections.

The values of the coefficients Gn can be determined by using (5.62). On substituting

(5.13) and (5.66) into (5.62) it is found that

114 CHAPTER 5

Bn cosh(γny), y ∈(a,b). (5.67) n=0

n=0

Multiplying (5.67) by sin[λm(y −a)], integrating over [a,b] and using the standard OR, we

get

2α ∞

b

Pmn = sin[λm(y −a)]cosh(γny)dy. (5.69) a

Now, the velocity condition (5.65) at x = 0 may be expressed as

X∞

i Bnsn cosh(γny) n=0

X∞

= iF`η`cosh(τ`y)−i n=0 Anηn cosh(τny), y ∈(0,a) (5.70)

j=0a)], y ∈(a,b).

On multiplying (5.70) by αcosh(γmy), integrating from 0 to b and using the OR

(5.18), it is found that

−iγm sinh(γmb) αF`η`R`m

Bm = E2 + −AnηnRnm

Dmsm Dmsm Dmsm n=0

BnPjmPjn

−, (5.71)

Dmsm

where

Gm = (b−a)µ2 −λ2m nX=0BnPmn, (5.68)

115

=− m m

` ` `m

Bm Dmsm

− E1

Dmsm n=0 Cn

ηnBqRnqRnm

− Dmsm n=0q=0 Cn

2iα2 ∞ ∞ B

− X X nPjmPjn . (5.73)

Dmsm

Note from (5.8) that E1 = 0 whereas the value of E2 can be obtained by multiplying

(5.73) with γm sinh(γmb), summing over index m and finally using (5.9).

Precisely, we get

mb)R`m

E2 =− S1 m=0 Dmsm

RnqRnm

. (5.74)

In order to proceed further, the eigenvalues λn must be specified. The choice of

eigenvalues is dictated by the behavior of the vertical membrane at y =b. We have

precisely two situations.

5.4. MODAL APPROACH

where ∞

E2 = X

Bnisnγn sinh(γnb) =ψ2xy(0,b). n=0

Now, invoke (5.21) to eliminate An from (5.71) to get

iγ sinh(γ b) 2αF η R

(5.72)

s

116 CHAPTER 5

5.3.1 ZERO DISPLACEMENT AT y=b

If the displacement is zero at y =b, then the appropriate choice is

nπ

b−a

Finally, Equations (5.21) and (5.73) along with the values of constants Ej, j = 1,2 are

truncated and solved numerically.

5.4 MODAL APPROACH

The modal approach is a little alteration of Galerkin one. The solution is obtained by

expressing the vertical membrane displacement in terms of the eigenfunctions for the

duct lying in x > 0 and then using the properties of duct modes. In this case it is assumed

that

W Gn cosh(γny), y ∈(a,b), (5.77)

where the modal coefficients Gn,n

= 0,1,2... are unknowns.

Recall that the non-dimensional pressure for the region x > 0 is given by (5.13). On

substituting (5.13) and (5.77) into (5.62) it is found that

Bn cosh(γny). (5.78) n=0

n=0

Multiplying (5.78) by cosh(γq y) and integrating over [a,b], we obtain

X∞ X∞

BnTnq, (5.79) nn=0

where

λn = b−a, n = 1,2,3,··· ,

5.3.2 ZERO GRADIENT AT y=b

If the gradient is zero at y =b, the appropriate choice is

(5.75)

(n

λn = +1/2)π

n = 0,1,2,··· , (5.76)

117

Tnq . (5.80)

Note that

γ2nTnq =γ2

qTnq +γn sinh(γnb)cosh(γqb)−γn sinh(γna)cosh(γqa)

−γq sinh(γqb)cosh(γnb)+γq sinh(γqa)cosh(γna). (5.81)

118 CHAPTER 5

Therefore, substituting (5.81) into (5.78) and rearranging, we obtain

X∞ E3 cosh(γqa) E4 cosh(γqb) E5γq sinh(γqa)

GnTnq 2

n=0 =− γq +µ2 − γ2q +µ2 − γ2q +µ2

X∞ BnTnq

, (5.82)

γ2q +µ2 n=0

where

E Wy(a), (5.83)

E Wy(b), (5.84)

E W(a) (5.85)

E W(b). (5.86)

Now, multiplying (5.82) by αcosh(γq y)/Dq, summing over q and interchanging the

orders of summation and, integration and using (5.19), it is found that

W(y) =−E3φ(1)(y)−E4φ(2)(y)−E5φ(3)(y)−E6φ(4)(y)

Tnq

+α n=0q=0 2 +µ2)Dq , y

∈[a,b], (5.87)

(γq

where

φ(1)(y) =χ(a,y), φ(2)(y) =χ(b,y), (5.88)

(5.89)

with

χ(ν,y) =αq=0 2 +µ2)Dq . (5.90)

119

(γq

Hence, an expression for W(y) is obtained in terms of of duct eigenfunctions. Now

substitute this together with (5.12) and (5.13) into the normal velocity condition

(5.37), to get

X∞ Bnsn cosh(γny) n=0

= F` ∞ n=0 Anηn cosh(τny), y

∈(0,a), (5.91)

Gj cosh(γj y), y ∈(a,b).

