aeem-728 introduction to ultrasonics lecture notes

8/9/2019 AEEM-728 Introduction to Ultrasonics Lecture Notes

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INTRODUCTIONTO ULTRASONICS

20-251-728

Peter B. Nagy, 2001


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Part 1 Introduction

1.1 What is ultrasonics?

Ultrasonics is a branch of acoustics dealing with the generation and use of (generally)

inaudible acoustic waves. There are two broad areas of use, sometimes called as the low- and

high-intensity applications. In low-intensity applications, the intent is to convey information

about or through a system, while in high-intensity applications, the intent is to permanently alter

a system. To some extent, the low- and high-intensity fields are also delineated by a frequency

range and power level. Thus, low-intensity applications typically involve frequencies on the order

of 106 Hz or higher and power levels on the order of milliwatts. High-intensity applications will

typically involve frequencies of 5 to 100 kHz and powers of hundreds to thousands of watts. In

actual fact, the total frequency range of all ultrasonic applications is enormous, ranging from 5 -

10 kHz to as high as 10 GHz. There are also applications, such as sonar, which are exceptions to

the previous categorizations, since intense power levels are involved in conveying information

via underwater sound.

Ultrasonic materials characterization is the most important application of ultrasonics in

aerospace engineering and engineering mechanics. Historically, ultrasonic nondestructive testing

(NDT) has been used almost exclusively for detecting macroscopic discontinuities in structuresafter they have been in service for some time. It has become increasingly evident that it is

practical and cost effective to expand the role of ultrasonic NDT testing to include all aspects of

materials production and application. Research efforts are being directed at developing and

perfecting NDT capable of monitoring (i) material production processes, (ii) material integrity

following transport, storage and fabrication, and (iii) the amount and rate of degradation during

service. In addition, efforts are underway to develop techniques capable of quantitative

discontinuity sizing, permitting determination of material response using fracture mechanics

analysis, as well as techniques for quantitative materials characterization to replace the

qualitative techniques used in the past. Ultrasonic techniques play a prominent role in these

developments because they afford useful and versatile methods for evaluating microstructures,

associated mechanical properties, as well as detecting microscopic and macroscopic

discontinuities in solid materials.

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The main difference between the basic methods of Ultrasonics and the more specialized

ones used in Ultrasonic NDE is in the approach to the elastic medium. In ultrasonics, the material

is usually assumed to be ideal (isotropic, homogeneous, linear, attenuation-free, dispersion-free,

temperature-independent, etc.) in order to study the basic laws of elastic wave propagation in

their simplest form. In ultrasonic NDE, real materials with more complex elastic properties

(anisotropy, inhomogeneity, nonlinearity, attenuation, dispersion, temperature-dependence, etc.)

are considered. The primary purpose of ultrasonic NDE is to understand the wave-material

interaction and assess the sought material properties from the observed deviation in the ultrasonic

response from that of an ideal, defect-free medium. The main topics to be covered in Ultrasonics

and Ultrasonic NDE are listed in the following table.

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Ultrasonics(high-frequency wavepropagation in elasticmedia)

Ultrasonic NDE(the propagation medium is animperfect medium, i. e., a realmaterial)

Wave-Material Interaction(special physical phenomena dueto interaction withimperfections)

isotropic anisotropic

texture columnar grains prior-austenite grains composites

anisotropy (orientation)

birefringence (polarization)quasi-modes (three waves)phase and group directionsresidual stress effect

homogeneous inhomogeneous polycrystalline two-phase porous composite

incoherent scattering noiseattenuationdispersion (weak)

linear nonlinear intrinsic (plastics)

damage (fatigue)

harmonic generationacousto-elasticity

crack-closure

attenuation-free attenuative air, water, viscous couplants polymers coarse grains porosity

absorption viscosity, relaxation heat conduction, scattering elastic inhomogeneity geometrical irregularity

dispersion free dispersive intrinsic (polymers) geometrical (wave guides)

relaxationresonancewave and group velocitypulse distortion

temperature-independent temperature-dependent nonlinearity residual stress (composites) phase transformation (metals) moisture content (polymers)

velocity changethermal expansion

no defects defects cracks, voids misbonds, delaminations

reflection, diffractionattenuation, velocity changescattering, nonlinearity

ideal boundaries

flat, smooth rigidly bonded interface

imperfect boundaries

curved, rough slip, kissing, partial, interphase

mode conversion

refraction, diffractionscattering

canonical wave types plane wave spherical waves harmonic

complex wave types apodization (amplitude) focusing (phase) impulse, tone-burst

beam spreaddiffraction lossedge wavesspectral distortion

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Elements of ultrasonic waves

In ultrasonics, one is interested in acoustic waves, either propagating or standing, in

solids, liquids and gases. It is of use, at the outset, to note the elementary characteristics of

waves, with more detailed analysis to follow. Recall the main features of a simple harmonic

wave, shown in Figure 1.1.

u x t A t x

c( , ) cos[ ( ) ],= − +ω ϕ (1.1)

where A denotes the amplitude, ω = 2π f is the angular frequency, where f is the cyclic

frequency, ϕ is the phase angle at x t = = 0, and c denotes the propagation (phase) velocity.

Here, u could represent longitudinal or transverse displacement of a string, particle velocity in a

solid, pressure wave amplitude in a gas, or a number other physical quantities. The basic wave

parameters of propagation, the wavelength λ and the period of vibration T are related through

c T f / / λ = =1 .

A

-A

x

λ

t = t 1

t 2

t 3

cu

Figure 1.1 Simple harmonic wave.

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In order to better facilitate algebraic manipulations needed to solve ultrasonic wave

propagation problems, we will use complex notation without explicitly indicating as such.

Equation (1.1) can be written as follows

u x t U e i k x t ( , ) ( )= ± − ω , (1.2)

where U is a complex amplitude that includes the phase term, and k is the so-called wave

number. In this notation, only the real part of the complex quantity corresponds to the actual

physical quantity, therefore the + and - sign conventions are equivalent.

It will be found that three basic types of wave may exist in a material, depending on

whether it is solid or fluid and depending on the nature of its boundaries. The wave types are

dilatational, shear , and surface waves. Propagation velocities will depend on the material, and

may range from 102

m/s to 104

m/s. The basic natures of the waves are shown in Figure 1.2.The dilatational wave (also called longitudinal or pressure wave) may exist in solids, liquids, and

gases, and is the familiar wave of acoustic theory. It is seen that particle motion is in the same

direction as the propagation direction. A shear wave (also called transverse or equivoluminal

wave), on the other hand, may exist only in a solid. It is seen that particle motion is at right

angles, or transverse, to the direction of propagation. These are the only two types of wave that

may exist in an extended media. If a free surface exists on a solid half-space, a surface (or

Rayleigh) wave may also propagate. Such a wave has a complicated particle motion at the

surface, and has an amplitude that rapidly decays away from the surface. A main point to

emphasize is that these waves are all well known from classical acoustic and elasticity theory. No

"mysterious" new waves are associated with ultrasonics.

Finally, the behavior of waves upon encountering surfaces and boundaries is another

fundamental aspect of wave propagation. The simplest situation is depicted in Figure 1.3a, where

a wave encounters a boundary at right angle or normal incidence. The interaction only involves

reflection of some of the wave and transmission of a portion, with the amount of energy in each

part depending on the material characteristics. A more complicated situation may arise,

particularly in solids, when the wave strikes at an angle, or at oblique incidence. What mayoccur, as shown in Fig. 1.3b, is that two types of waves are reflected for a single incident wave.

This phenomenon is known as mode conversion, and is illustrated for the case of a pressure wave

generating both pressure and shear waves. Yet another aspect is involved when waves encounter

edges. Complex scattering and diffraction of the waves may occur, similar to optics. This is

meant to be illustrated by Figure 1.3c.

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Longitudinal Wave:

Shear Wave:

Surface Wave:

Figure 1.2 Different wave modes in a solid material.

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ρ , c1 1

ρ , c2 2

Incident Wave Reflection

Transmission

a)

Liquid

Solid


θi θr

ShearTransmission

LongitudinalTransmission

θs

θd

b)


θi θr

c)

Edge Diffraction

Figure 1.3 Different types of acoustic wave interaction with material discontinuities:

reflection and transmission (a), refraction and mode conversion (b), and

diffraction and scattering (c).

