9/12/2009

1

ANDE Course - Guided WavesPart 1

September 11, 2009

Overview

• Recap

• Waves guided by physical boundaries of a medium– SH, Lamb, and Lamé waves

• Waves guided by an interface between two media– Rayleigh waves

– Love waves

– Stonely/Sezawa/Scholte waves

9/12/2009

2

Bulk waves: Longitudinal Waves

Bulk waves: Shear (V) Waves

9/12/2009

3

Bulk Waves: Surface Excitation

Guided Waves

• Unbounded plates

• Shells

• Rods

9/12/2009

4

Possible Guided Wave “Modes”

• Propagating waves

• Attenuating waves

• Non-propagating waves

• Steady state response

• Transient response

Frequency domain response and Transient response can be very different!

Method of Partial Waves sincos ikxikzAe

sincos ikxikzBe

sincoscos )( ikxikzikz eBeAe

down going wave

up going wave

Superposition of partial waves

Assume rigid boundary conditions at z = 0 and z = h (i.e., vertical component of displacement must vanish)

)1:(cos2

,1 21cos2 1 ni

n

hikenote

h

ncfgetweefrom

sin

1ccP

1

2

2

1 2;

1

hq

q

n

ccpn

Cut-off frequencies

Dispersion relation

h

c0

h

9/12/2009

5

Guided Shear (H) Waves

Partial wave pattern for transverse resonance analysis of SH wave propagation in an isotropic plate with free boundaries

SH waves do not mode convert during reflection from a traction-free surface

)(

02

2

2

2 )(;1 tkxi

zzz

T

z eyuut

u

cu

hyyuz ,0/

...)2,1,0(2

0sincos nn

qhqhqh

2

2

22

0

2

2

0

2

,0 kc

quqdy

ud

T

zz

22

2

2

22

2hk

n

c

h

T

kh

Tch /

Dispersion curves for SH modes in an isotropic plate with free boundaries

SH Dispersion Curves

0.0 1.0 2.0 3.0 4.0 5.00.0

1.0

2.0

3.0

Frequency (MHz)

Vgr

(m/m

s)

0.0 1.0 2.0 3.0 4.0 5.00.0

2.0

4.0

6.0

8.0

10.0

12.0

Vph (

m/m

s)

Symmetric SH

Anti-Symmetric SH

SH dispersion curves for Aluminium plate of 3mmThickness

9/12/2009

6

Love waves

Large transverse displacements were observed during Earthquakes

Love suspected that a layered earth may lead to this observation

Love Waves

2

222

2

222 1;1

0tan

TT c

ckq

c

ckqwhere

qdqq

,, Ty cu

,, Ty cu

)(

1

)(

2

)(

1

tkxizq

y

tqzkxitqzkxi

y

eeBu

eAeAu

0

0

zanduu

dz

zyzyyy

zy

z = -d

z = 0

z

x

Boundary conditions

Frequency equation

Love wave dispersion curves

kcT

kcT

22

222222 ;)(

TT

TTn

cc

ccccn

k

/c

9/12/2009

7

Symmetric and Antisymmetric Lamb Modes

)(

2

)(

1

)]sin()sin([

)]cos()cos([

kxti

kxti

eqzikApzpBu

eqzqApzikBu

mode Longitudinal component of z

Transverse component of z

Symmetric 0 even odd

Antisymmetric /2 odd even

A and B are arbitrary constants determined by boundary conditions

(a) About the median plane, longitudinal components are equal, transverse components are opposite

(b) About the median plane, transverse components are equal, longitudinal components are opposite

Rayleigh-Lamb Equation

modessymmetric4

)(

)tan(

)tan(

modessymmetric)(

4

)tan(

)tan(

2

222

222

2

antipqk

kq

ph

qh

kq

pqk

ph

qh

2

2

2

2

2

2

kc

q

kc

p

T

L

modetheofvelocityphasecc

k :;