5.4. MODAL APPROACH

Multiplying the above identity with αcosh(γmy) and invoking OR (5.18), it is found that

αF`η`R`m iE2γm sinh(γmb)

Bm = −AnηnRnm − smDm smDm n=0 smDm iα

no

+ E E smDm

iα3 X∞ X∞ BnTmqTnq

, (5.92)

− smDm n=0q=0 (γ2q +µ2)Dq

where

dy, j = 1,··· ,4. (5.93)

Finally using (5.21), we eliminate the coefficients An, and therefore (5.92) reduces to

iα X X BnTmq

s

s

s

s

s

s

120 CHAPTER 5

The above equation involves the constants E1 −E6 which can be found by using the

edge conditions (5.8)-(5.9) and (5.63)-(5.64). First of all, note that E1 = ψ1y(0,a)

= 0. Now,

multiplying (5.94) with γm sinh(γmb), summing over index m and using the edge

condition (5.9), it is found that

∞ γ X m sinh(γmb) ½na)Rnm ¾

E2S1 =−iα 2F`η`R`m −E1

m=0 smDm n=0 Cn

qRqmRqn iαTqmTqn )

+ m=0 smDm n Cq + (γ2q +µ2)Dq

+©

E E

ª (5.95) m=0 smDm

It remains to apply the appropriate condition to the edges of the vertical

membrane, for which two cases are considered herein. In each case, the displacement

at y = a is chosen to be zero, that is, W(a) = 0 may be expressed as

E3φ(1)(a)+E4φ(2)(a)+E5φ(3)(a)+E6φ(4)(a)

Tnq

=α n=0q=0 2 +µ2)Dq . (5.96)

(γq

5.4.1 ZERO DISPLACEMENT AT y=a AND y=b

For this set of edge conditions E5 =E6 = 0 from (5.63), which simplifies (5.96) to

∞ ∞ B (1) n cosh(γqa)Tnq

E3φ (a) . (5.97)

= = Dq

The equivalent condition at y =b yields

121

∞ ∞ Bn cosh(γq y)Tnq

E

. (5.98)

5.4.2 ZERO DISPLACEMENT AT y=a AND ZERO GRADIENT AT y=b

This set of edge conditions is defined in (5.64) and implies E4 = E5 = 0. Once

again, this simplifies (5.96) to

(1) (4) 2 X∞ X∞ Bn cosh(γqa)Tnq

E3φ (a)+E6φ (a) =α n=0q=0 2

+µ2)Dq . (5.99)

(γq

In addition, the zero gradient condition at y =b provides

∞ ∞ Bnγq sinh(γqa)Tnq

E3

6 y 2 . (5.100)

Each set of edge conditions leads to system of four equations in which E3−E6 are

unknowns. These together with (5.94) and (5.95) are truncated and solved

numerically.


In this section, Galerkin and modal solutions are discussed numerically. The unknowns

An and Bn for n = 0,1,2,

··· , can be calculated by solving simultaneously the truncated

system of equations in each case. The results are used to compare the scattered

powers in duct sections for both solutions. Moreover, each one of the solutions is used

to reconstruct the normal velocity condition (5.65).

Let us rewrite (5.65) after using the Lanczos filter as

V2(y) =V3(y), y ∈[0,b],

Here

(5.101)

N

V2(y) = X

Bnσbnisn cosh(γny),

y ∈[0,b], n=0

and V3(y) is defined piece-wise as

(5.102)

122 CHAPTER 5

V1(y), y ∈[0,a], V3(y)

=

W(y), y ∈[a,b].

(5.103)

5.5. NUMERICAL RESULTS AND DISCUSSION wherein

N

V1(y) =F`η`i cosh(τ`y)−i X

Anσ

naηn cosh(τny) (5.104) n=0

The Lanczos filters σna and σb

n can be obtained from (3.53) on setting p = a, ζn =τn and p =

b, ζn = γn, respectively. Note that the Lanczos filters have been used to remove the

oscillations appearing in the normal velocity fields.

ÂIV jM

FIGURE 5.18. The real part of normal velocities using Galerkin solution at matching interface.

ÁIV jM

j=3

j=2

0.2 0.4 0.6 0.8 1.0 y

0.4

0.2

0.2

0.4

j=3

j=2

0.2 0.4 0.6 0.8 1.0 y

1

2

3

4

5

123

FIGURE 5.19. The imaginary part of normal velocities using Galerkin solution at matching interface.

On using the solutions obtained by Galerkin approach the real and imaginary

components of normal velocity condition (5.103) are shown in Figures 5.18-

5.19. Similarly Figures 5.20–5.21 depict the real and imaginary parts of normal velocity

condition for Modal solutions.

ÂIV jM

FIGURE 5.20. The real part of normal velocities using modal solution at matching interface.

ÁIV jM

FIGURE 5.21. The imaginary part of normal velocities using modal solution at matching interface.

In Figures 5.22- 5.23, the reflected and transmitted powers and their sum are

plotted versus frequency with two sets of edge conditions. On taking both

j=3

j=2

0.2 0.4 0.6 0.8 1.0 y

0.2

0.2

0.4

j=3

j=2

0.2 0.4 0.6 0.8 1.0 y

1

2

3

4

5

124 CHAPTER 5

fundamental and secondary mode incidences, the power components obtained by

Galerkin approach are compared with those of Modal approach.