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Historical aspects

Since ultrasonics is a part of acoustics, its development, particularly in the early years, is

to some extent embedded in the broad developments in acoustics. However, the history of

classical acoustics can be traced back to Pythagoras in the 6th century B. C., while investigations

of high-frequency waves did not originate until the 19th century. The era of modern ultrasonics

started about 1917, with Langevin's use of high-frequency acoustic waves and quartz resonators

for submarine detection. Since that time, the field has grown enormously, with applications

found in science, industry, medicine and other areas. The following is meant to identify dates of

some of the major developments in ultrasonics.

DATES OF SOME MAJOR DEVELOPMENTS IN ULTRASONICS

1820 Wollaston made early observations of pitch audibility limits.

1830 Savart developed large, toothed wheel to generate very high frequencies.

1842 Magnetostrictive effect discovered by Joule.

1845 Stokes investigated effect of viscosity on attenuation.

1860 Tyndall developed the sensitive flame to detect high frequency waves.

1866 Kundt used dust figures in a tube to measure sound velocity.

1868 Kirchhoff investigated effect of heat conduction on attenuation.

1876 Galton invented the ultrasonic whistle.

1877 Rayleigh's "Theory of Sound" laid foundation for modern acoustics.

1880 Curie brothers discovered the piezoelectric effect.

1890 Koenig, studying audibility limits, produced vibrations up to 90,000 Hz.1903 Lebedev and coworkers developed complete ultrasonic system to study

absorption of waves.

1912 Sinking of Titanic led to proposals on use of acoustic waves to detect

icebergs.

1915 Langevin originated modern science of ultrasonics through work on

submarine detection.

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1921 Cady discovered the quartz stabilized oscillator.

1922 Hartmann developed the air-jet ultrasonic generator.

1925 Pierce developed the ultrasonic interferometer.

1927 Wood and Loomis described effects of intense ultrasound.

1928 Pierce developed the magnetostrictive transducer.

1928 Herzfeld and Rice developed molecular theory for dispersion and

absorption of sound in gases.

1929 Sokolov proposed use of ultrasound for flaw detection.

1930 Debye and Sears and Lucas and Biquard discover diffraction of light by

ultrasound.

1937 Sokolov invented an ultrasonic image tube.

1937 Dussik brothers made first attempt at medical imaging with ultrasound.

1938 Pierce and Griffin detect the ultrasonic cries of bats.1939 Pohlman investigated the therapeutic uses of ultrasonics.

1940 Firestone, in the United States and Sproule, in Britain, discovered

ultrasonic pulse-echo NDT.

1940 Sonar extensively developed and used to detect submarines.

1945 Piezoelectric ceramics discovered.

1948 Start of extensive development of power ultrasonic processes.

1948 Start of extensive study of ultrasonic medical imaging in the United States.

For more details on this subject, see Karl F. Graff, A History of Ultrasonics, in Physical

Acoustics, Volume XV (Academic Press, New York, 1982).

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1.2 Vibrations of a simple oscillator

The vibrational characteristics of the simple oscillator will be reviewed first. This will

provide the opportunity to emphasize certain vibrational characteristics of special interest in

ultrasonics. The simple wave equation will then be reviewed. This basic equation will be found

to be applicable to a wide range of ultrasonic wave propagation problems.

A simple, undamped mechanical resonator of mass m and spring constant k is shown in

Figure 1.4.

k m

Figure 1.4 Simple undamped mechanical resonator of a spring and a mass.

The equation of motion for the oscillator is

mu k u + = 0. (1.3)

where u denotes the displacement from the equilibrium position. Putting this in the form

,u uo ok

m+ = =2

0ω ω , (1.4)

leads to the solution

u B t C t o o= +cos sinω ω , (1.5)

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where ωo is the natural (angular) frequency of the free vibration of an undamped mechanical

resonator. This result may also be expressed in terms of an amplitude and a phase angle as

u D t o= −cos( )ω φ , (1.6)

where

D B C C

B= + =2 2 , tanand φ . (1.7)

If a dashpot is added to the oscillator, as shown in Fig. 1.5, the equation of motion is

simply

mu d u k u + + = 0 . (1.8)

This may be put in the form

uQ

u uo o+ + =ω

ω2 0 , (1.9)

where o k m2ω = / as before and the so-called quality factor Q is defined here as an impedance

ratio at the resonance frequency

k m

d

Figure 1.5 Simple damped mechanical resonator of a spring, a mass and a dashpot.

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Q X

R

m

d

k

d

mk

d

reactive

dissipative

o

o

= = = =ω

ω . (1.10)

The solution to Eq. 1.9 may take several forms depending on the value of Q. The so-called

“damped periodic” case (Q ≥ ½) is the most applicable to ultrasonics and to most vibration

situations. The solution, in exponential form, is

u e A e A et i t i t = +− −ζ ( )1 2Ω Ω , (1.11)

where

Ω = − =ω ζ ω

oo

Q Q1

1

422

and . (1.12)

Ω is the natural (angular) frequency of the free vibration of a damped mechanical resonator and

ζ is a decay constant. In terms of sine and cosine functions, the result is

u e B t C t D e t t t = + = −− −ζ ζ φ( cos sin ) cos( )Ω Ω Ω . (1.13)

The well-known pattern of free vibration of a damped oscillator is shown in Fig. 1.6. It may be

easily shown that the ratio between amplitudes exactly one period (T) apart is

e - t ζ

t

T

u

Figure 1.6 Exponentially decaying free vibration of a damped oscillator.

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u t

u t T e eT

Q( )

( )+= ≈ζ π / . (1.14)

This leads to an other more general definition of the quality factor through the “logarithmic

decrement,” ∆, given by

∆ =+

= =ln( )

( )

u t

u t T T

Qζ

π . (1.15)

Consider now the forced vibrations of simple oscillators. The case of an undamped

oscillator shown in Fig. 1.4 subjected to a harmonic forcing function exp (iω t ) results in the

governing equation

mu k u F eoi t + = ω (1.16)

or

u uF

meo

o i t + =2ω ω . (1.17)

The steady-state vibrational response to the harmonic forcing function is given by

u

F

k e

o

o

i t =

−12

2

ω

ω

ω . (1.18)

Noting that F o / k is simply the displacement of the spring-mass system under a static force F o,

the amplitude response may be written as

= −

u

st

o

δ ω

ω

1

12

2

. (1.19)

This response is shown in Fig. 1.7.

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From the result of Eq. 1.19 and Figure 1.7, it is seen that as ω → ωo , the amplitude

“blows up,” i.e., approaches infinity. This is the phenomenon of undamped resonance. When the

forced vibrations of a damped oscillator previously shown in Fig. 1.5 are considered, the

governing equation of motion becomes

0

1

2

3

4

0 0.5 1 1.5 2 2.5

ω / ωo

u

δst

Figure 1.7 Amplitude response of an undamped resonator.

uQ

u uF

me

oo

o i t + + =ω

ω ω2 . (1.20)

The steady-state response is given by

u

F

m

i Q

e

F

k

iQ

e

o

o o

i t

o

o o

i t =

− +

=

− +2 22

21ω ω

ω ωωω

ωω

ω ω . (1.21)

At the resonance frequency, the vibration amplitude δ δm st Q= and it is 90o off-phase with

the driving force. In other words, the peak amplitude at the resonance frequency is determined

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solely by the mechanical impedance of the dashpot δ ωm o oF d = / . After a bit of manipulation,

it is possible to express this in terms of a real amplitude and a phase angle as

u

Q

est

o o

i t =

−

+

−δ

ω

ω

ω

ω

ω φ1 2

22

2

2

2 21

/ ( ) (1.22)

and

φ

ω

ω

ω

ω

=

−

atanQ o

o

12

2

. (1.23)

The behavior of the amplitude and phase are shown in Fig. 1.8 for various degrees of

damping. It is quite evident that for decreasing values of the quality factor (increasing damping)

the sharpness of the resonance decreases. Put another way, for increasing damping, the

bandwidth of the oscillator increases. It is also seen that the change in phase angle is quite rapid

for small frequency changes at low damping, where the phase angle changes much slower for

heavy damping. A physical quantity of great interest in vibrations is the energy dissipation rate,

which offers another way to define the quality-factor of the system. The probably most basic

definition of this parameter is

QU

U

s

d

=2π

, (1.24)

where U s denotes the maximum stored energy and U d is the energy dissipated per each cycle

(this definition is essentially the same as Eq. 1.15). For the simple oscillator, the maximum

stored energy would be given by

s mo

U k m F

d = =

1

2 2

22

2δ , (1.25)

where δm is the maximum vibration amplitude at the resonance frequency as obtained from Eq.