2/0;)tan(

)tan(14 22

4

4

or

qhq

phpqk

cT

Alternate form

9/12/2009

8

Dispersion Relation

ccq

ccp

T

L

11

11

2

22

2

22

2

2

2

2

2

2

kc

q

kc

p

T

L

a) Wavenumbers p and q are real : c > cL > cT or k < /cL < /cT

b) q is real and p is imaginary : cT < c < cL or /cL < k < /cT

c) p and q are imaginary : c < c T < cL or k > /cT > /cL

ck

cT

cL

Low-Frequency Region: Symmetric Mode

)tan(

)tan(14 22

4

4

qhq

phpqk

cT

For modes without cut-off frequency, 0 as k 0

22

22222

4

4 114)(4

LTT cckpqk

c

LPTP

L

TT cccBoundsvelocityplatec

c

ccc 2:);(12

2

2

p and q are given by

11

12

11

122

2

222 211

c

cikpand

c

ck

c

ckq

T

P

Taking ph 0, qh 0, we find for the displacements u1 and u2

= 0

2

11

12222

2

2

11

12

22

2

1

12

1

12

1

kxc

ciqAx

qk

pikqAu

uniformc

cqA

qk

kqAu

1)(

11

12

1

2

khkh

c

c

u

hu

Ecvelocitybarrecall

EcP

02

;)1(

9/12/2009

9

Low-Frequency Region: Antisymmetric Mode

)tan(

)tan(14 22

4

4

qhq

phpqk

cT = /2

22222

22

2222

4

4

)(3

4

3/1

3/114 hkqpq

hp

hqqk

cT

khc

khccc

kc P

LTT

3/1

3

2 22hk

cP 2

3 parabolic as k 0

Taking ph 0, qh 0, we find for the displacements u1 and u2

22

22

2

222

22

222

222

1

qk

qkikAu

kxqk

qkkAx

qk

qkAqu

kh

u

hu

2

1

Thin plate analogue classical plate theory

Mode Displacement Structure

Wave structure of A0 mode across the thickness of an Al plate

Wave structure of S0 mode across the thickness of an Al plate

In-plane Out of plane

9/12/2009

10

Mode Cutoff Frequencies0cossin phqhAs k 0,

For the symmetric case, qh=n, n=0,1,2,…

T

TT

ncfdornfd

c

d

cqh

2

2

2

OR ph=n/2, n=0,1,2,…

For the antisymmetric case,

L

T

ncfd

andcn

fd

;

2

)12(

Dispersion curves for a traction-free isotropic Al plate

222

2

2

L

LL

ncfdor

nfd

c

d

cph

Group Velocity of Lamb Waves

dk

dcg

1

2

)()(

fdd

dcfdccc P

PPg

PgP ccfdd

dcas ,0

)(

0,)(

gP c

fdd

dcas

mode cutoffDispersion curves for a traction-free isotropic Al plate

9/12/2009

11

Non-propagating Modes

The dominant frequency measured by the impact-echo method corresponds to the zero group velocity point of the 1st order symmetrical (S1) Lamb wave of a free plate.

Consequently, the thickness resonance can be interpreted as a standing S1 wave field characterized by a high out-of-planeexcitability and an energy that does not propagate away in lateral direction.

2.0 4.0 6.0 8.0 10.00

20

40

60

80

Frequency-Thickness (MHz-mm)

An

g (

de

g)

A0

S0 A1

S1S2

A2 S3

A3 A4 S4

35.12°

19.29°

10.44°

SS 316; h = 2 mm; (VL) = 5700 m/s, (VS) = 3046 m/s, ρ = 7980 kg/m3

Pshoeprobe

inc

cc

90sinsin

Guided Waves and Snell’s Law

L wave Transducer on a water wedge

9/12/2009

12

Can we “measure” Dispersion Curves?

FFT

t (s)

Time domain

f (Hz)

Frequency domain

Experimental Validation

Aluminium Plate 6.5 mm

Transmitted using 3.5 MHz immersion transducer

Theoretical Transmission Spectra

PVDF

Received by 3.5 MHz

immersion transducer

Received by a large

aperture broadband

PVDF sensor

Measured Transmission Spectra

9/12/2009

13

Non-leaky Lamb Waves?

