FUNDAMENTALS OF DUCT ACOUSTICS

P. JOSEPH

FLUID DYNAMICS AND ACOUSTIC GROUP

WHY STUDY THE ACOUSTICS OF DUCTS

Ducts, also known as waveguides, are able to efficiently transmit sound over large distances. Some common examples:

• Ventilation ducts

• Exhaust ducts

• Automotive silencers

• Shallow water channels and surface ducts in deep water

• Turbofan engine ducts

WAVE EQUATION

The acoustic pressure p(x,y,z,t) in a source-free region of space

in which there is a uniform mean flow Ux in the x – direction

satisfies the convected wave equation:

DDt

c p2

22 2 0− ∇

⎡

⎣⎢

⎤

⎦⎥ = Dp

Dtpt

U pxx= +

∂∂

∂∂

where c is the speed of sound.

For simplicity, and without loss of generality, we shall only

consider solutions to the wave equation in the absence of

flow, Ux = 0.

SEPARABLE SOLUTIONS TO THE WAVE EQUATION IN CARTESIAN AND CYLINDRICAL COORDINATES

Cylindrical duct Rectangular duct

x

y

z

Lz

Ly

xθra

∇ = + ⎛⎝⎜

⎞⎠⎟ +

22

2 2

2

21 1∂

∂∂∂

∂∂

∂∂θx r r

rr r

∇ = + +22

2

2

2

2

2∂∂

∂∂

∂∂x y z

Assume harmonic (single-frequency) separable solutions of the form

( ) ( ) ( )θωα HrGAep tkxi −−= ( ) ( ) ( )zZyYAep tkxi ωα −−=

b

SEPARABLE SOLUTIONS TO THE WAVE EQUATION IN CARTESIAN AND CYLINDRICAL COORDINATES

(Continued)

Substituting into the wave equation and separating variables:

( )∂∂

∂∂

α2

22 2

2

21 1 0G

r rGr

k mr

G+ + − −⎧⎨⎩

⎫⎬⎭

=

∂∂ θ

2

22 0H m H+ =

022

2

=+ YkyY

y∂∂

∂∂

2

22 0Z

zk Zz+ =

Cylindrical duct Rectangular duct

Bessel’s equation of order m

GENERAL SOLUTIONS

General solutions to these second order equations are:

( )( )( )

( )( )222 1 ακ

θ

κ

κ

θ

−=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=

=

−

k

eH

rY

rJrG

im

m

m ( ) ( )( )

( ) ( )( )

( )Y y

k yk y

Z z k zk z

k k k

y

y

z

z

y z

=

=

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

+ = −

sincos

sincos

2 2 2 21 α

Transverse wavenumber is κ. Transverse wavenumber is22zy kk +

HARD-WALLED DUCT EIGENVALUE EQUATION

The component of particle velocity un normal to the hard-walled

duct vanishes:

∂∂Gr= 0 ∂

∂Zz= 0 ∂

∂Yy= 0

The solutions Ym(κr), sin(kzz) and sin(kyz) cannot satisfy these boundary conditions. Furthermore, only particular discrete values of transverse wavenumbers (eigenvalues) satisfy the boundary conditions given by.

( ) 0=′ bJ mnm κ ( )sin k Lny y = 0 ( )sin k Lnz z = 0

(prime denotes derivative of Bessel function)

MODAL EIGENVALUES

bjmnmn /′=κ k n Lny y y= π / k n Lnz z z= π /

Bessel functions of order m = 0, 1 and 2

J x0( )

J x1( )J x2( )

x

m/n 1 2 3 4 5 6 70 0 3.8317 7.0156 10.1735 13.3237 16.4706 19.61591 1.8412 5.3314 8.5363 11.7060 14.8636 18.0155 21.16442 3.0542 6.7061 9.9695 13.1704 16.3475 19.5129 22.67163 4.2012 8.0152 11.345 14.5858 17.7887 20.9725 24.14494 5.3176 9.2824 12.681 15.9641 19.1960 22.4010 25.58985 6.4156 10.5199 13.987 17.3128 20.5755 23.8036 27.01036 7.5013 11.734 15.268 18.6374 21.9317 25.1839 28.4098

