ELASTIC WAVES GUIDED BY ISOTROPIC LAYERS
Electrical
I-Heng Sun, B.Sc. (Eng.), (Nat. Taiwan)
M.Eng.
ABSTRACT
The properties of elastic surface waves propagatin'g in layered isotropic
media consisting of a fi lm on a substrate are described in this thesis. The generalized
Rayleigh waves, Love waves and Stoneley waves are discussed in detoil, and dispersion
curves of phase velocity, group ve locity, partide displacements and energy flow are
presented for a large number of Rayleigh and Love modes. Special attention is given
to the occurrence of the Rayleigh leaky mode. By means of a right-angle reflector
technique the velocity dispersion of elastic waves guided by thin films has been
measured for the first two Rayleigh modes for a number of material combinations and
film thicknesses. Surprisingly, the Rayleigh leaky mode appears to be easily excited
in these experiments. The results obtoined ore found to be in good ogreement with
the theoreticol predictions.
ELASTIC WAVES GUIDED SY ISOTROPIC LAYERS
I-Heng Sun, S.Sc.(Eng.), (Nat. Taiwan)
ELASTIC WAVES GUIDED BV ISOTROPIC LAVERS
by
I-Heng Sun, B.Sc. (Eng.), (National Taiwan)
Department of Electrical Engineering
McGtII University,
Montreal, Quebec.
€)
l
l-iieng Sim
lm
ELASTIC WAVES GUIDED BY ISOTROPIC LAYERS
by
I-Heng Sun, B.Sc. (Eng.), (National Taiwan)
A thesis submitted to the Faculty of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Moster of Engineering.
Department of Electrical Engineering,
McGill University,
Montreal, Canada.
September, 1'170 .
J
t
ABSTRACT
The properties of elastic surface waves prapagating in layered isotropic
media consisting of a film on a substrate are described in this thesis. The generalized
Rayleigh waves, Love waves and Stoneley waves are discussed in detail, and dispersion
curves of phase velocity, group velocity, parti cie displacements and energy flow are presented for a large number of Rayleigh and Love modes. Special attention is given to the
occurrence of the Rayleigh leaky mode. By means af a right-angle reflector technique
the velocity dispersion of elastic waves guided by thin films has been measured for the
fint
two.Raylel~
modes for a number of moterial combinations and film thicknesses. Sur-
prisingly the Rayleigh leaky mode appears to be easily excited in these experiments. The
results obtained are found to be in good agreement with the theoretical predictions.
ii
t
ACKNOWLEDGEMENTS
The author wishes to express his sincere gratitude to Dr. E.L. Adler
for his guidance and encouragement throughout this project.
Special thanks are due to Dr. G. W. Farnell, for providing the necessary
theory of Chapter Il and useful discussions l to Dr. S. McBride for his valuable
comments i to Mr. T.e. Tan for his helpful suggestions and discussions i and to
Mrs. P. Hyland for her excellent typing.
Thanks are also due to the Defence Research Board of Canada for financial
ossistance •
iii
...
TABLE OF CONTENTS
Page
ABSTRACT
ACKNOWLEDGEMENTS
ii
TABLE OF CONTENTS
iii
CHAPTER
CHAPTER
CHAPT ER
CHAPTER
•
CHAPTER
REFERENCES
.(..
.t\
INTRODUCTION
Il
DISPERSIVE ELASTIC WAVES IN LAYERED MEDIA
2.1
2.2
2.3
2.4
2.5
2.6
Governing Equations and General Solution
Generalized Rayleigh Waves
Love Waves
Stoneley Waves and Their Existence
Energy Flow Calculation
Rayleigh Leaky Modes
4
4
11
18
22
28
33
III
ITERATIVE SEARCH PROCEDURE AND RESULTS
36
3.1
3.2
3.2.1
3.2.2
3.2.3
The Search Techniques
The Computed Results
Dispersion of Surface Waves
Particle Displacement
Energy Flow
39
41
49
62
IV
EXPERIMENTAL· APPARATUS, TECHNIQUES
AND RESULTS
72
36
4.1
4.2
4.3
4.4
4.5
Introduction
Ultrasonic Reflectivity at Liquid-Solid Interface
Measurement Using A Ri~t-Angle Reflector
ether Preparations and Measurements
Error and Precisions of the Results
82
84
V
SUMMARY AND CONCLUSIONS
92
72
72
n
95
CHAPTER 1
INTRODUCTION
Elastic surface wave propagation in layered media is a subject which has
recently attracted interest. This is because there are many possible device applications
as discussed below and alsa because geophysicists have developed mathematical models
which permit the explanation of many seismic phenomena. The simplest model is a system
consisting of a semi-infinite solid substrate in contact with a layer of a different solid
material, and more complicated many layer modes have also been studied [1 ] •
Elastic waves can be utilized in passive electronic companents, especially
at microwave frequencies, such as delay lines, filters, resonators and other signal pro5
cessing devices [2]. Since, in solids, acoustic wavI velocities are a factor of 10
times lower than electromagnetic waves, large delays can be obtained without involving
prohibitively large companent sizes.
The geophysical models explain the transmission of surface waves through
stratified bodies, under some restrictions in density ratio ; the mode 1can interpret, for
ex~~p-I'~', the system of the Eurasian Continent and the Pacific Ocean [3 J.
Lord Rayleigh (4 J, in 1887 was the first to investigate the propagation of
woves along the free surface of a homogeneous, isotropic, elastic solid. Such a surface
permits the propagation of a strai~t-crested, non-dispersive surface wave having displocement components both in the direction of propagation and normal to the surface, now colled
a Rayleigh wave. These waves are choracterized by an energy concentration near the surface and they travel on the surface with a phase velocity which is slower than the bulk
transverse velocity.
2
ln 1935, Sezawa [3], a Japanese seismologist discovered that, in addition
to the Rayleigh surface wave, a second fundamental mode called a Sezawa wave, or the
~ mode, may also propagate in the layered structure.
Tolstoy and Usdin [5] explained
the differences between these two modes. Rayleigh modes correspond to symmetric modes
of a free-plate modified by the contact of thelower surface to the semi-infinite elastic
solid (Ml n mode)
j
the Sezawa modes correspond to a similar modification of the anti-
symmetric free plate modes (~n mode). Stoneley [6 J extended the Rayleigh wave theory
and showed that an elastic surface wave which is analogaus to the Royleigh wave may also
propagate in the interface between two solids. Scholte [7] confirmed Stoneleyls finding
and gave a general algebraic discussion of the existence of interface waves.