Lamb wave dispersion curves for steel, showing dominant in-plane displacement points

222

2

)(

4

)tan(

)tan(

modes,symmetricfor

kq

pqk

ph

qh

222

2

2

222

2

2

kkkc

q

kkkc

p

T

T

L

L

iqs

ipr

0)cosh()sinh(4

)sinh()cosh()(

2

22

shrhrsk

shrhsk

As r 0,

...2,1,0)(0)sinh( ninshsh n

Using that,

,...2,1for)/(1

)(

get we

0as

2

222

ncc

ncfd

rkks

LT

Tn

TL

2/0)(lim0

dzforzwzr

Amplitude of normal component for symmetric mode vanishes on the free surface

Poisson Ratio and Lamb Modes

Variation, with Poisson’s ratio in the range 0 - 0.49, of anti-symmetric Lamb modes in an isotropic free plate of thickness d. For each mode, the lower curve corresponds to =0 and the upper one to =0.49.

For an isotropic solid, Lamb modes can be described with only one parameter

Behavior of dispersion curves over the coincidence points. Dispersion curves of a given modecross the line F= √2K with the same slope.

Rigid solid: =0, =√2Fluid: =0.5, =∞

9/12/2009

14

Poisson Ratio and Lamb Modes

Variation, with Poisson’s ratio in the range 0 - 0.49, of symmetric Lamb modes in an isotropic free plate of thickness d. For each mode, the lower curve corresponds to =0 and the upper one to =0.49.

Behavior of dispersion curves over the coincidence points. Dispersion curves of a given modecross the line F=√2K with the same slope.

Rigid solid: =0, =√2Fluid: =0.5, =∞

Lamé Modes Lamé modes are particular solutions of the Rayleigh-Lamb equations for kT

2 =2k2,i.e., for a phase velocity c = cT√2.

2

2

2

2

2

2

kc

q

kc

p

T

L

222

222

kkq

kkp

T

L

22

22

1kp

kq

Boundary conditions are satisfied with kh=n+1/2 for the symmetric modes and kh=n for the anti-symmetric modes .

0)sin()()sin(2

0)cos(2)cos()(

22

22

qhAqkphikpB

qhikAphBqk

The Rayleigh-Lamb equation

Partial wave pattern for Lamé wave propagation. At a 45 angle of incidence there is no coupling of

the SV waves with the P waves.

The Lamé line is a locus for the roots of the Rayleigh–Lamb equation.

In a rigid solid, the Lamé solutions correspond to a constant longitudinal displacementpropagating at the velocity cL=cT√2

9/12/2009

15

Lamé ModesThese solutions exist for any positive value of and the normalized co-ordinates of their representative points are the equally spaced values:

K = m/2 and F = m/ √2,

where m is an odd or even integer for the symmetric or antisymmetric modes. The Lamé line corresponds to F=K √2

Special behavior at =0 rigid solid. (a) Symmetric Lamb modes: segments of the Lamé line belong to successive

modes. The change from mode Sn to mode Sn+1 occurs at the coincidence point of abscissa K=n+1/2 and gives rise to a discontinuity of the slope.

(b) This discontinuous behavior is not observed on anti-symmetric branches.

The acoustic energy is carried at a velocity equal to the projection of the shear wave velocity on the plate axis.

The group velocity cg=d/dk of Lamé modes is equal to cT / √ 2, i.e., half the phase velocity.

Rayleigh waves

9/12/2009

16

Rayleigh Waves: High-Frequency Limit of Lamb waves

iqqandippwherekqqpk

ph

qh

2222 )(4

)2

,0(1)tan(

)tan(

1

1

2

222

2

222

T

L

c

ckq

c

ckp

As k ,

Rayleigh surface wave – frequency equation

For such small wavelengths, the finite thickness plate appears as a semi-infinite medium

1

12.187.0

T

R

c

c

Stonely/Sezawa/Scholte Waves

Particle volcity field components at an interface between polycrystalline Al and tungsten

V’R < VS < V’s

< ’VS << V’s

9/12/2009

17

Additional References

• Karl Graff, Wave Motion in Elastic Solids, Dover (1991)

• J.L.Rose, Ultrasonic Waves in Solid Media, Cambridge University Press (1999)