Stationary values of the Bessel function

CUT OFF FREQUENCY

Earlier we saw that the transverse and axial wavenumbers of a

single mode are connected by the dispersion relationships

( )2/1 kbjmnmn ′−=α ( ) ( )α π πnynz y y z zn kL n kL= − +⎡⎣⎢

⎤⎦⎥

12 2

/ /

These expressions make explicit the existence of threshold

frequencies at frequencies below which α is purely

imaginary and the mode decays exponentially along the duct. The

mode is said to cut off, or evanescent.

ωmn mnck=

Cylindrical duct Rectangular duct

CUT OFF FREQUENCY

This cut-off frequency follows from the above as

bjc mnmn /′=ω ( ) ( )ω πnynz y y z zc n L n L= +⎡⎣⎢

⎤⎦⎥

−/ /

/2 2 1 2

In terms of cut-off frequency:

( )α ω ωmn mn= −12

/ ( )α ω ωnynz nynz= −12

/

Cylindrical duct Rectangular duct

MODE COUNT FORMULAE

At a single frequency only a finite number of modes N(kb) and

N(kLx) are cut on and able to propagate along the duct without

attenuation. The rest decay exponentially along the duct. In the

high frequency limit, ka →∞:

( ) ( )221

21 kbkbkbN +→ ( ) ( ) ( )21

xxx kLRkLRkLNππ

++

→

R L Ly x= ≥1where

Cylindrical duct Rectangular duct

MODE COUNT FORMULAE

A comparison of this mode-count formula for circular ducts with

the exact count (histogram) is presented below.N

umbe

r of p

ropa

gatin

g m

odes

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Non-dimensional frequency kb

180

160

140

120

100

60

40

0

20

80

( ) ( )221

21 kbkbkbN +→

MODES AND MODE SHAPE FUNCTIONS

In seeking a solution for the pressure field in a duct we obtained, not a single unique solution, but a family of solutions. The generalsolution is a linear superposition of these ‘eigenfunction’ solutions:

( ) ( )∑∑∞

=

−∞

−∞=

Φ=1

,,,ˆn

kximnmn

m

mnerAxrp αθθ ( ) ( )∑∑∞

=

−∞

=

Φ=11

,,,ˆnz

kxinxnynxny

nyezyAxzyp α

( ) ( ) θκθ immnmmn erJr −=Φ , ( ) ( ) ( )Φnynz ny nzy z k y k z, cos cos=

The resultant acoustic pressure in the duct is the weighted sum of fixed pressure patterns across the duct cross section. Each of which propagate axially along the duct at their characteristic axial phase speeds.

Cylindrical duct Rectangular duct

MODE SHAPE FUNCTIONS Φ

Cylindrical Duct Mode Shape Functions

+

++

+

+

+

+

+

+ +

+

+

+

+

+

+

+ +

+

++

-

-

-

-

-

-

-

-

- -

-

-

-

--

-

- -

--

m

n( ) =0ka 01

( ) =1.8412ka 11

( ) =3.0542ka 21 ( ) =6.0706ka 22 ( ) =9.9695ka 23

( ) =5.3314ka 12 ( ) =8.5363ka 13

( ) =3.8317ka 02 ( ) =7.0156ka 03

1 2 3

0

1

2

m

n

1

2

3

0 1 2

nb. ka used here. Should be ka.

MODE SHAPE FUNCTIONS Φ

Rectangular Duct Mode Shape Functions

n ny z= =0 0, n ny z= =0 1, n ny z= =1 1, n ny z= =1 2,

MODAL CUT - OFF RATIO

Earlier we saw that each mode the axial wavenumber is αmnk and the wavenumber in the direction of propagation is k (=ω/c)

By simple geometry, αmn equals the cosine of the angle φxbetween the local modal wavefront and the duct axis. It may therefore be interpreted as a measure of how much the mode is cut on.