Recently, elastic surface wave propagation in anisotropie media has been
theoretically investigated by Schmidt and Voltmer [8]. They studied wave propagation
in Cd S layers on a Fused Quartz substrate. A general numerical method for finding elastic
waves and pseudo-surface waves on an anisotropie crystal was developed by lim and Farnell
[9]. The sarne search method is applied in this thesis. The movable wedge technique of
launching and receiving surface wave signais WQS used by Adkins and Hughes [10] on
guidance structures consisting of thin Gold films deposited on Fused Quartz and a more compiete treatment is presented in their paper. Rollins et al [11, 12 J determined the elastic
surface wave velocity on law-index surfaces by an'ultrosonic reflection technique. The
sarne technique was used in showing the presence of the Rayleigh leaky mode on layered
media by Adler et al [13 J and is 0150 discussed in this thesis in Chapter IV. Mattews
and Van de Voort [14] have generated elastic surface waves (I.o~e modes) at UHF
01'
frequency on a nonpiezoelectric VAG (Yttrium Aluminum, Garnet) substrate. The structure of the transducer and the method of propagation are described extensively in 'fheir paper .
3
ln this thesis, a detailed study of the generalized Rayleigh, Love and
Stoneley waves for isotropic solid materials is carried out. A number of aspects which
have not previously received much attention are discussed, namely
j
the higher mode
structure, the energy distribution, and the problems of leaky waves. An experimental
technique is 0150 described and the results of measurements carried out for loyers of Silver
and Tin on Nickel and Aluminum presented.
ln the first part of this thesis the theory of the generalized Rayleigh, Love
and Stoneley waves is developed. The second part is devoted mainly to a description of
the computer iterative search procedure used and a discussion and summary of the computed
results obtained. The third part gives a description of the experimental work corried out,
the technique of measurement and also compares the results obtained with theoretical predictions. The summary and conclusions are stated in the finol section.
4
CHAPTER Il
DISPERSIVE ELASTIC WAVES IN LAYERED MEDIA
A theoretical analysis of the wave equation for the propagation of elastic
surface waves on free and layered surfaces is reviewed in this chapter. The formulation
of the generalized Rayleigh, Love and Stoneley wave solutions which are carried out in a
manner suitable for numerical calculation will be discussed extensively. Finally, in the
last two sections, bath energy flow and Rayleigh leaky modes are discussed.
2.1
Governing Equations and General Solutions
For isotropie and anisotropie media, the generalized Hook1s Law for homo·
geneous, non-dispersive media is given by :
=
Ciikl \1
T..
'1
=
the stress tensar ,
'1
where
( i, i, k, 1 = l, 2, 3).
T..
=
C
iikl
Ski
=
(2.1 )
the elastic stiffness tensor,
the strain tensor •
Note: Cartesian tensor notation and the summation convention for repeated indices ;s
used throughout.
Since the layer and the substrate are homageneous, the elastic tensor elements
ore constant and sotisfy the symmetry relation
....
= ~Iii •
(2.2)
5
The independent constants are therefore reduced from eighty one to twenty one for anisotropic media [15]. This number is. further reduced if the crystal has axes or planes of
symmetry, i.e. there are only two independent constants for isotropic media. The strain
tensor relating to the parti cie dlsplacement vector U is given by
\1
1
=
aUk
2' ( TXj +
oUI
0Xk),
(2.3)
Where U., are the displacement components measured along the Cartesian axes x. to
1S
1
which the elastic tensor is referred .
According to Newton's Law of motion, the equation of motion for a material
of density
J and neglecting the body force is given by
or ..
_"
a
(2.4)
x.
1
The left-hand side of the above equation is the mcss per unit volume times the acceleration,
and the right-hand side is the force applied on a unit volume. Equations (2.1) and (2.3)
substituted into (2.4) gives the equation of motion or wave equation
(2.5)
C
iikl
OxiOx I (i,i,k,1 = 1,2,3).
The above equation applies to a perfectly elastic, homogeneous, anisotropic medium and
also in the absence of body forces and piezoelectric effects, i.e. it is 0150 certainly volid
for isotropi c media.
ln the following, the general solution of the surface wave will be discussed for
the free surface and forthe layer on a half-space.
6
The coordinate, system for the free surface-wave problem of interest here will
be taken with x as the outward normal to the traction-free surface of the medium which
3
occupies the half-space x
3
~
0 shown in Figure 2.1. The axis xl is chosen in some
convenient direction in the surface as the propagation direction. For the layer problem,
the coordinate origin is chosen on the interface as shown in Figure 2.2.
Unattenuated solutions of the wave equation are sought which decay with
depth from the surface or interface and which are "Straight-crested Il in that there is no
dependence of the displacement at any depth on the distance measured in the x direction
2
perpendicular to the sogittal plane (xl and x plane of Figure 2.1 and Figure 2.2) .
3
That is, the solutions are written in the form
(2.6)
or compactly
(2.6)
U. = a.exp [ik (l.x. -vt)]
1
1
1 1
where i = 1, 2, 3 and 1 = 0 ,
2
which simultaneously sotisfy the wave equation.
wr.ere
U. =
Component of displacement along x. ,
'"' =
a.
the eigenvector along axis x. ,
k =
the wave number = 2. / ). ,
ikl3 =
depth decoy constant,
1
1
v
=
1
1
..
the phase velocity along k •
Equation (2.6) substituted into (2.5), gives
7
o
.
.
FIGURE 2.1. COORDINATE SYSTEM FOR THE SURFACE WAVE •
8
h
Direction of
_____~-_r---......-~r___---. . . .io_-Propogation
)
FIGURE 2.2. WAVE - GUIDE GEOMETRY.
9
Sv2) a k = 0
i.e.
( rjk
where
rjk
= li Il Cijkl
and
6
=
jk
- 6 jk
C
(2.7)
fOf
-1
k
foc
=
k•
Equation (2.7) has a non-trivialsolution only if the determinant of the coefficients vanishes ,
that is, the secular equation
(2.8)
To be called surface or layer waves, the quantity 13 in each of the terms of the solution
must be such that the amplitudes of ail the displacement components vanish as x .. 3
IX) •
Conceptually, the x - dependence is regarded as part of the "ampIl tude Il of the term
3
and the wave-like properties are taken to be contained in the common propagation part of
the terms. Thus the propagation vector is always assumed to be parallel to the surfoce even
though there may be a real part in the 13' Similarly 1 the planes of constant phase are
perpendicular to the surface and to the so-defined propagation vector 1 but the amplitude
of the term varies in the x - direction over a plane of constant phase •
3
ln an isotroplc salid, the only non-zero terms in the matrix form of the stiff"I!:i;
constants are
Cil - C22 = C33 ,
and
'Ïnd
C44 = CSS
C12
= C21
1
= C
= C66 = 2' (Cil
I3 = C31 = C23 = C32
- CI_
2)
when substituted for the appropria te constants in Equation (2.7) 1 gives
10
Sv2
o
o
o
2
B-J v
o
o
o
A-
where
=
A
C -
a
2
= 0 (l.9)
J v2
212
Cll + C44 13 = Cll + 2 (C ll - C12 ) 13
212
B = C44 + C44 13 = 2 (Cil - C ) (1 + 1 )
12
3
212
C = C44 + Cll 13 = ~ (C ll - C12) + Cil 13
o
1
= (C 12 + C44 ) 13 = 2(C + C ) 13'
ll
12
To stress the separation into two types of solutions, Equation (2.9) can be rewritten as
2
A- j v
0
0
2
j v
C-
0
0
0
al
0
a
3
J
B-
2
v
= 0 • (l.10)
a
2
For Equation (2.10) to have a non-trivial solution requires that the determinant of the
coefficients vanish, i.e" the secular equation equals zero.