In Figure 5.25, for the secondary mode incident (which cuts-on at 553 Hz), zero

displacement condition at y = a and zero gradient condition at y

= b the 5.5. NUMERICAL RESULTS AND DISCUSSION

FIGURE 5.22. A comparison of the power components versus frequency for Galerkin solution

(solid and dashed line) and Modal solution (dotted on solid and dashed lines) subject to the fundamental mode incidence and zero displacement condition at y=a and y=b.


(solid and dashed line) and Modal solution (dotted on solid and dashed lines) subject to the secondary mode incidence (which cuts-on at 553Hz and zero displacement condition at y =a and y=b.

power components are compared versus frequency for the Galerkin and Modal

solutions. The figure shows a perfect match.

P-ref

PB

P-trans

0 100 200 300 400 500 600 700 Frequency

0.2

0.4

0.6

0.8

1.0

P-ref

PB

P-trans

600 700 800 900 1000 Frequency

0.2

0.4

0.6

0.8

1.0

Power

125


(solid and dashed line) and Modal solution (dotted on solid and dashed lines) subject to the

fundamental mode incidence, zero displacement condition at y=a and zero gradient condition at y=b.

FIGURE 5.25. A comparison of the power components versus frequency for Galerkin solution (solid and dashed line) and Modal solution (dotted on solid and dashed lines) subject to the

secondary mode incidence (which cuts-on at 553Hz), zero displacement condition at y=a and zero gradient condition at y=b.

P-ref

PB

P-trans

0 100 200 300 400 500 600 700 Frequency

0.2

0.4

0.6

0.8

1.0

Power

P-ref

PB

P-trans

600 700 800 900 1000 Frequency

0.2

0.4

0.6

0.8

1.0

Power

ACOUSTIC SCATTERING IN ELASTIC

PLATE BOUNDED DUCT WITH ABRUPT

HEIGHT CHANGE

This chapter is a continuation of Chapter 5. We retain the same geometrical configuration

of the waveguide as in Chapter 5 except that the upper surfaces are now assumed to be

elastic plates instead of membranes. Precisely, we consider a waveguide consisting of

two semi-infinite duct sections of different heights connected through a vertical strip.

The upper surfaces are assumed to be elastic plates whereas the lower surface is taken

to be a rigid wall. Two types of vertical strip are taken into account: 1) rigid vertical strip

and 2) vertical elastic plate. The geometry of the waveguide is illustrated in Figure 6.1.

The scattering of fluid-structure coupled waves in the waveguide is studied and the

associated boundary value problems are derived. The solutions of the scattering

problems are obtained through mode-matching (MM) and Galerkin techniques. The later

one is a little alteration of the MM technique wherein the displacement of the vertical

elastic plate is chosen to be à priori known satisfying the elastic plate conditions. This

approach is relatively simple and has been recently applied to the problems involving the

vibration of plates. Refer, for instance, to the vibration analysis of free rectangular plates

by superpositionGalerkin method [69] and Galerkin approximation of the Iris problem

[89]. The reflected and transmitted powers in both problems are also discussed.

The chapter is organized as follows. In next Section 6.1, the governing boundary value

problem in two dimensions with rigid vertical strip is stated. The MM solution is

formulated in Section 6.1.1. The problem with elastic plate vertical strip is stated in

Section 6.2. The solution of this problem using Galerkin approach is developed in section

6.2.1. A selection of comparative numerical results are presented and discussed in

Section 6.3.

107

6.1 PROBLEM WITH RIGID VERTICAL STRIP

Consider a two-dimensional waveguide occupying the regions (−∞,0]×[0,a]∪ [0,∞)×[0,b]

where b > a. Let (x,y) be the two dimensional Cartesian plane coordinates. At x = 0, these

regions are joined by means of a vertical strip along the line segment a ≤

y ≤b. The

acoustically rigid lower wall of the waveguide is aligned with R×0. The upper walls are

127

assumed to be elastic plates. Moreover, let the waveguide be filled with a compressible

fluid of density ρ and sound speed c. The physical configuration of the problem is shown

Figure 6.1.

FIGURE 6.1. The physical configuration of the elastic plate bound waveduide with rigid vertical strip.

Consider an incident time harmonic wave propagating from negative x−axis towards

x = 0. On striking the interface the incident field generates reflected and transmitted

waves. Let ω be the angular frequency and k =ω/c be the fluid wave number. Then, we

define the non-dimensional coordinates using the length scale k−1 and time scale ω−1 as in

Chapter 2. Consequently, the non-dimensional velocity potential

= ψ1(x,y), (x,y) ∈(−∞,0)×[0,a],

ψ(x,y) (6.1)

ψ2(x,y), (x,y) ∈(0,∞)×[0,b],

satisfies the Helmholtz equations ¡ 2 ¢

∇ +1 ψ= 0, (6.2)

where a = ka, b

= kb and, ψ1 and ψ2 are the velocity potentials in first and the second duct

sections respectively.

The non-dimensional horizontal rigid surface is expressed by the conditions

∂ψj

0, R 0, j 1,2. (6.3)

∂y = × =

128 CHAPTER 6

6.1. PROBLEM WITH RIGID VERTICAL STRIP

Since the upper walls of both duct sections are assumed to be the elastic plates, we

impose the non-dimensional elastic plate boundary condition

µ ∂4 ¶

0, (6.4)

with the convention that the condition is applied at (−∞,0)×a for j = 1 and at (0,∞)×b for

j = 2, whereas the parameters µ1 and α1 are defined in Equation

(2.14).