A more common index of ‘cut on’ is specified by the cut-off ratio ζ defined by mnmn k κζ =

k

αmnk

κmn

φx

CATEGORIES OF MODAL BEHAVIOUR

ζmn < 1. Mode is cut-off and decays exponentially along the duct. Pressure and particle velocity are in quadrature and zero power is transmitted.

ζmn = 1. Mode is just cut on (or cut-off) and propagates with infinite phase speed (and zero group velocity). No modal power is transmitted.

ζmn > 1. Mode is cut on and propagates at an angle to the duct axis. Transmitted modal sound power . Axial phase speed greater than c and group velocity less than c.

( )mnζ/1cos 1−

CATEGORIES OF MODAL BEHAVIOUR

ζmn >> 1.

ζmn > 1.

ζmn = 1.

AXIAL PHASE SPEED

Each mode propagates axially along the duct as . The axial modal phase speed cmn is given by

pmni kxmn∝ −e α

mnmn

mn ck

c ααω /==

( )ccmn

mn

=−

1

12

ω ω/

ω ωmn/

c mn/c

The axial phase is infinite at the cut-off frequency, tending to c as the frequency approaches infinity.

CIRCUMFERENTIAL PHASE SPEED IN A CIRCULAR DUCT

At a fixed position along the duct, points at the wall of constant phase are given by . The circumferential phase speed at the wall is therefore

( ) θωθη mtt −=,

ma

tac mn

ω∂∂θ

θ ==

A property of Bessel functions Jm for large m is that . Combining this result with the cut on condition gives

k a mm1 ≥ω / c km> 1

ωamc

> 1 and hence cc mn >θ

For propagation, the circumferential (as well as axial) modal phase speed modes must be supersonic.

MODAL RADIATION FROM HARD WALLED CIRCULAR DUCTS

The acoustic pressure in the far field of a semi-infinite circular hard walled duct may be expressed as

( )tRp ,,, φθ

( ) ( ) 1;e,,,,, >>=+−−

kRR

akDAtRptiikRim

mnmnmn

ωθ

φφθ

MODAL RADIATION FROM HARD WALLED CIRCULAR DUCTS

Some representative directivity plots (in decibels) for the (m,n)=(40,3) mode at three frequencies is presented below

ka = 60, =20ζ ka = 137, =1.1ζka = 95, =2ζ

( )=(40,3)m,n

MODAL RADIATION FROM HARD WALLED CIRCULAR DUCTS:SIMPLE RULES

• The angle φx of the principal radiation lobe equals which is identical to the axial propagation angle within the duct.

• Modal radiation becomes progressively weaker as the frequency approach cut-off from above tending to zero exactly at cut-off.

• No major or minor lobes occur in the rear arc.

• Zeros (or nulls) in the radiation pattern occur at angles . Angles of the minor lobes occur roughly mid-way between the angles of the zeros. The number of zeros and minor lobes increase roughly as the frequency squared.

• Symmetrical angles exist θs beyond which modal radiation is extremely weak. These are referred to as shadow zones (or cones of silence) and occur at

( )mnx ζφ /1cos 1−=

= ′ ≠−cos ,1 j j nmj

kams1sin−=φ

LINERS FOR THE ATENUATION OF DUCT BORNE NOISE

Reactive and dissipative attenuators

Sound transmission in ducts may be attenuated, either by reflection by means of the introduction of impedance discontinuities (REACTIVE ATTENUATORS), or by the conversion of sound energy into heat (RESISTIVE OR DISSIPATIVE LINERS).

Many duct attenuators combine both attenuation mechanisms. Reactive attenuators are most effective at low frequencies (less than 100Hz).

REACTIVE AND DISSIAPTIVE ATTENUATORS

Incidentwave

Incidentwave

Strongreflection

weakreflection

weaktransmission

weaktransmission

REACTIVE SILENCER

DISSIPATIVE SILENCER

DISSIPATIVE ATTENUATORS MAY BE RESONANT

OT NON-RESONANT

LOCAL AND EXTENDED REACTION

Liners are either locally reactive, where the wave motion is confined to the normal to the surface, or bulk reacting, where the wave are free to propagate in the liner.

LOCALLY REACTING LINER

BULK REACTING LINER