1.22. The energy dissipated per cycle may be found by carrying out the integral

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d o

T o

oU F t u t d t

F

d = =Re{ ( )} Re{( )}

π

ω

2

, (1.26)

where we took advantage of the fact that the driving force and the velocity are in phase at the

resonance frequency. The quality factor can be expressed as

Qm

d

mk

d

o= =ω

, (1.27)

which is the same as or previous definition. This relates two different measures of system

damping.

The Q of a system may also be found from the so-called half-power (-3 dB) points of

the resonance curve. These correspond to the two points on the curve where the amplitude is

δm / 2 . Forgoing the somewhat lengthy manipulations, the result is

Q f

f f

o o≈−

=−

ω

ω ω1 2 1 2(1.28)

where f 1 and f 2 are the half-power frequencies. The quantity f 1 - f 2 = B is the system

bandwidth.

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1.3 Wave propagation

Ultrasonics involves the propagation of acoustic waves. Therefore, it is necessary to

understand the basic features of propagating waves and some of the mathematical equations

governing simple cases of wave propagation. The propagating harmonic disturbance is a good

place to start. Successive instants in the propagation of harmonic wave were shown in Fig. 1.1. A

simple harmonic propagating wave can be described as follows

u x t A t x

c A t k x( , ) cos[ ( )] cos( )= − = −ω ω , (1.29)

where k is the so-called wave number which is related to the reciprocal of wavelength

k = 2π / λ and is introduced mainly for convenience in writing wave propagation expressions.

The phase velocity c is meant to rigorously define the velocity of the wave as the speed with

which two successive points of constant phase move past a certain point. This serves to

distinguish it from other types of velocities associated with waves, such as the group velocity. A

propagating wave may be described by several equivalent expressions. Thus,

u A k x t A k x c t = − = −cos( ) cos[ ( )]ω

= =− − Ae Aei k x t i k x ct ( ) ( )ω . (1.30)

These all may be considered expressions for a rightward propagating harmonic wave. A leftwardwave would simply be given by a sign change, as u = A cos (kx + ω t ) .

The relationship between wave propagaton and standing wave vibrations in a system may

be shown by superimposing two oppositely propagating waves. Thus, using simple trigonometric

identities, it may be shown that

u A

k x t A

k x t = + + −2 2

cos( ) cos( )ω ω = A k x t cos( ) cos( )ω . (1.31)

This latter expression describes the vibrations of a system with well-defined nodes and antinodes,

as shown in Fig. 1.9 (node means a point, line or surface of a vibrating body that is free from

vibratory motion).

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u

x

ca)

u

x

c

f ( x - c t )

cos [ k ( x - c t ) ]

b) f ( x - c t ) f ( x - c [ t + dt ] )

Figure 1.10 Propagation of (a) an arbitrary pulse and (b) a harmonic wave packet.

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1.4 Longitudinal wave propagation in thin rods

We now will investigate the development of a very basic equation governing the

propagation of waves in many mechanical and electrical systems. In other words, any number of

systems could be used as the starting point for derivation of this equation. The case of a thin

elastic rod will be used here, as shown in Fig. 1.11.

dx x

u

a)

b)

dx

σ ∂σ

σ + dx∂ x

Figure 1.11 A thin rod (a) with coordinate x and displacement u of a section and (b)

the stress acting on a differential element of the rod.

The equation of motion of an element of the rod shown in Fig. 1.11b is

− + + =σ σ ∂σ

∂ρ

∂

∂ A

xdx A Adx

u

t ( )

2

2 , (1.35)

where A is the cross-sectional area and ρ is the mass density. This reduces to

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∂σ

∂ρ

∂

∂ x

u

t =

2

2 . (1.36)

For an elastic material, we know that

σ ε= E (1.37)

where ε is the strain in the material. This quantity is in turn defined by

ε ∂

∂=

u

x . (1.38)

(In elementary mechanics, the strain is defined by ε = ∆ / . In the present development,

∆ ∆u ≈ , ∆ x ≈ , leading to ε = ∆ ∆u x / . Taking this to differential form would be

ε = d u d x / . Since u = u (x, t), this becomes the partial derivative, ε ∂ ∂= u x / .)

Substitution of (1.38) in (1.37) and this in (1.36) gives the following equation for waves

in thin rod

E u

x

u

t

2

2

2

2

∂

∂ρ

∂

∂= , (1.39)

where E denotes Young's modulus. This may be put in the more usual form,

2

2 2

2

2

1∂

∂

∂

∂ ρ

u

x c

u

t c

E

oo= =, . (1.40)

This equation governs the one-dimensional propagation of longitudinal elastic waves in a

thin rod. It is usually termed simply the wave equation because it represents the situation for so

many problems in mechanical and electrical media. In mechanical systems for example, it covers

transverse waves in strings, torsional waves in rods, and one-dimensional compressional and

shear waves in extended media. Equally numerous situations may be defined in electromagnetics.

It may be shown that a propagating harmonic wave represents a solution of the wave

equation. Thus, substitution of

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u A k x ct = −sin[ ( )] (1.41)

into (1.40) leads to c = co. In other words, the propagation velocity of the wave must be co. It

may also be shown that the arbitrary pulse from f (x-co t ) also satisfies the wave equation

1.5 Vibrations of a finite-length rod

Instead of looking at the propagation of harmonic waves in, essentially, an infinite,

distributed system such as the thin rod of the previous section, consider the vibrations of a finite

system. Using the thin rod as an example, the governing equation would still be (1.40). Suppose

the rod to be undergoing free vibrations, so that

u x t u x t ( , ) ( ) sin( )= ω . (1.42)

Putting this in (1.40) gives

2

22 0

∂

∂

ωu

xk u k

co+ = =, . (1.43)

This has the following solution

u x C k x D k x( ) sin ( ) cos( )= + . (1.44)

To determine C and D a statement of the conditions (or boundary conditions) at the ends of the

rod must be made. Suppose both ends are completely free then we would have a free-free rod :

At x = 0, , the stresses are zero. Since σ ∂ ∂= E u x / , this is equivalent to

∂

∂

u

x= 0 at x = 0, . (1.45)

From (1.44), have

∂

∂

u

xk C k x D k x= −( cos sin ) . (1.46)

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Then, at xu

x= =0 0,

∂

∂ gives C = 0. At x

u

x= = ,

∂

∂0 gives

D k sin = 0 . (1.47)

This leads to

k n n = =π 1 2, ,... (1.48)

or

no

non c f

n cω

π= =

,

2(1.49)

These are the natural frequencies of the rod, and lead to the following vibrational modes

u x t Dn

x n( , ) cos( ) , ,...= =π

where 1 2 (1.50)

The first few modes are shown in Fig. 1.12. Note that the distributed system has an infinite

number of modes.

x

n =

n =

n =

u

1

2

3

Figure 1.12 First three modes of vibration of a free rod.

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1.6 Wave types in solids and fluids

Only a relatively few types of waves may exist in continuous infinite media. In an

extended fluid medium, only a pressure (or dilatational or longitudinal) wave may exist, while in

a solid, both pressure and shear (or transverse or equivoluminal) waves may exist. The existence

of a free surface on a solid brings in the possibility of a surface wave. The basic natures of these

waves have been previously shown in Fig. 1.2. Some additional details will now be noted.

Generally, the acoustic wave equation of Equation 1.40 can be written for any types of

polarization in the same form

2

2 2

2

2

1∂

∂

∂

∂

u

x c

u

t = , (1.51)

where u here is the general acoustic displacement in an arbitrary polarization direction and c

is the phase velocity that can be determined as follows

cstiffness

density= . (1.52)

These equations govern all types of one-dimensional wave propagation. Of course, for a

thin rod aligned with the x-direction (σ σ y z= = 0),

σ ε x x E = (1.53)

and the longitudinal wave velocity is given by the previously derived result (Eq. 40)

c E

rod =ρ

. (1.54)

In a thin rod, the material is free to move in the lateral direction according to the Poisson effect

as shown in Figure 1.13. In comparison, for a thin plate parallel to the x-y plane (ε σ y z= = 0),

the stiffness in the x-direction is increased by the Poisson effect

σ ν

ε x x E

=−1 2

, (1.55)

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where ν is Poisson's ratio, and the longitudinal velocity

c E c

c platerod

rod =−

=−

≈ =( )

.1 1

1052 2 ν ρ ν

ν(for 0.3) (1.56)

is also somewhat higher.

Figure 1.13 Vibration pattern of a longitudinal wave propagating in a thin rod.