Moreover, as the matching interface 0×(a,b) is taken to be a rigid vertical strip, we

have

(6.5)

The continuity conditions of the fluid pressure and the normal velocity are given by the

equations

∂x 0, 0×(a,b).

In addition, the edge conditions are required at the points where two elastic plates

are connected with vertical rigid strip. These conditions determine how two plates are

connected with each other. In this section, we assume that the plates are clamped, that

is, at the edges displacement and the gradient are zero. Therefore, we have

ψ1y(0,a) =ψ2y(0,b) = 0, (6.8)

and

ψ1xy(0,a) =ψ2xy(0,b) = 0. (6.9)

Therefore, the scattering problem in the elastic plates bounded waveguide is given

by the boundary value problem

and

ψ1(x,y) =ψ2(x,y), ∀(x,y) ∈0×(0,a), (6.6)

∂ψ1

∂ψ2 = (x,y), 0×(0,a), (x,y) ∂x (6.7)

129

¡∇2 +1

¢ψ= 0,

ψψψ122xx===ψψ02,,1xχ0

×(0,a),

(−∞,0)×[0,a]∪(0,∞)×[0,

b],

0×(0,a),

0×(0,b),

0×(a,b),

ψjy = 0, R×0, (6.10

)

³³∂∂∂∂xx4444 ´´ 0, (0−,∞∞,0))××ba,,

ψ

1y(0,a) = 0 =ψ2y(0,b),

ψ1xy(0,a) = 0 =ψ2xy(0,b),

where χ is the characteristic function, that is, for any domain D ⊂Rd

1,

χD(x) =

0,

x ∈D,

x ∈Rd \D. (6.11)

6.1.1 MODE MATCHING SOLUTION

In this section, we resolve the boundary value problem (6.10) using mode-matching

technique. In order to do so, we describe the velocity potentials in terms of the

eigenfunction expansions in respective duct sections as

ψ1(x,y) = F` An cosh(τny)e−iηnx, (6.12)

Bn cosh(γny)eisnx. (6.13)

The first term in (6.12) is the incident wave having amplitude F` =p

α/C`η` chosen to

ensure unit incident power. The counter ` assumes the value 0 to indicate the

fundamental mode incidence and 1 for the first higher mode incidence.

q q

The non-dimensional wave numbers ηn = τ2n +1 and sn = γ2

n +1 are either positive real

or possess a positive imaginary part. Moreover, the eigenvalues τn and γn, for n = 0,1,2,···

, are the roots of the dispersion relation K(ζn,p) = 0 where

K(ζn,p) :

=³¡ζ2

n +1¢2 −µ4

1´ζn sinh(ζnp

)−α1 cosh(ζnp

) = 0. (6.14)

130 CHAPTER 6

The roots τn and γn can be found numerically. Note that the corresponding eigenfunctions

are non-orthogonal but satisfy the orthogonality relation

p

αcosh(ζny)cosh(ζmy)dy 0

=δmnÅn ), (6.15)

We elaborate that in (6.14)-(6.16), ζn = τn, λn = ηn, p =

a and Ån =Cn for the left hand duct section at (−∞,0)×[0,a] whereas ζn = γn, λn = sn, p = b and

Ån = Dn for the right hand duct section (0,∞)×[0,b].

The unknown coefficients An and Bn are the complex amplitudes of nth reflected and

transmitted modes. In order to determine the amplitudes An and Bn, we substitute

(6.12) and (6.13) in the continuity condition (6.6). In fact, we have

F` Bn cosh(γny), y ∈[0,a]. (6.17) 6.1. PROBLEM WITH RIGID VERTICAL STRIP

Now, multiplying (6.17) by α1 cosh(τmy), integrating over y from 0 to a and using OR (6.15)

for x < 0, it is found that

τm sinh(τma)n o α1 X∞

Am =−F`δm`+ e BnRmn, (6.18)

CmCm n 0

where

α1p αcosh(ζnp)sinh(ζnp) Ån := + +2λ2

n sinh2(ζnp).

2 2γn (6.16)

where

=

Rmn

a

=cosh(τmy)cosh(γny)dy, 0

(6.19)

e1 = ψ1yyy(0,a), (6.20)

e2 = ψ1y(0,a). (6.21)

131

Similarly on substituting (6.12) and (6.13) into (6.7), multiplying by αcosh(γmy),

integrating over y from 0 to b and using OR for x > 0, we arrive at

Bm =α1F`η`R`m + γm sinh(γmb)nee4o

Dmsm Dmsm

AnηnRnm, (6.22)

eBnsnγn sinh(γnb) =−iψ2xy(0,b). (6.24) n=0

Remark that the constants e1 −e4 can be found by enforcing the edge conditions.

Indeed, from the clamped edge conditions (6.8)-(6.9) it is trivial to get e2 = 0 = e4.