~
A - J v
IMI
"',
<1.
=
0
2
0
0
C
-j
2
y
1
a
1
a
a
1
B - j y2
=
o.
(2. 11 )
11
2
ln the case where 13 = 0, which produces a cubic equation in v , the three roots are
the squares of the velocities for the three bulk waves. The eigenvectors of the three
solutions form a mutually perpendicular triad, one displacement vector being longitudinal
and the other two be ing transverse.
Clearly, the condition of Equation (2.11) is sotisfied when either the upper
subdeterminant or the lower subdeterminant of Equation (2.11) is zero. This implies that
there are two inclependent sets of solutions. One of them will-give the solution where the
particle displacement is confined to the sogittal plane while the other is in the transverse
direction perpendicular to the sagittal plane. These two indepenclent solutions will be
defined later on as generalized Rayleigh and Love waves. Adetailed discussion of these
solutions is given in the next two sections.
2.2
Generalized Lossless Rayleigh Waves
The generalized Rayleigh waves are surface acoustic waves that propagate in
a wave guide composed of a thin -film in perfect mechanical contact with a substrate having
different acolAtic properties whose amplitudes decrease rapidly with distance fram the interface.
The propagation of the generalized Rayleigh wave is taken along the xl axis,
the displacement having cOf'lllonents bath in the direction of propagation and narmal to the
surface, i.e. xl and x direction. The generalized Rayleigh waves will be termed
3
simply the Rayleigh waves in the following sections. The wave-guide geometry is shown
in Figure 2.2 •
12
~
mentioned in the preceding section, it is clearly seen that Rayleigh waves
exist when the upper subdeterminant of Equation (2.11) is equal to zero, i.e.
=0
(2.12)
122
2" (C ll - C12) + Cll IJ - jv
which simplifies to
Equation (2,lJ) is regarded here as a quadratic equation in IJ with the
phase velocity v as a parameter. The four roots are expressed as IJ"), where
n = 1, 2, J, 4 and the Ir)are either real, or occur in complex conjugate pairs.
For the isotropie substrate, the roots Iying in the upper-holf of the complex
.
plane correspond to waves which become infinite os xJ - - CI), these solutions of the
wove equation can be ignored because they do not satisfy the stipulated condition thot the
displocements vanish at large depths. In general, there are two lower -holf -plane (negative imaginory port) roots which give propagating terms satisfying the conditions at infinite
depth.
From Equation (2.1J), the lower-half-plane rootswill be:
2
I~) = - i J( 1 - v/ 2 )
J
v
t
1-
.e.
2
1(2)
J(1
v
/ 2)
-i
J =
vI
(2.14)
13
v = [ (C - C ) 12 j
ll 12
t
where
1j ]
vI = [ C11
]
1/2
1/2
= [C 44/j ]
1/2
ore the velocities of the transverse and longitudinal bulk waves in the substrate,
respectively.
Substituting the new!y found values of
I~n) into Equation (2.7),
the eigen-
vectors corresponding to the lower - half -plane roots of Equation (2.14) ore chosen os
follows :
(1 )
_ 1(1)
al =
3
(1 )
O
2
=
0
(1)
0 =
3
Note the se ore perpendicular, i.e.,
0(2)
1
=
0(2)
2
=
0
0(2)
3
=
1(2)
3
~ (1)
(2.15)
• êi (2) = 0 •
From the above, a linear combination of terms of the form given in Equation
(2.6) con be formed to obtoin a general solution for the complete set of the displacements
in the substrate :
2
U
j
[•
(n)
]
L Cn a(n)
j exp 1 k (Xl + 13 x - v t) , j =1, 3 •
3
=\
(2.16)
n=1
Where C ore wei~ting factors which will be determined From the boundary n
".
condi tians.
14
Similarly, the layer subdeterminant is set equal to zero, and we obtain the
ge ne ra 1solution for the displacements in the layer.
6
" _ \'
(n)
[•
(n) _
]
U - L Cn 0j exp 1k (xl + 13 x3 v t) ,
j
n=3
We take 011 of the four roots in the layer, where
where
(2.18)
= -i
1(5)
3
=
i J(1 _ v /.;2) = _ 1(3)
t
3
1(6)
3
=
i J (1 _ v / v2 ) = _ 1(4)
.3
l
J(1 _ v / v21 )
2
2
"
"
,. 1/2
v" = [(C - C )/2j]
I2
ll
t
" =
(2.17)
I~n) are as follows :
2
1(4)
3
VI
j = 1-, 3.
"
• 1/2
/] ]
44
= [C
"
• 1/2
[C / J ]
ll
are the volocities of the transverse ~nd longitudinal bulk wave in the layer, respectively.
And also the eigenvectors corresponding to the roots of Equation (2.18) ore written os
follows :
1
=
_ 1(3)
3
°1
0(3)
=
0
°2
0(3)
2
(3)
°3
=
0(5)
=
1
(4)
(4)
(4)
°3
1(3)
3
(6)
°1
=
=
0
=
1(4)
=
3
(2.19)
15
(5)
°2
=
(5)
°3
=
(6)
0
°2
(6)
°3
=
0
=
_1(4)
3
The expressions for the parti cie displacements given in Equations (2.16) and
(2.17) show the exponential de..crease in amplitude with depth in the x - direction, and
3
the parti cie motion is elliptically polarized. It con also be shown that the skin depth or
depth of penetration of waves decreases with increasing frequency in the same manner os
the skin depth behaviour of electromagnetic waves.
Having already discussed how to obtain the general solution for the complete
set of the displacements in the layer and the substrate, it remains to consider the boundarycondition for the Rayleigh waves. The problem is now to determine the weighting factors
C (n =1, 2, 3, 4, 5, 6) from the boundary condition.
n
For the interface, the displacements in the layer and in the substrote must be
continuous, thus
i = l, 3 .
U. = U.
1
1
(2.20)
The stress must also be continuous in the interface, thus
T3i = T3i
"'.
' at x3 = 0
i = 1, 3 .
(.L21)
The last boundary condition is thot the stress at the traction -free surface of
the layer must he set to zero, i.e.
16
T = 0
3j
= l,
(2.22)
3.
. The weighting factors C can now be found from these boundary - condition
n
equations, (2.20), (2.21) and (2.22) which can be written [8] [C ] = O. The
n
secular determinant 1 8 1 formed by the weighting factors in the se equations must be set
to zero in order to have a non-trivial solution. The boundary - condition determinant is a
6 x 6 determinant, it can be simplified to Equation (2.23) , which is shown be low.