For a longitudinal pressure wave propagating in an infinite medium shown in Figure 14.a

ε ε y z= = 0, therefore the Poisson effect further increases the stiffness

σ ν

ν

ε ν

ν νε x x x

E E =

−−

= −

+ −1

2

1

1

1 1 22( )

( )( )(1.57)

and the longitudinal or dilatational wave velocity

c E

c cd rod rod = −

+ −=

−

+ −≈

( )

( )( )

( )

( )( ).

1

1 1 2

1

1 1 2116

ν

ν ν ρ

ν

ν ν(1.58)

is even higher.

Equation 1.57 can be also written with Lamé's constants as follows

x xu

xσ λ µ ε λ µ

∂

∂= + = +2 2 . (1.59)

where λ is a Lamé constant and µ is the shear modulus (often denoted by G). In an extended

elastic solid (a so-called "infinite medium"), the pressure wave velocity may be also expressed in

terms of the elastic constants of the medium and the density as

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d c = +λ µ

ρ

2. (1.60)

As a final note on pressure waves, the propagation velocity of acoustic waves in a gas can

be obtained from the so-called gas equation p RT = ρ . Here T is the (absolute) temperature

and R denotes the gas constant that is the ratio of the universal gas constant and the average

molecular weight. For an adiabatic process p K = ργ so that the bulk modulus

B p K = =ρ∂ ∂ρ γ ργ / . The sound velocity is given again by d c B= / ρ as follows

d o

c p

RT = =γ

ργ , (1.61)

where

po = static (ambient) pressure

γ = c c p v / is the specific-heat ratio.

x u

σ x

σ y

σ x-

σ y-

y

Figure 1.14 Particle motion and stresses for pressure waves.

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An example of transverse wave propagation is the flexural vibration propagating in a thin

rod shown in Figure 1.15. The well-known differential equation governing the bending

deformation of a thin rod is

Figure 1.15 Flexural vibration of a thin rod.

I E x

q∂

∂

4

4

v = , (1.62)

where I is the moment of inertia for the cross-sectional area, v is the transverse displacement,

and q is the distributed load intensity for a unit length. In our case, the load is entirely due to

inertia forces accelerating the beam

q At

= − ρ ∂∂

2

2

v

, (1.63)

where A denotes the cross-sectional area. For harmonic vibrations of v v o = −sin[ ]k x t ω ,

∂ ∂4 4 4 / x k = and ∂ ∂ ω2 2 2 / t = − , therefore Equations 1.62 and 1.63 can be combined as

follows

I E k A4 2= ρ ω . (1.64)

The phase velocity is then

ck

I E

A f = =

ω ω

ρ

24 (1.65)

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proportional to the square-root of frequency. For a rectangular bar of height h, the flexural

velocity can be written as

c E h

c h f rod

= =2 2

4

1205373

ω

ρω. , (1.66)

where c E rod = / ρ is the previously discussed longitudinal wave velocity in the thin rod.

Similarly, for a cylindrical rod of diameter d , I A d / / = 2 16 and c c d f rod = 0 5. ω . This

wave mode is also the limiting case of the lowest-order asymmetric Lamb mode in very thin

plates (Lamb waves are elastic waves propagating in a solid plate with free boundaries, which

will be discussed in more detailed in Chapter 5). However, for thin plates, the phase velocity can

be calculated from Equation 1.66 by substituting E / ( )1 2− ν for E .

Another example of transverse wave propagation is the case of shear waves in infinite

solid media, which is illustrated in Fig. 1.16.

xv

τ yx

τ yx-

τ xyτ xy-

Figure 1.16 Particle motion and stresses for shear waves.

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The expressions relating shear stress τ xy , shear strain γ xy and transverse particle displacement

v are

τ µ γ γ ∂

∂ xy xy xy

x

= =, v

. (1.67)

The case of shear waves is also governed by the wave equation,

2

2 2

2

2

1∂

∂

∂

∂

v v

x c t s= . (1.68)

The velocity of propagation of shear waves is

sc = µ

ρ . (1.69)

It may be shown that the ratio of the two wave velocities in an isotropic solid, for which

E = +2 1( ) ν µ , the ratio of cd and cs depends only on Poisson's ratio:

d

s

c

c=

−

−

2 2

1 2

ν

ν . (1.70)

Finally, the velocity of surface waves may be noted. There is not a simple formula for this

velocity. However, an approximate expression is given by

c c R s≅ +

+

087 112

1

. . ν

ν . (1.71)

Acoustic Impedance

The relationship between stress σ, displacement u, and particle velocity v for a

propagating wave is of interest. As an example, let us consider a dilatational wave propagating in

an infinite elastic medium:

u x t Ae xi k x t ( , ) ( )= − ω , (1.72)

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v x x i k x t x t

u

t i Ae( , ) ( )= = − −

∂

∂ω ω , (1.73)

and

x x i kx t u

x Ai k eσ λ µ

∂

∂λ µ ω= + = + −( ) ( ) ( )2 2 . (1.74)

The ratio of the pressure (or negative stress) to the particle velocity is called the acoustic

impedance. For a dilatational wave propagating in the positive direction,

Z c Ai k e

i Ae

cd x

x

d i kx t

i k x t d = − = =

−

−

σ ρ

ω

ρω

ωv

2 ( )

( )

. (1.75)

The product of density and wave velocity occurs repeatedly in acoustics and ultrasonics and is

called the characteristic acoustic impedance (for a plane wave). As it will be seen later, it will be

the impedance that acoustically differentiates materials, in addition to the moduli and density.

The densities, velocities and acoustic impedances of a number of materials are summarized in

Table 1.1.

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Table 1.1 Densities, acoustic velocities and acoustic impedances of some materials.

Material Density

[103 kg/m3]

Acoustic velocities

[103 m/s]

Impedance

[106 kg/m2s]

ρ long. cd shear cs Z cd d = ρ

Metals

Aluminum 2.7 6.32 3.08 17

Iron (steel) 7.85 5.90 3.23 46.5

Copper 8.9 4.7 2.26 42

Brass 8.55 3.83 2.05 33

Nickel 8.9 5.63 2.96 50Tungsten 19.3 5.46 2.62 105

Nonmetals

Araldit Casting Resin 1.25 2.6 1.1 3.3

Aluminum oxide 3.8 10 38

Glass, crown 2.5 5.66 3.42 14

Perspex (Plexiglas) 1.18 2.73 1.43 3.2

Polystyrene 1.05 2.67 2.8

Fused Quartz 2.2 5.93 3.75 13

Rubber, vulcanized 1.4 2.3 3.2

Teflon 2.2 1.35 3.0

Liquids

Glycerine 1.26 1.92 2.4

Water (at 20oC) 1.0 1.483 1.5

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1.7 Wave Dispersion

Dispersion means that the propagation velocity is frequency-dependent. Since the phase

relation between the spectral components of a broadband signal varies with distance, the pulse-

shape gets distorted and generally widens as the propagation length increases. Figure 1.17 shows

schematically the distortion of a unipolar pulse caused by dispersive wave propagation.

input pulse

= 0

< 0

δδω

c> 0

δδω

c

δδω

c

Figure 1.17 Pulse distortion caused by dispersive wave propagation.

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Group Velocity

Generally, the pulse distortion due to dispersion has to be determined by spectral

(Fourier) analysis. Figure 1.18 illustrates the dispersive wave propagation of a tone-burst. In the

case of relatively narrow band "tone-bursts", the effect of dispersive wave propagation can be

approximated by the concept of different phase and group velocities.

phase

velocity

group

velocity

Figure 1.18 Dispersive propagation of a tone-burst.

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where the first high-frequency term is called carrier wave and the second low-frequency term is

the modulation envelope. This shows that the propagation velocity of the carrier is the phase

velocity

c

k k k

p =+

+≈

ω δω

δ

ω2

2

, (1.78)

and the propagation velocity of the modulation envelope is the group velocity

ck k

g = →δω

δ

∂ω

∂ . (1.79)

The characteristic equation of a certain wave mode provides the relationship between c p

and k . Then, the group velocity can be easily calculated from c p (k) as

c c k c

k g p

p= +

∂

∂ . (1.80)

Very often the phase velocity is given in the form of c p( )ω . Then, the group velocity can be

calculated as follows

cc

c

cg p

p

p=

−1 ω ∂

∂ω

. (1.81)

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Spectral Representation

In the case of dispersive wave propagation,

f t x

c f t

x

c( ) (

( ))− −becomes

ω . (1.82)

Let us assume that f (t ) is known at x=0 . Its Fourier transform can be written as

F { ( )} ( ) ( ) exp( ) f t F dt f t i t = = −−∞

∞ω ω . (1.83)

The inverse Fourier transform can be used to obtain the signal in the time domain again

F -1{ ( )} ( ) ( ) exp( )F f t d F i t ω

πω ω ω= =

−∞

∞1

2 . (1.84)

According to the shift theorem,

F { ( )} ( ) exp( ) f t t F i t p p− = −ω ω , (1.85)

therefore the frequency spectrum of the signal after dispersive propagation over a distance of x

is

F x F i x

cF i x k ( , ) ( , ) exp[

( )] ( , ) exp[ ( )]ω ω ω

ωω ω= − = −0 0 . (1.86)

It should be mentioned that essentially the same approach based spatial rather than temporal

frequency representation is often used in two- and three-dimensional wave propagation problems,

too.