Moreover, we multiply (6.18) by ηmτm sinh(τma), sum over index m and finally recall (6.9)

to obtain

2F η τ sinh(τ a) α ∞ ∞ B η τ sinh(τ a)R

S1 = . (6.26) m=0 Cm

Similarly, by multiplying (6.22) with γm sinh(γmy) and summing over index m, we get

mb) X∞

γ2m sinh2(γmb)

S . (6.28)

Dmsm

The expressions for e1 and e3 together with (6.18) and (6.22) lead to the system of

equations that can be solved numerically after truncation.

where

e3 =

∞ X 3

Bnsnγn sinh(γnb) =−iψ2xyyy(0,b), n=0 (6.23)

e1 = ` ` ` ` − 1 X X n m m m mn ,

S1 S1 m=0n=0 Cm

where

(6.25)

where

e3 = F`η`R`m − AnηnRnm,

S2 m=0 Dmsm n=0

(6.27)

132 CHAPTER 6

6.2 PROBLEM WITH VERTICAL ELASTIC PLATE

In this section, we consider a variant of the problem discussed in Section 6.1. Suppose

the vertical interface 0 × [a,b] between the two duct sections is an elastic plate instead

of a vertical rigid strip. The geometric configuration of the problem is illustrated in Figre

6.2.

FIGURE 6.2. The geometry of the elastic plate bounded waveguide with abrupt height change.

The new boundary value problem is obtained when rigid strip condition

(6.5) is replaced by elastic plate condition

µ d4 4¶

dy 4 −µ1 W(y)+α1ψ2(0,y) = 0, y ∈[a,b], (6.29)

where W(y) = ψ2x(0,y) is the elastic plate displacement. It is worthwhile precising that the

condition (6.29) will affect only the continuity condition (6.7) of normal velocity and the

rest of the equations still remain valid. The continuity condition of normal velocity now

takes the form

= ψ1x, 0×(0,a)

ψ2x (6.30)

W(y), 0×(a,b).

In addition, four edge conditions are required at the vertical membrane ends to

completely determine the fluid velocity potential ψ. In the sequel, we consider clamped

connection at y = a and y =b, that is,

W(a) =W(b) = 0, (6.31)

Wy(a) =Wy(b) = 0. (6.32)

Recall from Chapter 5 that the MM technique renders poorly convergent solutions to

problems with flexible vertical strips unlike the case of those with

133

6.2. PROBLEM WITH VERTICAL ELASTIC PLATE

rigid vertical strips. Albeit, the MM technique combined with the Galerkin approach

provides a viable tool to obtain the solution to such problems. In the next section, we

resolve the aforementioned problem using a Galerkin approach blended with a MM

technique.

6.2.1 GALERKIN SOLUTION

Since the equations described for rigid vertical strip problem except (6.5) and (6.7)

remain unchanged, therefore, the solution described in (6.12)-(6.21) can also be

adopted here. Precisely, the values of Bn for n = 0,1,2,

··· , have to be calculated now using

(6.29)-(6.30) subject to appropriate edge conditions.

In order to find an expression for vertical elastic plate displacement, we substitute

(6.13) into (6.29) to get

µ d4 4¶ ∞

dy 4 −µ1 W Bn cosh(γny), y ∈[a,b]. (6.33)

By solving ordinary differential equation (6.33), we obtain

W(y) =a1 cos(µy)+a2 sin(µy)+a3 cosh(µy)+a4 sinh(µy) (6.34)

, y ∈[a,b], (6.35)

where the constants ai,i = 1

,··· ,4, can be found by using the edge conditions (6.31)-(6.32).

Using (6.35) in the edge conditions, one may get the values of ai by

M 1 1 1 1 , (6.37)

−sin(µ1a) cos(µ1a) sinh(µ1a) cosh(µ1a

−sin(µ1b) cos(µ1b) sinh(µ1b cosh(µ1b)

a1

where

A = M−1R, (6.36)

cos(µ1a)

cos(µ b) sin(µ1a) cosh(µ1a) sin(µ b)

cosh(µ b) sinh(µ1a)

sinh(µ b)

134 CHAPTER 6

a2

A a3 R 1 P a) . (6.38) n n

n

µ1 n 0 γ4 µ41 a4 αµ11 P∞n==0 Bnγnγsinh(n4n−−µ41γnb)

Now the continuity condition (6.30) of normal velocity can be used to find

Bn. Therefore, by substituting (6.12)- (6.13) into (6.30), multiplying by α1 cosh(γmy) and

integrating from 0 to b, it is found that

=−iγm

sinh(γmb) 2 + α1F`η`R`m

Bme3 +(γm +2)e4

Dmsm iα o − a1Λ1m +a2Λ2m +a3Λ3m +a4Λ4m

Dmsm

i BnTmn

− AnηnRnm + ,

(6.39)

Dmsm n=0 Dmsm n

where

b

Tmn =cosh(γmy)cosh(γny)dy,

(6.40) a

b

Λ1m =cos(µ1y)cosh(γmy)dy,

(6.41) a

b

Λ2m =sin(µ1y)cosh(γmy)dy,

(6.42) a

b

Λ3m =cosh(µ1y)cosh(γmy)dy,

(6.43) a

b

Λ4m =sinh(µ1y)cosh(γmy)dy,

(6.44) a

and the constants e3 −e4 are defined in (6.23)-(6.24).

s n

135

Finally, note from the edge condition (6.9) at y = b that e4 = 0. The constant e3 can be

found by multiplying (6.39) with γm sinh(γmy), summing over m and

using (6.8), that is

mb)n X∞ o

e3 = F`η`R`m − AnηnRnm

S2 m=0 Dmsm n=0 iα1 ∞ γm

sinh(γmb)n o

a1Λ1m +a2Λ2m

+a3Λ3m +a4Λ4m S2 m=0 iα21 X∞ X∞ Bnγm sinh(γmb)Tmn

+

.