The roots of the transcendental determinant (Equation (2.23) governs the
propagation of the Rayleigh waves. It can be seen that the phase velocity depenos on the
wave number, the waves are thus dispersive, and Equation (2.23) is henceforth referred to
as the dispersion relation which results in an infinite number of modes with increasing layer
thickness or frequency. The first mode is the fundamental Rayleigh mode. When the
layer is very thin, this wave has displacements and phase velocity characteristics of a pure
Rayleigh wave on the substrate. Moreover, "there is a limiting linear relationship between
the phase velocity and wave number, that is, the velocity vs. wave number for the fundamental Rayleigh type wave is linear for small wave-numbers" [16]. The second mode is
the fundamental $ezawa mode which was first discovered by Sezowa. The fundamental
Sezowa mode and ail hig,er order modes exhibit a cut-off behoviour at the shear wave velo·
city of the substrote. Ali of the above assumes that the layer loads the substrate, i.e., the
velocity of shear waves in the layer is less than the velocity of shear waves in the substrate.
On the 'other hand, no higher order modes exist if the velocity of shear waves in the layer
is geater thon the velocity of shear waves in the substrate or the film stiffens the substrate
·>
[1 J.
_ 1(1)
3
1(2)
3
C
44
(1
2 C 44
- 1(1)2)
3
1 (1)
3
2 C
44
1(2)
3
(2)2
CIl 13
o
o
o
o
-
+C 12
- ê 44
1(3)
3
- 1
1
_ 1(4)
3
(1-
_ 2 CA 1(3)
443
ê
44
2
[1 -
ê44
_ 2
(3)2)
3
If)2]eikl~)h
1(3) ikl (3)h
3 e
3
ê
_ 1(3)
3
-ê
1(4)
44 3
ê
1(4) ikl (4)h
44 3 e
3
12 ]e
44
2
ê
[ê 11 1(4)2+
...
3
C
1(4)
3
- 1
(4)2
- C 11 13
- C 12
2
- 1
2
(1 _ 1(3)2)
3
ê
ê44 1(3)
3
- C 11 13
(1 -
_ 2
...
(3)2)e -iklf)h
(4)2"
- C 12
ê
44
ikl (4)h
3
1(4)
44 3
-2ê
1 (3) -ikl (3)h
44 3 e
3
[ê
44
L(4)2 + C
11"'3
1(4) ikl (4)h
e
3
]
-ikl (4)h
12 e
3:
o .
(2.23)
......,
lB
The group velocity of Ray:eigh waves like any other wave is obtained from
the basic relation,
(2.24)
where v is the Rayleigh wave phase velocity and k = w / v •
R
R
2.3
Love Waves
Love waves are
0150
surface acoustic waves that propagate in a wave guide
composed of a loyer on a half-spoce hoving different acoustic properties. These waves
are transverse ond they bri ng only shear stresses into action. The particle displocement
vector is perpendicu!ar to the propagation direction xl ond is oriented in the x direc·
2
tion, see Figure 2.2, or, in other words, it possesses one displacement component, parollel
to the surface and perpendicular to the direction of propagation.
From the definition, it is c1ear that Love waves exist in the case where the
lower subdeterminant of Equation (2.11) is equal to zero, i.e.
(2.25)
Equation (2.25) is regorded here os a second order equation in 13 with the phase velocity
v os a parame ter whase two roots con be expressed 05
....
I~n),
where n =l, 2, and the
I~n)
ore either real, or 0 eomplex eonjugote pair .
For the isotropie substrate, we ehoose the lower-half-plone root for the some
rcasan os the generolized Rayleigh woves. From Equation (2.25), the lower-holf-plone'
19
root will be :
(2.26)
I~) into Equation (2.7)
Substituting the value of
the eigenvector corres-
ponding to the lower-half-plane root of Equation (2.26) will be taken as :
(2.27)
From the above, we can obtain the displacement component in the substrate
for the Love wave •
(2.28)
where Cl is also the weighting factor • Similarly, we can obtain the displocement
solution for the layer:
(2.29)
The two layer roots are as follows :-
(2.30)
1(3) =
3
2
i J(1 _ v /
il )= _ 1(2)
t
3
And the eigenvectors corresponding to the roots of Equation (2.30) are taken os :
0(3)=1.
2
20
The boundary-condition for the Love wave problem will be simpler thon the
generalized Rayleigh wave. Since there is only one displacement component which is
perpendiculor to the sagittal plane, for the interface, Equation ~.20) will be simplified
as
at ><'3
and Equation
~.21)
=0
(2.31 )
will be simplified os
~.32)
at x = 0 •
3
The stress at the traction-free surface will be
f32
at x = h •
=0
(2.33)
3
The weighting factors Cl ' C , C can be found from the boundary2 3
condition equations (2.31), (2.32) and (2.33), the determinant formed by coefficients
of the weighting factors must be set to zero in order to have a non-triviaholution. The
boundary-condition determinont is a 3 x 3 dt!terminant, it con be written :
-1
-1
C I~')
44 3
- ê4431(2)
- C44 13
0
.. (2) ikl (2)h
C 13 e 3
44
.. (3) ikl (3)h
C 13 e 3
44
;f "
4'
Since If) = - If)
Equation (2.34) can be simplified to :
..
(3)
- o.
(2.34)
21
--- ."'
~-
- 1
- 1
C I~)
443
-ê
" 1(2)
C44 3
0
ikl (2)h
e 3
-e
1(2)
44 3
=0
-ikl (2)h
3
or
(2.35)
We may write Equation (2.35) in the form
(2.36)
Equation (2.36) is referred to as the dispersion relation of the Love wave which consists
of an infini te number of modes due to the periodicity of the tangent function. Furthermore,
Equation (2.36) has a real value only when the transverse velocity in the substrate is greater
thon the transverse velocity in the layer. Therefore, a necessary condition of existence for
Love waves is that the propagation velocity of transverse wave in the layer must be smaller
thon the propagation velocity of transverse wave in the substrate [17] •
When the higher order Lave wave velocity equals the shear-wave velocity of
the substrate, that velocity is the cut-aff velocity, since v = v ' Equation (2.36) can be
t
sirrplified to
(2.37)
22
Equation (2.37) is the cut-off relation for the Love wave. It can also be seen that there
are an infinity of cut-off points due to the periodicity of the tangent function.
From Equation (2.37), we obtain the cut-off frequency as follows :
where n = 0, 1, 2, ••••
nv
f
t
co
=--2~--
v
2 h.j(~
(2.38)
- 1)
v
t
The group velocity of Love waves is obtained by differentiating Equation (2.36),
which leads to
dw
w
where k = -
v
2.4
(2.39)
= IT =
v
g
w
k =t
vt
Stoneley Waves and their Existence
Asurface wave may propogate olong the interface plane between two semi-
«)
~)
infinite solids of different elastic properties in contact. Such waves are called Stoneley
woves
r6J. Stoneley waves con exist only if the mate rial properties of the two semi-inflnite.
23
T
solids satisfy certain restrictive conditions which wi Il be discussed in the following,
Cansider Iwo semi -infinite media that ore in contact along the plane x :1: 0
3
of Figure 2.2. The general expressions for the displacement compone nt and stresses ore a5
for Rayleigh waves. Since the wave propagates along the interface between th\!
IWO
solid
media, the boundary conditions of Equations (2.21) and (2.22) ore still valid.