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Material versus Geometrical Dispersion

There are two main causes of dispersive wave propagation of ultrasonic fields. First,

inherent material behavior such as relaxation in polymers, which is best described by a

characteristic time constant. The degree of the dispersion is generally rather weak and the

dispersion is dependent on the ratio of this time constant to the time period of the ultrasonic

vibration. Second, geometrical effects such as in the case of dispersive Lamb waves when the

dispersion is determined by a characteristic dimension. In this case, the degree of the dispersion

can be very high and the dispersion is dependent on the ratio of this dimension (e. g., plate

thickness) to the acoustic wavelength.

As an example of inherent material dispersion, Figure 1.20 shows the sound velocity as afunction of frequency in polyethylene [M. O'Donnell et al. J. Acoust Soc. Am. 69, 696 (1981)].

Frequency [MHz]

V e l o c i t y [ k m / s ]

2.6

2.7

2.8

0 2 4 6 8 10

Figure 1.20 Sound velocity as a function of frequency in polyethylene.

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To a much less degree than the flexural mode, the longitudinal mode propagating in a thin

plate also becomes dispersive as the frequency increases due to the Poisson effect. As an example

of geometrical dispersion, Figure 1.21 shows the phase and group velocities of the lowest-order

symmetric Lamb mode as functions of frequency in a thin aluminum plate.

Normalized Frequency

N o r m a l i z e d V e

l o c i t y

0

0.5

1

1.5

2

0 1 2 3 4 5 6

phase

group

Figure 1.21 Phase and group velocities of the lowest-order symmetric Lamb mode as

functions of frequency in a thin aluminum plate (the velocities are

normalized to the shear velocity and the normalized frequency is k sd ).

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Part 2 MATHEMATICAL

FORMALISMS FOR

ACOUSTIC WAVEPROPAGATION

Acoustics is the study of time-varying deformations, or vibrations in elastic media. It is

concerned with material particles that are small but yet contain many atoms. Within each particle

the atoms move in unison. Therefore, acoustics deals with macroscopic phenomena and is

formulated as if matter were a continuum. Structure at the microscopic level is of interest only

insofar as it affects the medium's macroscopic properties. When the particles of a medium aredisplaced from their equilibrium positions, internal restoring forces arise. It is these restoring

forces between particles, combined with the inertia of the particles, which lead to oscillatory

motions of the medium. To formulate a mathematical description of these vibrations, which may

be either traveling waves or localized oscillations, it is first necessary to introduce quantitative

definitions of particle displacement , material deformation, and internal restoring forces.

displacement-strain relation (deformation)

stress-strain relation (constitutive equation)

balance of momentum (Newton's Law)

⇓

equation of motion (wave equation)

⇓

wave field (displacement, displacement potential,

velocity, velocity potential, stress, etc.)

Notation:

position vector x e e e( )= + +1 1 2 2 3 3 x x x

displacement vector u

strain matrix ε

stress matrix τ

stiffness tensor C

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ONE-DIMENSIONAL PROBLEM

a) there is only one spatial coordinate x e= 1 1 x

b) there are three spatial coordinates but the field parameters change only in

one dimension ∂

∂∂

∂ x x2 30= =

Displacement-strain relation

ξ( ) ( ) x x u x= +

ξ ∂

∂( ) ( ) ( )

( ) x dx x dx u x dx x dx u x

u x

xdx+ = + + + ≈ + + +

ε ξ ξ ∂

∂=

+ − −≈

[ ( ) ( ) ] x dx x dx

dx

u

x

Stress-strain relation

τ ε= C

Balance of momentum (without body force)

τ τ ρ ∂∂

( ) ( )( )

x dx x dx u x

t + − = 2

2

∂τ∂

ρ∂

∂ xu

t =

2

2

Equation of motion

∂

∂

ρ ∂

∂

2

2

2

2

u

x C

u

t =

Wave field

u f t x

c= −( ) , where c

C =

ρ

2-2


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THREE-DIMENSIONAL PROBLEM

Physical problem: differential equations

Mathematical description: vector notation

indicial notation

differential equations (with abbreviated notation)

Solution: plane wave technique, Christoffel's equationpotential technique

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VECTOR NOTATION

Vector operator nabla

∇ ≡ + +e e e11

22

33

∂∂

∂∂

∂∂ x x x

Gradient of a scalar field

∇ϕ ≡ ≡ + +grad x x x

ϕ ∂ϕ

∂∂ϕ∂

∂ϕ∂

e e e11

22

33

Divergence of a vector field

∇ ⋅ ≡ ≡ + +ψdiv x x x

∂ψ ∂

∂ψ ∂

∂ψ ∂

1

1

2

2

3

3

Curl of a vector field

∇ × ≡ ≡ − + − + −ψcurl x x x x x x

e e e13

2

2

32

1

3

3

13

2

1

1

2

( ) ( ) ( )∂ψ ∂

∂ψ ∂

∂ψ ∂

∂ψ ∂

∂ψ ∂

∂ψ ∂

Laplace operator:

∇ = + +22

12

2

22

2

32

∂

∂

∂

∂

∂

∂ x x x

Laplacian of a scalar field

∇ ≡ ≡ + +22

12

2

22

2

32

ϕ ϕ ∂ ϕ

∂

∂ ϕ

∂

∂ ϕ

∂

div grad

x x x

Laplacian of a vector field

∇ ≡ ∇ + ∇ + ∇2 12

1 22

2 32

3e e eψ ψ ψ

2-4


45/147

INDICIAL NOTATION

In a system of fixed rectangular Cartesian coordinates

x = + +e e e1 1 2 2 3 3 x x x

Free index

ui where the subscript or index assumes the values of i = 1 2 3, ,

Summation convention: repeated index implies summation

vector: u e e e e= + + =1 1 2 2 3 3u u u ui i or simply ui

vector by vector u v⋅ = + + =u v u v u v u vi i1 1 2 2 3 3

Partial differentiation is denoted by a comma

scalar ∂

∂u

xui

ji j= ,

vector ∂∂

∂∂

∂∂

∂∂

ue e e e

x x x x j j j j= + + =1

12

23

3u u u ui i j, or simply ui j,

Gradient of a scalar

grad f f f f

f i i= + + =e e e e1 2 3∂∂

∂∂

∂∂ x x x1 2 3

, or simply f i,

Divergence of a vector

div u u u

ui iu = + + =∂∂

∂∂

∂∂

1 2 3

x x x1 2 3,

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DIFFERENTIAL EQUATIONS IN THREE DIMENSIONS

Displacement-strain relation

vector notation ε = ∇s u

indicial notation εij i j j iu u= +½( ), ,

differential notation ε ∂

∂

∂

∂iji

j

j

i

u

x

u

x= +½( )

Six independent strain components

ε ∂

∂ ε

∂∂

ε ∂

∂111

122

2

233

3

3

= = =u

x

u

x

u

x, , ,

ε ε ∂

∂∂∂12 21

1

2

2

1

= = +½ ( )u

x

u

x

ε ε ∂

∂∂∂23 32

2

3

3

2

= = +½ ( )u

x

u

x

ε ε ∂∂

∂∂31 13

3

1

1

3

= = +½ ( )u

x

u

x

Stress-strain relation (constitutive equation):

vector notation τ ε= C :

indicial notation τ εij ijkl klC =

τ ε ε ε

ε ε ε

ε ε ε

ij ij ij ij

ij ij ij

ij ij ij

C C C

C C C

C C C

= + +

+ + +

+ + +

11 11 12 12 13 13

21 21 22 22 23 23

31 31 32 32 33 33

2-6


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48/147

ABBREVIATED NOTATION

ε =

ε ε ε

ε ε ε

ε ε ε

11 12 13

21 22 23

31 32 33

τ =

τ τ τ

τ τ τ

τ τ τ

11 12 13

21 22 23

31 32 33

Stiffness matrix

τ

τ

τ

τ

τ

τ

ε

ε

ε

ε

ε

ε

11

22

33

23

31

12

11 12 13 14 15 16

12 22 23 24 25 26

13 23 33 34 35 36

14 24 34 44 45 46

15 25 35 45 55 56

16 26 36 46 56 66

11

22

33

23

31

12

2

2

2

=

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

Stress-strain relations for isotropic materials (Hooke's Law)