(6.45)

Dmsm

Therefore, (6.45) together with (6.18) and (6.39) can be exploited to obtain a numerical

solution by truncation.


In this section, the aforementioned systems are truncated upto N terms and are solved

numerically. The convergence of the solution can be ensured by reconstructing the

matching conditions and by validating the power balance identity (PB). The expressions

for reflected power, transmitted power and power balance identities (5.28)-(5.29) used

in Chapter 5 remain effective for this problem

136 CHAPTER 6

as well. In fact, this can be confirmed by using the eigenfunction expansion (6.12)-

(6.13) together with (2.32) and (2.34). Thus, the reflected power (P-ref) and

transmitted power (P-trans) are given by

1 XJ1 2

Pref = |Am| Cmηm, (6.46) α1 m=0

and

1 XJ2 2

Ptrans = |Bm| Dmsm, (6.47)

α1 m=0

where J1 (resp. J2) is the number of cut-on modes in the left (resp. right) hand duct

section.

As discussed in previous chapters, the choice of F` allows to normalize the incident

power to unity. Therefore, the sum of reflected and transmitted powers must also be

unity, that is,

Pref +Ptrans = 1, (6.48)

which is the statement of conservation of energy.

For each graph presented in this section, the elastic plate is taken to be of

aluminum, with thickness h = 0.0006m and density ρp = 2700 kg m−3. The Poisson’s ratio

and Young’s modulus are taken to be E = 7.2×1010Nm−2 and ν= 0.34; whereas ρ = 1.2kg m−3

and c = 344ms−1. The two duct sections of the waveguide

are fixed at heights a = 0.06m and b

= 0.085m.

In order to check the validity of the pressure condition (6.6), ℜeψj(0,y) and

ℑmψj(0,y) for j = 1,2 are plotted against y; refer to Figures 6.3 and 6.4. Clearly, both the

curves match exactly in the region 0×[0,a] in agreement with transmission condition

(6.6).

Similarly, in Figures 6.5–6.6, the real and imaginary components of the normal

velocities ℜeψjx(0,y) and ℑmψjx(0,y) for j = 1,2 are delineated. It is observed that both

the real and the imaginary parts of ψ2x(0,y) are approximately 0 along the rigid vertical

strip at a ≤

y ≤ b, whereas, they are in agreement with corresponding components of

ψ1x(0,y) at the aperture 0 ≤

y ≤ a. Therefore, we conclude that condition (6.7) is satisfied.

6.3. NUMERICAL RESULTS AND DISCUSSION 137

Note that the Lanczos filters, discussed in detail in the previous chapters, are used to

remove the oscillation in the velocity field.

For the second problem, the region 0×[a,b] is considered to be an elastic plate.

Figures 6.7–6.8 show the continuity condition (6.30) of the normal velocities along the

vertical elastic plate 0×[a,b]. It can be seen that the normal velocity along the elastic

plate is equal to the plate displacement as considered in (6.30).

Note that a very little fluctuations in the normal velocity field are observed which

exhibit no particular change in the power propagation curve. In Figures 6.5-6.6, the

power components are compared. Note that the Figures 6.3-6.8 are ÂIΨjM

FIGURE 6.3. The real component of pressures versus y at the matching interface.

ÁIΨjM

FIGURE 6.4. The imaginary component of pressures versus y at the matching interface.

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.080

0.090

0.095

0.100

0.105

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.08

0.07

0.06

0.05

0.04

138 CHAPTER 6

plotted for N = 180 and frequency 700Hz. By considering the structure-borne mode

incidence (` = 0) and the fluid-borne mode incidence(` = 1) the scattered powers are

plotted against frequency(Hz); see Figures 6.9-6.10.

These graphs show the power propagation curves against frequency for rigid and

plate vertical strip. For chosen identical duct configurations and clamped edges the

maximum of the incident power goes on reflection while the structureborn mode

incident is taken in account (see Figure 6.9). However at the precise value of frequency

i. e., f = 191Hz the reflected and transmitted powers level ÂIΨjxM

FIGURE 6.5. The real component of normal velocities versus y at the matching interface.

ÁIΨxjM

FIGURE 6.6. The imaginary component of normal velocities versus y at the matching interface.

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.0

0.2

0.3

0.4

0.5

j=1

j=2

0.2 0.4 0.6 0.8 1.0 y

0.0

0.2

0.3

0.4

0.5

0.6

0.7

6.3. NUMERICAL RESULTS AND DISCUSSION 139

each other. This is indeed be the point where the right hand duct section start

propagating. But once frequency crosses this point the duct goes again in totally

reflected mode. But the inverse behavior of power propagation is noted in Figure 6.10

where the fluid-born mode is incident. Clearly 80% of the power goes on reflection as

the secondary mode cuts-on that is at f = 191Hz. It sharply decreases on increasing

frequency and after at f ≥ 300Hz the duct behave in maximum propagating mode. Also

it is observed that the rigid/plate vertical problem govern the same power balance

curves where the power balance iden-

FIGURE 6.7. The real part of W(y) (dashed line) and ψ2x (0,y) (solid line).