Equations (2.21) and (2.22) result in four homogeneous equations for the
weighting factor C (n = l, 2, 3, 4). In order that this system have nontrivial solutions,
n
the boundary-condition determinant of the coefficient must vanish, i.e.
_ 1(1)
3
1(2)
3
C
44
n- 13(1)1
2 C 1(2)
44 3
-1
- 1
_ 1(4)
3
- ê44
(2)2
C 1 +C
11 3
12
2 C 10)
44 3
_ 1(3)
3
2
=
0
ê
1(4)
(1 - 1(3)1 2
3
44 3
ê
1(3)
44 3
,. (4)2"
- Cll 13 - C12
(2.40)
whcle
1(1)
3
=
- i -Al
2
v
5/ 2 )
Vt
2
1(2)
3
=
- i ytl
- v5/v~ )
2
1(3) 3 -
_ 1(5)
3
=- i
./(1 -
~/v2)
t
2
of'
04>
1(4)
3
=
v
_ 1(6) = - i ./(1 _ 5/ 2)
vI
3
24
V
s
is the phase velocity of Stoneley wave and
V is the velocity of the upper-half medium.
Since the wave propagates along the boundary and decays in bath media, we must have
v <v
s
t
and
v
5
<v .
(2.41)
t
ln the above relations, the roots are chosen based on the fact that we take the negotive
imoginory part for the lower-half medium and the positive imaginary part for the upperhalf medium in order to satisfy the boundary-condition requirement that ail displacement,
components should vanish at infinite distance from interface. Clearly, Equation (2.40)
is most difficult to interpret. Stoneley [6] considered the case where the two media were
incompressible in order to determine the nature of the roots of Equation (2.40). He resorted to the special case where vI ' v '
t
vI
and
vt for the two media were approximately
equal, i.e.
v ..,
t
V1
t
and
leijing
this transforms the left hand side of Equation (2.40) into a function of X. Thus a root of
the cquation f (X) / X2 = 0 always exists betweerî X = 0 and X = 1. In other words,
a positive reol root always exists for the quantity Vs / v ' This root is alwoys less thon
t
unity, indicating thot the velocity of the Stoneley wove in the interface is less thon that of
the shear waves in either unbounded media as Equation (1.41) mentioned. Since that time,
~veral
other papers have been published giving further investigation of this boundory deter-
25 .
minant equation [7, 18, 19]. Schotle [7] using a general algebraical treatment of the
~undary-condition
determinant, gives Iwo equations :
(~ N)2t(2 - 0)2 - 4 J (1 -)"0)(1 - o)} - (~/~'){2 (2 - 0) - 4 J (1 -'0)(1 - 0)
(1.42)
and
where
c.J
= (r.)
P
P=
2
JI
~
,
V'= I+2ii
pl =
1
~I
_
V -
r+Tjj1
~I
J:r
1
Note that }. = C , ~ = 2 (CIl - ).) are Lamé constants,
12
is the density and prime
represent the upper-medium porameters.
Stoneley waves con exist for every value of
~ 1~I
and
JIJI
which 1ies
hetween the two curves of Equations (1.42) and (1.43), drown in Figure 2.3. The
Iwo curves ore close Iy connected with each other
fll
j
if we change
into each other 1 Equations (1.42) and (1.43) will
0150
J, ).,
~
and JI,
). 11
interchange. Curve A hos
therefore the same connection with the upper medium os curve B hos with the lower medium.
$ezowo and Kani [18)
0150
obtained these curves by numerical co~utation.
26
,
3
A
8
2
2
3
FIGURE 2.3. STONELEY WAVE BOUNDARY CURVE ( >, /!J = >,I/!JI
:f,
1) •
27
Achenbach [1] investigated the existence of 5toneley waves for bath welded
and smooth contact between the two media. For smooth contact, the baundary-condition
determinant can be simplified to :
where
RI = (2
R =
2
_,,~2 -4 (1 _ E~
,,2) 1/2 (1 _,,2) 1/2,
(2 - K2 ,,2) _ 4(1 _ E~K2,,2)1/2 (1 _ K2 ,,2) 1/2 ,
v
17
=
G
=
.;. = dimensionless phase velocity ,
v
t
shear modules,
.. 2
.. 2
v
v
2
t
2
t
K =
G1 = :2
Tv
vI
t
2
v
2
t ,
G2 = "'1
vI .
and 1 is the layerls porameter •
When A > 0, thef.. is a root in the region 0 < 17 < 1. Moreover, because bath RI
ss
and R are monotonic in ", there is only one root in that region. Conversely, if
2
A < 0 , there are no roots ot ail, and 5tone ley waves are not possible. For welded conss
;
2
tact, by setting 51 = -AI - s/v 1= 0, the boundory-condition determinont is :
t
G
A5b = P (
where
G2 2
Q (" = 1) + (G") R (,,= 1)
2
1
I
,,= 1) - -G2
P =
Q =
52 =
1 - ql 52 '
2
(I+S )(2-q I 51) -4q252+2ql 52'
2
2
vs/,,2
JI _ vS/v 2
ql = JI VI
t
(1.45)
28
Equation (2.45) will also give the limitin~condition for existence of Stoneley waves for
welded contact which is analogous to Scholte's result.
Thus the main condition for the existence of Stoneley waves is that the velocity
of shear waves for two media be nearly equol. Table 1 [t 9] shows some material combinations for which Stoneley woves are possible.
Since Stoneley waves are the special case for Rayleigh waves when the layer
thickness to wave-Iength ratio is very large
j
for K < l , the phase velocity of the
lowest mode then approaches the velocity of Rayleigh surface waves in the layer. The
second mode approaches the velocity of Stoneley waves, if Stoneley waves can exist for the
combination of material praperties$hich are considered. This is due to the condition of
Equation (2.41). If the Stoneley waves cannot exist, the phase velocity becomes in the
high frequency limit the velocity of shear waves in the layer.
2.5
Energy Flow Calculation
The transport of energy for the elastic surface waves can be specified bya
vector whose components are [4].
E. = (R eT .. ) (R e O.)
1
Il
1
(2.46)
which gives the energy flow per unittime per unit area across a surface normal to this vector.
For the Rayleigh waves case, when the time dependence type solution is
ossumed in Eqvation (2.6) , the time-overage of the above vector is [4]
TABLE
1.
THE
FOR
STONELEY
WHICH
A
PARAMETER
STONELEY
FOR
WAVE
THOSE
IS
COMBINATIONS
POSSIBLE
Medium
(2)
-~----snear-wave
Medium
Aluminium
(1 )
Antimany
Cabalt
Capper
Magnesium
Nickel Tungsten
Steel
Alnica Duralumin O.08C
0.991
Aluminium
0.997
Antimany
0.992
Magnesium
2140
0.987
0.997
Nickel
2970
0.986
1728
0.982
Rhodium
0.998
0.992
Titanium
0.968
Tungsten
3005
0.982
0.996
2088
0.947
0.996
0.965
2810
0.997
Zinc
2220
0.996
Alnico
0.992
3033
0.991
Beryllium
2395
0.998
Duralumin
Monel
0.997
Steel O.OSC
0.988
SteeIO.38C
0.987
Note
3020
0.993
0.999
Platinum
3185
2897
0.999
Copper
Velacity in
Medi"", (1)
m / sec.