In indicial notation, τ λ ε δ µ εij kk ij ij= + 2 , where δij is the so-called Kronecker delta:

δ δij iji j= = =1 0if and else. In full details,

τ λ µ ε λ ε λ ε11 11 22 332= + + +( )

τ λ ε λ µ ε λ ε22 11 22 332= + + +( )τ λ ε λ ε λ µ ε33 11 22 332= + + +( )

τ τ µ ε12 21 122= =

τ τ µ ε23 32 232= =

τ τ µ ε31 13 312= =

Stiffness matrix in abbreviated notation

τ

τ

τ

τ

τ

τ

λ µ λ λ

λ λ µ λ

λ λ λ µ

µ

µ

µ

ε

ε

ε

ε

ε

ε

11

22

33

23

31

12

11

22

33

23

31

12

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

2

2

2

=

+

+

+

2

2

2

2-8


49/147

Stress-displacement relation

τ λ ∂

∂

∂

∂

∂

∂

µ ∂

∂11

1

1

2

2

3

3

1

1

2= + + +( )u

x

u

x

u

x

u

x

τ λ ∂∂

∂∂

∂∂

µ ∂∂22

1

1

2

2

3

3

2

2

2= + + +( )u

x

u

x

u

x

u

x

τ λ ∂∂

∂∂

∂∂

µ ∂∂33

1

1

2

2

3

3

3

3

2= + + +( )u

x

u

x

u

x

u

x

τ τ µ ∂∂

∂∂12 21

1

2

2

1

= = +( )u

x

u

x

τ τ µ ∂∂

∂∂23 32

2

3

3

2

= = +( )u

x

u

x

τ τ µ

∂

∂

∂

∂31 133

1

1

3= = +( )

u

x

u

x

Balance of momentum

vector notation ∇ ⋅ + =f uρ

indicial notation τ ρij j i i f u, + =

differential notation∂τ∂

∂τ∂

∂τ∂

ρ∂∂

11

1

12

2

13

31

21

2 x x x f

u

t + + + =

∂τ∂

∂τ∂

∂τ∂

ρ∂∂

21

1

22

2

23

32

22

2 x x x f

u

t + + + =

∂τ∂

∂τ∂

∂τ∂

ρ∂∂

31

1

32

2

33

33

23

2 x x x f

u

t + + + =

2-9


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THREE-DIMENSIONAL WAVE EQUATION

vector notation ∇ ⋅ ∇ = −C u u f : s ρ

For isotropic materials,

( ) λ µ µ ρ+ ∇ ∇ ⋅ + ∇ = −u u u f 2

indicial notation ( ) &&, ,λ µ µ ρ+ + = −u u u f j ji i jj i i

detailed differential equation form

( )( ) ( )λ µ ∂

∂

∂∂ ∂

∂∂ ∂

µ ∂

∂

∂

∂

∂

∂ρ

∂

∂+ + + + + + = −

21

12

22

1 2

23

1 3

21

12

21

22

21

32

21

2 1u

x

u

x x

u

x x

u

x

u

x

u

x

u

t f

( )( ) ( )λ µ ∂∂ ∂

∂

∂

∂∂ ∂

µ ∂

∂

∂

∂

∂

∂ρ

∂

∂+ + + + + + = −

21

1 2

22

22

23

2 3

22

12

22

22

22

32

22

2 2u

x x

u

x

u

x x

u

x

u

x

u

x

u

t f

( )( ) ( )λ µ ∂∂ ∂

∂∂ ∂

∂

∂µ

∂

∂

∂

∂

∂

∂ρ

∂

∂+ + + + + + = −

21

1 3

22

2 3

23

32

23

12

23

22

23

32

23

2 3u

x x

u

x x

u

x

u

x

u

x

u

x

u

t f

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PLANE WAVE SOLUTIONS, CHRISTOFFEL'S EQUATION

u p k x= − A ei ( t)ω

amplitude A

angular frequency ω

polarization unit vector p

wave vector k d= k

propagation unit vector d

wave number k k k k = + +12

22

32

sound velocity ck

= ω

In details,

u A p e ei t i k d x d x d x1 11 1 2 2 3 3= − + +ω ( )



( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

λ µ µ ρ λ µ λ µ

λ µ λ µ µ ρ λ µλ µ λ µ λ µ µ ρ

+ + − + +

+ + + − ++ + + + −

=

d d c d d d d

d d d d c d d d d d d d d c

p

p p

1 12

1 2 1 3

1 2 2 22

2 3

1 3 2 3 3 32

1

2

3

0

00

Nontrivial (non-zero) solution requires that the determinant be zero, which provides the

characteristic equation for any propagation direction d. The solutions of this characteristic

equation are frequency-independent (nondispersive) longitudinal (dilatational) and shear waves.

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Since the material is isotropic, d e= = = =1 1 2 31 0( , )d d d can be assumed without loss of

generality.

λ µ ρ

µ ρµ ρ

+ −

−−

=

2 0 0

0 0

0 0

0

0

0

2

2

2

1

2

3

c

c

c

p

p

p

Longitudinal (or dilatational) wave

cd = +λ µ

ρ2

and p p2 3 0= =

Shear (or transverse) wave

cs = µ

ρand p1 0=

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53/147

Potential decomposition of the displacement vector, Helmoltz Theorem:

u = ∇ϕ + ∇ × ψ

Displacement from scalar and vector potentials

u x x x

11

3

2

2

3

= + −∂ϕ∂

∂ψ ∂

∂ψ ∂

u x x x

22

3

1

1

3

= − +∂ϕ∂

∂ψ ∂

∂ψ ∂

u x x x

33

2

1

1

2

= + −∂ϕ∂

∂ψ ∂

∂ψ ∂

For an isotropic solid, the stress components can be written in terms of the displacementpotentials as follows

τ λ ∂ ϕ

∂

∂ ϕ

∂

∂ ϕ

∂µ

∂ ϕ

∂

∂ ψ ∂ ∂

∂ ψ ∂ ∂11

2

12

2

22

2

32

2

12

23

1 2

22

1 3

2= + + + + −( ) ( ) x x x x x x x x

τ λ ∂ ϕ

∂

∂ ϕ

∂

∂ ϕ

∂µ

∂ ϕ

∂

∂ ψ ∂ ∂

∂ ψ ∂ ∂22

2

12

2

22

2

32

2

22

23

1 2

21

2 3

2= + + + − +( ) ( ) x x x x x x x x

τ λ ∂ ϕ

∂

∂ ϕ

∂

∂ ϕ

∂

µ ∂ ϕ

∂

∂ ψ ∂ ∂

∂ ψ ∂ ∂33

2

1

2

2

2

2

2

3

2

2

3

2

22

1 3

21

2 3

2= + + + + −( ) ( )

x x x x x x x x

τ τ µ ∂ ϕ∂ ∂

∂ ψ

∂

∂ ψ ∂ ∂

∂ ψ

∂

∂ ψ ∂ ∂12 21

2

1 2

23

22

22

2 3

23

12

21

1 3

2= = + − − +( ) x x x x x x x x

τ τ µ ∂ ϕ∂ ∂

∂ ψ

∂

∂ ψ ∂ ∂

∂ ψ

∂

∂ ψ ∂ ∂23 32

2

2 3

21

32

23

1 3

21

22

22

1 2

2= = + − − +( ) x x x x x x x x

τ τ µ ∂ ϕ∂ ∂

∂ ψ

∂

∂ ψ ∂ ∂

∂ ψ

∂

∂ ψ ∂ ∂31 13

2

1 3

22

32

23

2 3

22

12

21

1 2

2= = − + + −( ) x x x x x x x x

Separation of the wave equations

For an isotropic material, the balance of momentum equation was previously written as (without

body forces)

( ) &&λ µ µ ρ+ ∇ ∇ ⋅ + ∇ =u u u2 .

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DILATATIONAL WAVES

no shear deformation ε ε ε12 23 31 0= = =

τ τ τ12 23 31 0= = =

Dilatational wave in an infinite solid

no lateral strain ε ε22 33 0= =

axial stress τ λ µ ε11 112= +( )

lateral stress τ τ λ ε22 33 11= =

longitudinal velocity cd

= +λ µ

ρ2

In a fluid, µ = =0 0( )cs and the compressional wave velocity is cd = λ

ρ.