FIGURE 6.8. The imaginary part of W(y) (dashed line) and ψ2x(0,y) (solid line).

0.85 0.90 0.95 1.00 1.05

0.0015

0.0010

0.0005

0.0005

0.0010

0.0015

0.0020

0.85 0.90 0.95 1.00 1.05

0.002

0.001

0.001

140 CHAPTER 6

tity (6.48) remains identically satisfied. This behaviour is comparable to the previous

study by Nawaz and Lawrie [67] where they chosen soft/backed flange vertical strip.

FIGURE 6.9. The comparison of the power components versus frequency (Hz) for the fundamental mode incidence (which cuts-on at 191Hz). The dotted lines show the results of Galerkin approach.

FIGURE 6.10. The comparison of the power components versus frequency (Hz) for the secondary mode incidence (which cuts-on at 191Hz). The dotted lines show the results of Galerkin approach.

P-ref

P-trans

PB

0 100 200 300 400 500 600 700 Frequency

0.2

0.4

0.6

0.8

1.0

Power

P-ref

P-trans

PB

200 300 400 500 600 700 800 900 Frequency

0.2

0.4

0.6

0.8

1.0

Power

142 CHAPTER 7

7 CONCLUSION

The model problems discussed in this thesis demonstrate how a particular class of

boundary problem related to acoustic scattering at step discontinuity in duct or

channel may be solved using different semi-analytic techniques. Each one of the

approaches has its own limitations and strengths. The mode-matching (MM)

technique together with an appropriate form of the orthogonality relation (OR) is

viable tool to solve many interesting physical problems. It incorporates both the

structural discontinuity and the singularities at the corners or edges. The use of

appropriate OR enables to express the edges in terms of algebraic equations that can

be found in a straightforward way after truncation. Practically, such systems

convergence very rapid. Refer for instance to the investigation by Abrahams and

Wickham [91] which substantiates that these systems are in fact l2. On contrary, the

case of hard/hard channel leads to a poorly convergent solution due to presence of

stronger singularity at the edge; refer, for instance, to Warren at el. [35]. The leading

term in this case is canceled out during MM process thereby affecting the convergence

rate. Indeed, the convergence is inversely proportional to the strength of the corner

singularity of the fluid velocity potential [39]. However, it is not the case with the

flexible panel/duct. The convergence of two specific examples are studied through the

exact formulation of (An,Bn) by Wiener-Hopf technique by Lawrie and Abraham [51]

and later on a detailed analysis is performed by Lawrie [59]. These studies establish

the point-wise convergence for the case of flexible panels. Furthermore, it is exactly

reflected in modeled problems with rigid vertical strip wherein the moderate values

of truncation parameter N lead to highly convergent results even for extreme behavior

of infinite systems. On the other hand, the problem with flexible vertical strip includes

the contributions of Dirac masses to impose the extra conditions at the edges of

vertical flexible wall. This involvement of Dirac masses leads to a poorly convergent

solution even at the moderate level of truncation. In order to avoid Dirac masses, the

vertical flexible strip is defined explicitly and the solution is obtained via Galerkin

approach blended with MM technique as

121

in [69]. It works well to incorporate the problems with flexible vertical strip. The

chapter wise observation, concluding remarks and future work predictions are given

below:

Chapter 2 comprises the introductory details of the class of problems included in

the thesis. A brief overview of the MM techniques is provided from the perspective of

generalized ORs.

In Chapter 3 the model problem describing the scattering of fluid -structural

coupled wave at a flanged junction and step discontinuity is discussed. The problem

resembles very closely to a dissipative device wherein a sound absorbing material

containing lines are held by means of a welded or pivoted flange. The envisaged

applications of the problem are in HVAC [67, 92]. The canonical problem is solved for

two sets of edge conditions by using MM and low frequency (LF) approaches and a

range of numerical investigations has been provided. The reflection and transmission

effects of structure-borne and fluid-borne energy fluxes have been depicted. It is

shown that the structure-borne fundamental mode incidence in the presence of flange

and zero displacement condition alleged the LF solution only in low frequency regime,

that is, f < 450Hz (see Case-I and Case-II), however for zero gradient condition it has

given surprisingly an accurate solution as compared with MM solution. It is also

concluded that the fundamental mode and secondary mode incidences optimized the

power transmission. The LFA solution fails to reconstruct the matching conditions at

interface. Therefore, it should not be used for the case where the actual information

of pressure or velocity is required. However, since the LFA solution works well for

structure-borne fundamental modes only, it can be used as an initial approximation of

the solution. The MM technique not only provides a benchmark solution to the

problem but also highlights many interesting scattering features. A comparative study

of both the solutions is provided. It is worthwhile precising, that the MM solution has

some limitations while incorporating the singular behavior of the velocity potential at

the tip of the flange as y →d. The issue has been addressed in recent study by Nawaz

and Lawrie [67], wherein the authors modeled the velocity potential at the tip of the

flange by using the multi-term Galerkin approximation involving Gegenbauer

polynomials [92, 93, 94]. However, the MM solution copes well with the discontinuity

in the normal velocity. Small oscillations are observed in the normal velocity that can

be explained by Gibbs phenomenon and can be removed by Lanczos filters thereby

rendering useful information. The method used here is applicable for the case of

multiple discontinuities in different waveguide structures where the number of

resulting equations by the MM method may be large. This method lends itself to a

variety of extensions to the theory such as an elastic plate can be used in place of

membrane together with diverse sets of edge conditions (for example, clamped, pin-

jointed and pivoted) whichever is appropriate. 123

Chapter 4 deals with the propagation and scattering of acoustic wave in a

rectangular waveguide involving a step discontinuity and change in materiel

properties. A membrane attached with the mouth of an expansion chamber can

effectively reduce the transmission of low-frequency noise in ducting system; refer,

for instance, to [6, 7, 8, 9]. The investigation is motivated by the applications of these