2108
0.973
0.995
Cobalt
Steel
O.41C
0.977
0.999
0.995
The velocity of the Stoneley wave is the shear wave ..elocity multiplied by the apprapriate factor.
3065
2785
3118
3172
~
-0
30
*}
{ •• U.
•
P. = • ~1 ReT
I/.II
(2.47)
1
which is analogous to the "time-averaged Poynting ve.ctor" of elp.ctromagnetic theory
and will be referred to here as the "Poynting vector".
From Equation (2.16), we get
.* .
\'
*
U. = 1 k v L C a.
m
1
(m) *
m 1
e
- i k 1 (m) *x
3
3
(2.48)
=
I
m
• - i k 1 (m)\
A1 e
3
3
m
where Ai = •1 k V C* a.(m) *
m
m 1
Equation (2.3) substituted into (2.1) gives
1
T..
11
=
2 Cijkl
=
i kC
ijkl
=
S~
LCn a~ 1; e ikl;nl x3
n
\' . 'kl (n)
) Si el 3 x3
L
n
where
aUk aui
( iij" + rr~)
= i kC
(2.49)
n
ijkl
C
n
a~ I~
•
Equations (2.48) and (2.49) substituted into Equotion (2.47) , thus gives the Poynting
vector fotthe substrate of the generalized Royleigh -Noves:
IA~~eikx3(I~n)-I~m)*)
Pi = ReL
n, m
(2.50)
31
C C* (m) * (n) (n)
where AQ) = 2"1 k2 v C
a k al
nm
ijkl n m aj
n, m = 1, 2
i
= 1, 2, 3 •
For the isotropie layer problem, P = P = 0 at ail depths, thus the power flow in the
2
3
substrate of the Rayleigh waves is :
W1 =
Jo Pl d x3 = ReL'""' '\L Anm(1)
-al
= Re
JO
ikx (1 (n) _I(m)*)
e
3 3
3
d x3
n,m-al
LI
(2.51 )
n,m
n, m = 1, 2 .
Similarly, we eanobtain the Poynting veetor in the layer:
(2.52)
0 < x s: h
3
where
(I)
1 2
(m)* (n) (m)
Anm = 2' k v Cijkl Cn C; ai
ak al
A
n, m = 3, 4, 5, 6
and the power flow in the layer·
o
(2.53)
n, m
n, m = 3, 4, 5, 6 .
32
l'
The IIcross-termsll,
min, in the summations of Equations
(2.51) and
(2.53), that is the products coming from different terms in the displacement of Equations
(2.16) and (2.17) are non-zero. In theJsotrqpic lossless case, the wave is only propagated in th,! xl direction, therefore W '
1
W1 are the only non-zero terms.
The ratio of W1/W 1 gives energy flow in the layer as compared to that
in the substrate. The direction of energy flow is parai lei to propagation vector. Its
amplitude decreoses exponentially with depth.
The Love wave energy flow is similar to the generolized Rayleigh woves, and
these woves are transverse. Since the Love wave energy flow is treated in here by the
some procedure os the generalized Rayleigh woves, the solution can be obtoined os follows :
Substrate Energy Flow:
(2.54)
Layer Energy Flow :
(2.55)
"
W1/W gives the loyer to substrate energy ratio of the Love waves. The propagation
1
energy is located in the loyer and in that part of the substrate thot is close to the surface.
115 amplitude also decreases expanentially with depth.
33
2.6
Rayleigh Leaky Modes
Rayleigh leaky modes are elastic surface waves which are attenuated in the
direction of propagation. Their phase velocities are higher than the transverse wave
velocity, i.e., above the cut-off velocity. The presence of leaky modes has been demonstrated in the experimental measurement which will be discussed in Chapter IV.
The computation procedure is modified by adding on imoginary port to Il of
parti cie displacement compone nt corresponding to an attenuation of the wave in the direction
of propagation.
i.e.
(2.56)
Il = 1 + j a
where a k is the attenuation constant.
Hence, the partide displacement solution for the substrate of Equation (2.16)
will be written as :
2
U =
j
LCna~n)exp [ik (11 xI + l~n)X3·vt)J,
j=l, 3
n=1
where Il is defined as Equation (2.56), the roots
a ~n) ( i = l, 2, 3) will be as :
1
I~n)(n = l,
2) and the eigenvectors
(2.57)
34
,.
1-
1(1)
3
=
1(2)
3
=
·2
_ i J(l2 _ v /
1
t
2
~ i J(12 _ v /
1
1
(1)
al
=
_1 (1)
3
a (2)
1
=
Il
(1 )
a
2
=
0
a(2)
2
=
0
(1)
3
=
Il
(2)
a
3
=
1(2)
3
a
l)
l)
(2.58)
Similarly, the particle displacement solution of the layer of Equation (2.17) will 0150 be
modified as :
6
Ûj =
L cnat) exp[ik (11 Xl
+
l~n)'x3
- vt)]
n=3
(2.59)
j =l, 3
The roots 1~n) ( n =3, ••••• 6) and the eigenvector a
t) (
j = ), 2, 3) being
modified as follows :1(3)
3
=
2
-i J(12 _ v /v 2)
)
t
, (4)
3
=
_i
a (3)
1
=
a(3)
2
1(5)
3
=
_ 1(3)
3
1(6)
3
=
_ 1(4)
3
_ 1(3)
3
(4)
a)
=
1)
=
0
(4)
a
2
=
0
a(3)
3
=
1)
(4)
a
3
=
1(4)
3
(5)
a)
=
1(3)
3
(6)
a)
=
1)
=
0
O
=
.0
=
1)
(5)
O
2
0(5)
3
2
J( ,2 _ v / v2 )
1
1
(6)
2
, o~6)
=
_ ,(4)
3
(2.60)
35
The roots Ir) of Equation (2.58) will no longer be purely negative
imaginary. In fact, they can have real.and positive as weil as negative imaginary components. Since lm (II) are < < l ,
I~)
is essentially unaffected, is still imaginary
if v < vI ' and corresponds to a layer wave term in the displacement solution given by
Equations (2.16) and (2.17), with an attenuation constant given by ka.
I~) however
has a significant real part corresponding to a wave term in the displacement solution having
a wave port of the form [20]:
exp [i k Re
(Il ))'~ + i k XI]
~.61)
We can see that this is a wave tilted at an angle tan -1 Re (If)) to the Xl' x
2
plane radiating into the substrate and propagating with a velocity v ' the bulk transverse
t
velocity.
The important point about this case is that the positive imaginary compone nt .
is a growth term which represents the extraction of energy from the
surfac~ Jnto
the bulk
wave. This part of the wave carries the energy flow oway from the surface, ond radiates
into the bulk wave in general.
36
CHAPTER III
ITERATIVE SEARCH PROCEDURES AND RESULTS
3.1
The Search Techniques
Equation (2.23) is an implicit function of v Oayer wave velocity) and hk
(thitkness x wave number). It is complex and cannot be reduced to an explicit form.