Dilatational wave in a thin rod

lateral strain ε ε22 33 0= ≠

axial stress τ λ µ ε λ ε11 11 222 2= + +( )

no lateral stress τ τ λ ε λ µ ε22 33 11 222 0= = + + =( )

Poisson's ratio ν ε

ελ

λ µ= − =

+22

11 2( )

Young's modulus E = = +

+= =

τε τ τ

µ λ µλ µ

11

11 022 33

3 2( )

sound velocity c E

crod s

= = +ρ

ν2 1( )

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Dilatational wave in a thin plate

Poisson effect ε ε22 330 0≠ =but

axial compression τ λ µ ε λ ε11 11 222= + +( )

lateral compression τ λ ε λ µ ε22 11 222 0= + + =( )τ λ ε λ ε33 11 22= +

strain ratio − =+

>εε

λλ µ

ν2211 2

plate stiffness K = = +

+= =

τε τ ε

µ λ µλ µ

11

11 022 33

4

2

( )

sound velocity c K

c plate s

= =−ρ ν2

1

Under hydrostatic pressure ( )τ δij ij p= − ,

ε ε ε12 23 31 0= = =

τ τ τ λ µ ε11 22 33 113 2= = = +( )

The bulk modulus is defined by the volume contraction

p B= − + +( )ε ε ε11 22 33

B = +λ µ2

3

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Part 3 REFLECTION AND

TRANSMISSION OF

ULTRASONIC WAVES

Nearly all applications of ultrasonics involve the interaction of waves with boundaries.

Nondestructive testing, medical imaging and sonar are ready examples of this. Even basic studies

of material properties, usually involving the attenuation of waves, require in the final analysis

accounting for boundary interactions.

3.1 Reflection/transmission - normal incidence

The simplest situation of reflection and transmission occurs when waves are impinging

normal to the surface. In Fig. 3.1, the case of a longitudinal wave incident on the interface

between two media is shown. This situation may be described mathematically in terms of three

propagating waves

ρ , c1 1

ρ , c2 2


Transmission

Figure 3.1 Reflection and transmission of an acoustic wave at normal incidence to a

plane boundary.

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u A k x t i i= −sin ( )1 ω , (3.1)

u A k x t r r = − − +sin[ ( )]1 ω , (3.2)

u A k x t t t = −sin( )2 ω . (3.3)

The amplitude of the reflected and transmitted waves may be found by noting that the

displacements and stresses must be the same (continuous) at the interface. Thus, for x = 0, it is

required that

u u ui r t + = and τ τ τi r t + = . (3.4)

This leads directly to the result

R A

A

c c

c cd

r

i

= =−

+2 2 1 1

1 1 2 2

ρ ρ

ρ ρ(3.5)

and

T A

A

c

c cd

t

i

= =+

2 1 1

1 1 2 2

ρ

ρ ρ . (3.6)

This gives the ratio of the displacement amplitude. More commonly, the stress (or pressure)

amplitudes are given. Thus,

R c c

c c Rs

r

id = =

−

+=

τ

τ

ρ ρ

ρ ρ2 2 1 1

1 1 2 2

(3.7)

and

T c

c cT s

t

id = =

+≠τ

τρ

ρ ρ2 2 2

1 1 2 2

, (3.8)

where R and T are known as the reflection and transmission coefficients. It is seen that these

results are in terms of the respective acoustic impedances of the materials.

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Illustration of the reflection and transmission at various interface combinations are worth

considering. For steel-water , we have ρs sc = ×46 5 106. kg / m s2 and

ρw wc = ×15 106. kg / m s2 . From Eqs. 3.7 and 3.8, one obtains R = − 0938. and

T = 0 063. . The interpretation of this result is that the amplitude of the reflected stress wave is

0.938 (or 93.8%) that of the incident amplitude. The negative sign indicates that the reflected

wave is 180o out of phase with the incident wave. Thus, when the incident wave is compressive,

the reflected wave is tensile and vice-versa. The transmitted pressure amplitude is but 6.3% of

the incident amplitude. This reflection/transmission situation is shown in Fig. 3.2a. For water-

steel, by using the previous values for ρs cs, , ρw , and cw, we obtain from Eqs. 3.7 and 3.8

R = 0938. and T = 1938. . Thus, the reflected wave amplitude is nearly the same as the

incident amplitude, where the transmitted (stress) amplitude is nearly twice the incident

amplitude, as shown in Fig. 3.2b.

pi

pr

pt

pt

pr

pi

a)

b)

steel water

water steel

Figure 3.2 Sound pressure values in the case of reflection from (a) a steel-water and

(b) a water-steel interface at normal incidence.

The preceding result may appear strange, as though conservation of energy were being violated.

However, both wave amplitude and wave velocity determine the time rate of flow of energy (i. e.,

power) at the interface. In terms of power, there should be a net balance. That is,

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P P Pr t i+ = (3.9)

should be satisfied. The power per unit area (i. e., intensity) will be given by I = − τv , where

τ ∂ ∂ ∂ ∂= =C u x u t 11 / / and v . Using Eqs. 3.1-3.3 to calculate Pi, Pr , Pt and substituting in the

power balance expression shows it to be satisfied.

Special cases: Suppose media 2 is vacuum, so that 2 2 0ρ c = . One obtains

Rd free = −1, T d

free = 2 , Rs free = −1, and T s

free = 0

indicating a simple phase reversal of the incident wave. Suppose media 2 is infinitely rigid, so

that 2 2ρ c → ∞ . Then from Eqs. 3.5 through 3.8 one obtains

Rd clamped = 1, T d

clamped = 0 , Rsclamped = 1, and T s

clamped = 2

There is thus no phase reversal of the incident displacement wave.

The case of shear waves normally incident on a boundary may also be considered.

However, a small subtlety arises. If the two media are bonded together, then conditions at the

interface would be that i r t i r t v v v + = + =, τ τ τ , and expressions of the form Eqs. 3.5 trough

3.8 would be obtained, with all velocities merely being changed to shear wave velocities.

However, the more common cases of a fluid-solid interface or of two solids separated by a thin

film of lubricant would prevent transmission of shear waves across the interface.

The case of waves normally incident on a layer sandwiched between two media is the

next step of complexity and represents a situation frequently arising in ultrasonics. Reflection at

and transmission through an elastic layer exhibit strong frequency dependence associated with

resonances in the layer. One of the simplest approach to describe this problem is applying the

impedance-translation theorem to the layer [See for example, L. M. Brekhovskikh, Waves in

Layered Media (Academic, New York, 1980) pp. 23-26]. The impedance-translation theoremsays that the input impedance Z input of a layer can be calculated from the loading impedance

Z load presented by the medium behind the layer and the acoustic impedance Z o of the layer

itself as follows:

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Z Z Z i Z k d

Z i Z k d input o

load o o

o load o

= −

−

tan( )

tan( ). (3.10)

Although this theorem is well known and widely used in several area, such as electrical

engineering, it is very instructional to derive it from the boundary conditions prevailing at thetwo interfaces. Let us write the stress distribution in the layer in the following general form:

τ( ) exp( ) exp( ) x A i k x A i k xo o= + −+ − , (3.11)

which is the sum of a forward and backward propagating plane wave. A+ and A- are the

complex amplitudes of the two waves and we omitted the common exp(- iω t ) term. The

velocity distribution is given by

v ( ) [ exp( ) exp( )] x x

i Z A i k x A i k x

o oo o= − = − − −+ −

∂τ ∂ωρ

/ 1 . (3.12)

The input impedance of the layer is

Z Z A A

A Ainput o= − =

+

−+ −

+ −

τ( )

( )

0

0v , (3.13)

where the ratio of the complex amplitudes A+ and A− can be determined from the condition

that

Z d

d Z

A e A e

A e A eload o

i k d i k d

i k d i k d

o o

o o= − =

+

−

+ −−

+ −−

τ( )

( )v . (3.14)

A

A

Z e Z e

Z e Z e

load i k d

oi k d

load i k d

oi k d

o o

o o

+

−

− −

= +

−(3.15)

Substitution of (3.15) into (3.13) yields

Z Z Z k d i Z k d

Z k d i Z k d input o

load o o o

o o load o

= −

−

cos( ) sin( )

cos( ) sin( )(3.16)

which is identical with the previously given form of (3.10).