144 CHAPTER 7

structures as a component of a modified silencer for heating ventilation and air-

conditioning (HVAC) ducting systems. The associated boundary value problem is

solved by the MM technique. The discussion based on numerical results and physical

aspects of elastic plate and membrane bounded ducts has been presented in detail

whereas the dimensions of the parameters were consistent with that of a typical HVAC

duct [7]. It is observed that in case of fundamental mode incidence the use of pin-

jointed edge conditions contributed in minimizing the power transmission as

compared to the clamped edge conditions. However, for secondary mode incidence

the rates of power distribution in duct sections are increased. It is worthwhile

mentioning that the conservation of power and matching interface conditions

guarantee the validity of the solution.

Note that the power distribution for only fundamental mode/plane wave

incidence for membrane/rigid bounded duct are already available, for example in,

[35]. However the current study focuses on the elastic plate bounded inlet duct with

two different incidence modes, that is, fundamental mode incidence and secondary

mode incidence. Albeit, the wall conditions and the physical edge conditions are

generally different yet the power distribution behavior for the fundamental mode

incidence is consistent with [35], whereas the power distribution behavior for

secondary mode incidence is consistent with the results presented in [67]. Therefore,

the results are in good agreement with already available results in [13, 35, 67].

In Chapter 5, two variations of a model problem involving the scattering of fluid-

structure coupled waves in a waveguide with abrupt change in height have been

considered. In the first case, consistent with previous results [35, 95, 90], the truncated

modal solution preserves the power balance. In the second case the truncated modal

solution does not preserve the power balance at least for small N. However, the energy

conservation law holds for N →∞. Recall that, even though the Dirac mass can be

represented in terms of the eigenfunctions γn for n = 0,1,2,

··· , [35], the series

representation is divergent. Consequently

ψ2Nxyy(0,y)+µ21ψ2Nx(0,y)−α1ψ2N(0,y)

=6 2E3δ(y −a)+2E4δ(y −b), y ∈(a,b),

and, therefore the error term IN in the power balance is non-zero. However, when N

→∞, (5.34) holds and IN → 0. It follows that, for this problem the accuracy of power

balance is reliable indicator for the accuracy of the modal solution. Thus, the effect of

the vertical membrane is not confined to the convergence of the power balance. The

reflected and transmitted components of power vary significantly for the rigid and

membrane strip and at the moderate level of truncation the solution is poorly

convergent.

Two solution techniques are used as an alternative in the aforementioned

scenario. The first one is the standard Galerkin method wherein the displacement is

represented as a generalized Fourier sine series. The the second approach (referred to

as the modal approach) makes use of the properties of the duct modes to construct a

Modal representation for the displacement of the vertical membrane. The Galerkin

approach, although conceptually simpler, requires a different set of basis functions for

each set of edge conditions applied to the vertical membrane. Further, for more

complicated conditions the eigenvalues cannot be expressed explicitly and must be

found numerically. In contrast, the Modal approach can deal with a range of edge

conditions without modification. However, a major draw back of the Modal approach

is that requirement of zero membrane displacement condition at y = a. Precisely, the

technique can only be used for a set of edge conditions in which the vertical membrane

displacement is zero at y = a, indeed, since the function ψ(3)(y) exists only for a ≤ y ≤b.

The Model approach is useful for the problems in which it is expected that the

functions equivalent to ψ(j)(y) for j = 1,··· ,4 will have better convergence although there

are still some restriction on the edge conditions that can be exploited.

In Chapter 6, the acoustic scattering of fluid-structure coupled waves in elastic

plate bounded duct with different material properties and step discontinuity is

discussed. The problem with acoustically rigid vertical strip is solved again using the

MM technique whereas the solution of the problem with elastic plate vertical strip is

obtained by means of Galerkin approach. In order to solve the problem involving

vertical elastic plate with MM technique the main difficulty is the involvement of the

Dirac masses and their derivatives to impose the edge conditions at vertical plate

edges. Thus, vertical plate defined by ψ2xyyyy

=2δ(y −a)e1 +2δ(y −b)e2 +2δ0(y −a)e3 y ∈(a,b),

leads to a poorly convergent MM solution at moderate level of truncation due to

strong singularities at the corners of the vertical elastic plate. However, alternative

Galerkin approach not only avoids the use of Dirac mass but also provides an excellent

scope to impose different edge conditions. The problem is solved for the clamped

connection of edge conditions that describes both the displacement and gradient to

be zero at each edge. Thus a minimum of power is transported in vertical elastic plate

and is comparable with rigid vertical strip. For both the fundamental and secondary

mode incidences, the results obtain via rigid and elastic plate are consistent with those

in [67].

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