Due to the fact that there is no explicit expression for the layer velocity, a numerical
~!larch
technique is necessary to find out the velocity in a layered half-space system. The
advent of high-speed digital computers made the numerical solution of the general directions
on isotropie and anisotropie layered problems more practical. The search technique used
is es:;entially the sa me as that of Lim and Farnell (9, 21]. The general computer strategy
lies in choosing successive values of phase velocity in the secular equation of Equation (2.8)
unti 1one is found for which the boundary-condition determinant vanishes. This approoch
has produced a solution of the form of Equations (2.16) and (2.17) for each of a very wide
ronge of conditions investigoted. This range is divided more or less at fixed intervols and
the magnitude of the boundary condition determinant is plotted by the computer a~ a function
of assumed velacity. The Flow charn used in finding the Rayleigh, Love and Stoneley
waves are illustrated in Figures 3..1 and 3.2 , respective Iy. The fundamental mode wi Il
uluolly lie between the independent transverse wave velocity of the two media. The detailcd search mokes use of the modified Golden Section [22], whose definilioh is "division
of a segment into two unequol parts so thot the ratio of the whole to the larger is equal to
the ratio of the lorger to the smoller". If R is the ronge and X is the length From one
end, then
R
X
X = R=X
(3.1 )
37
SUBSTRATE C ,
11
_LAYER
..
CIl'
C ,v'
CALCULATE . BULK
12 t
..
.. h,k.--.+--I WAVE VELOCITY
C12 , vt
vI AND vI
ASSUME v IN
SECULAR EQUATION
ASSUME v IN
SECULAR EQUATION
r-d
14-~_.....,
Il AT!'
ROOTS
CALCULATE
EIGENVECTORS
APPL YBOUNDARY
CONDITION TO
SET - UP 9 x 9
APPLY BOUNDARY
CONDITION TO
SET - UP 6 x 6
..,Il..
MODIFIED
GOLDEN SECTION
Illt
EIGENVECTORS
DETERMINANT
EVALUATE
BOUNDARY
CONDITION
DETERMINANT
~04
DETERMINANT
EVALUATE
BOUNDARY
CONDITION
DETERMINANT
NO
~S
CALCULATE AND PRINT
ROOTS, COEFFICIENTS
DISPLACEMENTS AND v
R
CALCULATE AND PRINT
ROOTS, COEFFICIENTS
DISPLACEMENTS AND v
R
FIGURE 3.1. THE RAYLEIGH WAVE SEARCH PROCEDURE.
MODIFIED
GOLDEN
SECTION
38
GIVEN : C , C12 , vt '
ll
..
..
..hk
Cll,CI2,Vt
I----..t
CAlCULATE BUlK
WAVE VElOCITY
vI AND vI
12 -- 0
ASSUME v IN
SECUlAR EQUATION
CALCULATE ROOTS
CAlCUlATE
EIGENVECTORS
MODIFIED
APPl Y BOUNDARY
CONDITION TO
SET - UP
GOLDEN
SECTION
DETERMINANT *
EVALUATE
BOUNDARY
CONDITION
DETERMINANT
IS
NO
YES
CAlCULATE AND PRINT
ROOTS, COEFFICIENTS
DISPLACEMENTS AND v
* For love waves: 3 x 3 Determinant. For Stoneley waves: 4 x 4 Determinant.
FIGURE 3.2. THE LOVE AND STONELEY WAVE SEARCH PROCEDURE.
39
Solving Equation (3.1) gives
x
= 0.618 R
The reduction of the search interval is such that after K iteration, the interval R is
K
given by
(3.2)
A flow chart for the Modified Golden Section is shown in Figure 3.3. Note that the
procedure stops if the correction to the velocity is less than the prescribed tolerance.
The program was written in FORTRAN IV and used single precision on an
IBM 360/75 computer. The number of iterations required to find the phase velocity
depends on the required accuracy of the velocity. For example, a Rayleigh wave velocity
calculated to four significant figures requires about 15 iterations. The time taken to find
a Rayleigh wave velocity depends on the accuracy of the 13 roots of the quadratic Equation
(2.13). In general, the velocity is calculated to an accuracy of four significant figures.
This is justified by the fact that most of the elastic constants tabulated are given to this
degree of accuraey [23]. On the IBM 360 /75 computer, the execution time taken
for 15 iterations is about 3
l'V
5 seconds for the Rayleigh wave velocity. The computa-
tion time for the bulk wave velocities, eigenvectOl's, the weighting factors, displacements,
et cetera, is negligible compared to the time taken by the velocity search process.
3.2
The
C~uter
Results
The existing computer pro~am con be used to obtoin numericol solutions for
any combination of moterials fOI' any of the Rayleigh and Love modes. A cO"1llete illustra-
40
i
••
GIVEN
XI, X 2
RANGE 0 = X 2 - X 1
CALCULATE
X 3 = X 1 + 0.3820
X 4 = X 2 - 0.3820
EVALUATE 80UNDARY
CONDITION DETERMINANT
AT X3, X4 ' A = F (X 3)
8 = F (X 4)
RETURN
A >8
A >8
A <8
X2
X4
8
0
X3
A
SET
= X4
= OLD X 3
=A
=X2-XI
= X 1 + 0.382
= F (X 3)
SET
XI = X 3
X2 = X4
o =.
X2 - X 1
SET
XI = X 3
X 3 = OLD X 4
A = 8
0 =X2-XI
X4 = X2-0.382D
8 = F (X 4)
FIGURE 3.3. FLOW-CHART FOR THE MODIFIED "GOLDEN SECTION".
8, X4
41
tion of the nature of Rayleigh and Love waves for ~ < v combination is obtained from
t
the results of calculations for a Gold layer on Fused Quartz shown here. This combination
is a useful one being currently used in many experiments. The numerical results obtained
with our programs are indistinguishable from the curves given by other investigators [10,16, 17J.
The numerical results are discussed and described in three sections dealing
with the surface wave dispersion, group velocity, parti cie displacement and the energy
flow.
3.2. 1
Dispersion of Surface Waves
The dispersion of the first few Rayleigh and Love wave modes has been cal-
culated as a function of layer thickness. It is interesting to nate that the distinction
between these waves and the common Rayleigh wave on a free surface is that they are dispersive. The value of the stiffness constants for bath layer and substrate materials are taken
from Anderson [23].
Figure 3.4 shows the velocity dispersion curves for the first five Rayleigh
modes and the first four Love modes. The curves are of velocity versus h k (thickness x wave
number). Since the Gold layer Ioc:'ds the Fused Quartz substrate, i.e., it is heavier and
less stiff, an infinite number of modes are possible.
The dashed line of Figure 3.4 represents the Rayleigh waves. The first mode is
the fundomental Rayleigh mode. When h k is small, i.e., a very thin layer, the phase velocity is characteristic of a pure Roylei~ wave and it con be seen from Fi~re 3.5, that there
is a smoillinear region for small wave-numbers. The velocity falls steadily with increasing
V
~
m,
l'
\
/':.