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The reflection coefficient of the layer can be easily obtained from (3.7) as

R Z Z

Z Z

input

input

=−

+

1

1

(3.17)

from Z Z load = 2 . In the simplest case of Z Z 2 1= , the reflection coefficient turns out to be

R i k d Z Z

i k d Z Z Z Z

o o

o o o

= −

+ −

tan( )( )

tan( )( )

212

212

12, (3.18)

while the transmission coefficient can be calculated from the law of energy conservation as

T R= −( )12

. (3.19)

From Equations 3.18 and 3.19, the moduli of the reflection and transmission coefficients

can be written as follows

R k d

k d

o

o

=+

ξ

ξ

sin( )

sin ( )2 2 1(3.20)

and

T

k d o

=+

1

12 2ξ sin ( ) , (3.21)

where ξ = −½ Z Z Z Z o o / / 1 1 is a measure of the impedance contrast between the layer and

the surrounding host materials.

The general situation is shown in Fig. 3.2a, where repeated reflections occur within the

layer until a steady reflection, transmission state is reached. Not only do the material impedances

enter, the ratio of layer thickness to acoustic wavelength ( d/ λo ) strongly influences the result,

too. The particular cases of steel and Plexiglas plates in water are shown in Fig. 3.3b.

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a)

ρ , c1 1


ρ , c2 2

Transmission

ρ , co o

Thickness / Wavelength

T r a n s m i s s i o n C o e f f i c i e n t

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 1 1.25

Plexiglas

Steel

b)

Figure 3.3 (a) Schematic diagram of reflection at and transmission through a layered

medium and (b) specific cases of steel and Plexiglas plates in water.

3-7


64/147

Equation 3.20 can be used to answer one of the basic questions of ultrasonic

nondestructive evaluation concerning the reflectivity of thin cracks in solids. As an example,

Figure 3.4 shows the reflectivities of air-filled and water-filled cracks in steel as functions of the

frequency-thickness product [J. Krautkramer and H. Krautkramer, Ultrasonic Testing of

Materials (Springer, Berlin, 1977) p. 29]. For very thin cracks,

limd

o R k d →

=0

ξ , (3.22)

i. e., the reflectivity is proportional to the product of impedance mismatch, frequency, and layer

thickness.

log {Frequency x Thickness [MHz mm]}

R e f l e

c t i o n C o e f f i c i e n t

0

0.2

0.4

0.6

0.8

1

-10 -8 -6 -4 -2 0

air gap water-filled crack

Figure 3.4 The reflectivities of air-filled and water-filled cracks in steel as functions

of the frequency-thickness product.

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One of the most important consequence of the impedance-translation theorem of Eq. 3.10

is the impedance matching capability of a single layer. When the layer thickness is an odd

multiple of the quarter-wavelength in the layer material, i. e., d n= +( )2 1 4λ / , the input and

load impedances are related through

Z Z

Z input

o

load

=2

. (3.23)

This means that perfect matching (total transmission and zero reflection) can be achieved even

between widely different impedances if a quarter-wavelength matching layer of Z Z Z o = 1 2

acoustic impedance is applied at the interface. Let us denote the center frequency where the layer

thickness equals to one quarter-wavelength by f o. In the vicinity of this center frequency,

sin( ) , cos( ) ,k d k d f f

f o o

o

o

≈ ≈ = −

12

and where∆ ∆ π

, (3.24)

and the reflection coefficient can be approximated as follows

R r

r i r

i r

r ≈

−

+ −

≈ −1

12

1

2

∆

∆ , (3.25)

where r Z Z = 2 1 / denotes the impedance ratio between the two media to be matched. The

energy transmission coefficient through the matching layer can be approximated as

T r

r energy ≈ −

−1

1

4

22

∆( )

. (3.26)

Figure 3.5 shows the energy transmission coefficient through a quarter-wavelength matching

layer between quartz (typical transducer element) and water.

Of course, good matching is limited to the vicinity of the center frequency. The relative

bandwidth (inverse quality factor) can be approximated as

1 4 2

1

18

12 1

Q

f f

f

r

r

r

r o=

−≈

−≈

−π ( )

., (3.27)

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66/147

where f 1 and f 2 are the half-power (-6 dB) points. In the previously given example of quartz

coupled to water, the relative bandwidth is reasonably wide at 69 %. In the case of larger

impedance differences, the bandwidth where good transmission occurs is much lower. For

example, Figure 3.6 shows the energy transmission coefficient through a quarter-wavelength

matching layer between steel and water where the relative bandwidth is only 33 %.

It can be also seen from Equation 3.10 that whenever the layer thickness is equal to an

integer multiple of the half-wavelength, i. e., d n= λ / 2 , the input impedance is equal to the

load impedance and the presence of the layer does not affect the transmission and reflection

coefficients of the interface between the two surrounding media.

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E n e r g y T r a n s m i s s i o n

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5

exact

approximate

unmatched

Figure 3.5 Energy transmission coefficient through a quarter-wavelength matching

layer between quartz and water.


E n e r g y T r a n s m i s s i o n

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5

exact

approximate

unmatched

Figure 3.6 Energy transmission coefficient through a quarter-wavelength matching

layer between steel and water.

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3.2 Reflection/transmission - oblique incidence

A more general situation of reflection and transmission of waves at an interface occurs

when the incident wave strikes at an oblique angle. A large number of possibilities exist,

depending on the combinations of solid, fluid and vacuum of the two media and, if the incident

media is a solid, whether the incident wave is pressure or shear wave. There are two somewhat

opposite approaches to handle this complexity. One can start from the simplest case of

longitudinal wave interaction with a fluid-fluid interface and build up build up the complexity

step-by-step by introducing solid on one side then on the other. We shall follow another approach

by giving formal solution for the most general solid-solid interface for an arbitrary incident wave

then simplify the resulting formulas for the simpler cases. This approach was adapted from B. A.

Auld Acoustic Fields and Waves

(John Wiley & Sons, New York, 1973) Vol. II, pp. 21-38.

General case: In the most general case, either a longitudinal or a shear incident wave interacts

with a solid-solid interface. This situation is shown in Figure 3.7.

θdi

solid 1

I d Rd

Rs

T d

solid 2

T s

θsi I s

z

y

θs1

θd 1

θs2

θd 2

Figure 3.7 General acoustic wave interaction with a solid-solid interface.

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From Snell's Law,

sin sin sin sin sin sinθ θ θ θ θ θdi

d

si

s

d

d

s

s

d

d

s

sc c c c c c1 1

1

1

1

1

2

2

2

2

= = = = = . (3.28)

The particle displacement amplitudes of the incident, reflected, and transmitted longitudinal

waves are I R T d d d , , and , respectively. Similarly, the particle displacement amplitudes of the

incident, reflected, and transmitted shear waves are I R T s s s, , and . Only two stress components

are relevant to the boundary conditions:

τ λ ∂

∂ λ µ

∂

∂ yy

z yu

z

u

y= + +( )2 (3.29)

and

τ µ∂

∂

∂

∂ zy

y zu

z

u

y= +( ), (3.30)

where µ ρ λ µ ρ1 1 12

1 1 1 122= + =c cs d , , µ ρ λ µ ρ2 2 2

22 2 2 2

22= + =c cs d , and .

The boundary conditions require that both normal and transverse velocity and stress components

be continuous at the interface:

u u

u u

u u u u

u u u u

y y

z z

yy yy

zy zy

yd

yd

ys

ys

zd

zd

zs

zs

yyd

yyd

yys

yys

zyd

zyd

zys

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) (

2 1

2 1

2 1

2 1

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1

0

0

0

0

−

−

−

−

=

− + − +

− + − +

− + − +

− + −

τ τ

τ τ

τ τ τ τ

τ τ τ

or

) ( )

( )

( )

( )

( )+

=

τ

τ

τ zys

yi

zi

yyi

zyi

u

u

2

, (3.31)

where the incident wave can be either longitudinal ( I d = 1, I s = 0) or shear ( I s = 1, I d = 0).

Equation 3.31 can be written by using the displacement amplitudes as follows

a a a a

a a a a

a a a a

a a a a

R

T

R

T

b

b

b

b

c

c

c

c

d

d

s

s

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

1

2

3

4

1

2

3

4

=

or (3.32)

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70/147

depending on whether longitudinal or shear wave incidence is considered. The matrix elements

aij, bi, and ci can be easily calculated from simple geometrical considerations:

a =

− − −

−− − −

− − −

cos cos sin sin

sin sin cos coscos cos sin sin

sin sin cos cos

θ θ θ θ

aeem-728 introduction to ultrasonics lecture notes

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