48~
4400
\
- - broken line = Rayleigh Wove
"\_\\
--solid
(!)
4000 1
3600
3200
"-
~.'L
\
~
'" '"
"'"" ---- ---
~
"---
2400
2000
~
Fused Quartz
""
v
--1...-.
---- ---- ---- ----
----
-
====----
1600
1200
'"
= Love Wave
\
\
2800
li ne
Gold on
1
L1
1
=
2
3 h k (Thickness x wave m/rffbers)
OrSPERSION CURVES FOR RAYLEIGH AND LOVé' MODES~
::
1
o
FIGURE
3.4.
VELOCITY
it
v
-1
4.6
,...,
."..
v
(rn/s)'
....
"
~9(y')
4400
Gold on Fused Quartz
V
t
4000
3600
3200
2800
2400
2000
1600
V
1200
t
~
~
o
0 .02
FIGURE
0 .04
3.5.
PHASE
0 .06
0 .08
VELOCITY OF
THE
O. 1
0 • 12
O. 14
O. 16
O. 18
0 .20
hk (thickness x wave-numbers)
FUNDAMENTAL RAYLEIGH WAVES
FOR A
VERY THIN
0 .22·
LAYER.
44
h k, ~nd is less than the transverse wave velocity of the layer for h k greater than 3.6.
Then it asymptotes ta the layer Rayleigh velocity [16]. The Sezawa mode has a slight
hump in the curve shape between h k = 2.0 and h k = 3.6. The curves for the
higher order modes tend to be smoather in the range considered. The velocity of ail the
higher order modes asymptotes to the layer shear wave velocity since Stoneley waves do not
exist for this material combination [20J. The Sezowa mode and ail the higher order modes
exhibit a eut-off behaviour at the transverse wave velocity of the Fused Quartz substrate.
At v > v ' the waves are no longer lossless, but become leaky waves which ore shown,.
t
R
in Figure 3.4 os dash-dotted lines. The attenuation constant as defined in the above
chapter, Section 2.6, is about 10
-2
l'V
10
-3
nepers per wavelength, for example, for the
Se zowa mode :
hk
Attenuation
Constant
(nepers per wave length)
•
-2
0.12
1.318 x 10
0.14
1.256 x 10
0.16
1.162 x 10
0.18
1.004 x 10
0.20
8.164 x 10
-2
-2
-2
-3
At 5 MHz
(db / cm)
At 50 MHz
(db / cm)
1.14
Il.4
1.12
Il .2
1.07
10.7
0.97
9.71
0.819
8.19
The above results correspond to the circled points in Figure 3.4 on the leoky part of the
Sezowa mode curve. It can be seen that the attenuation constants'decrease
05
the phase
velocity approaches the cut-off velocity. This relat10n con 0150 be shaNn in the following
figure.
45
1.6 x 10
-2
c
o
o
'Z
~
c
j!
~
0%
10%
20%
% Above v
t
It should be pointed out that the presence of the leaky wave is usually clearly indicated
in the computed print out by a minimum in the value of the boundary condition determinant
at some v > v • 9y putting in the attenuation constant into the displacement equations
t
(2.16) and (2.17), this minimum can be made arbitrarily 'small. The velocity is 50 unaffected by this procedure that locating the minimum without including attenuation is
sufficient to calculate the leaky wave velocity.
46
The solid line of Figure 3.4 is the velocity dispersion for Love waves and
are similor to the Rayleigh waveS; Only four modes are shown for the Love waves in the
range considered. The third and the fourth modes 'are much smoother than the first two
modes. A$ mentioned in the pr'eceding chapter, the condition for existence of Love waves
is that Equation (2.36) must be positive, i.e., real solutions exist only for
vt < v < vt •
Therefore, there will be no leaky mode.
Figure 3.6 shows the group velocity curves for Rayleigh waves. Note that
the group velocity of Rayleigh waves is always lower thon the phase velocity except at the
cut-off point where the group and phase velocities are equal. In particular, the group
velocity drops very sharply to a level just below
vt before asymptoting to vR(Rayleigh
wave velocity of layer) for larger layer thickness. In each case, at least one minimum
or kink value value of group velocity exists as the layer thickness is increased [2] •
Figure 3.7 shows the group velocity curves for Love waves. These are
similar to the Rayleigh wave curves, as far as their rapidly decreasing part is concerned.
But the Love wave curves do not have any kinks and are mu ch smoother. The slopes of
Love waves vanish at cut-off for 011 modes. This can be proved by differentiating v with
respect to w in Equation (2.36). The algebra is simple but tediolJs and is not included in
th is thesis.
Except for the first Rayleigh mode, the Rayleigh and Love waves group
velocity will approach the transverse wave velocity of the layer as h k increases. This
can be seen by considering Equation (2.24), as h k
for ail curves. (Where
v~
~
CX)
in Equation (2.24) v
g
is the transverse wave velocity of the layer) •
04
v~
f'"
v
V
4100
3700
3300
2900
~
(m;~=-.:-
r---tt-
1\
\
U-\
~\
2100
J-
1'.
\.
\
t\:
.- 1'-
~
........
Gold on Fused Quartz
t
"-
'" "
"'" ~
\
t\
2500
-
~
~
'---
-------
,--
~
1700
~
-
~
'-.........
...............
---- ---- --
---- -------- -
----- --------
1300
Phase
Velocity
Group
Velocity
~
----
~
c=:
===-= ct:.........:::
~
900
700
o
2
FIGURE
3.6.
GROUP
VELOCITY
CURVES
FOR
3
4
h k (thickness x wave-numbers)
RAYLEIGH WAVES.
4.6
'"
v (rn/;:)~ ~
4100
3700
3300
~
"'\
\
W \
2500
L
2100
1- \
1700
~
\
Ir-
2900
1
1300
\
,.....
Gold on Fused Quartz_·_·_._ __
Vt
\
\
\
~
~
Phase
~
~
Velocity
~
Group
Velocity
""
~
~
~
'----
~
. . . . . ---____....:::=-_
~
----
----- -----
--- --
,
\
\
\:
~
~_~
<:
(
Vt
"
900
600
o
2
FIGURE 3.7.
3
h k (Thickness x wave numbers)
GROUP VELOCITY CURVES
FOR
lOVE WAVES •
4
4.6
~
49
ln j;jgwre 3.4, the phase velocity can be redrawn in terms of velocity versus
c.J
x h lfrequency x thickness} as shown in Figures 3.8 and 3.9. In this representation
c.J
h is a truly independent variable 50 that this graph can be used to obtain directly the ,
velocity of a given layer structure at any frequency. We can also see that ail the modes
have the phase velocity decreasing with increasing frequency.
ln Figures 3.10 and 3.11, the curves are shown for the case when
vt
>v i
t
onlyone mode exists. Note that the Stoneley waves do not exist for the S. on ZnO
1
combination of materials shown in graph (Figure 3.10). For these of the graph in Figure 3.11,
the Rayleigh wave velocity coincides with Stoneley wave velocity for h k > 7.0 , the
Stoneley wave velocity is 2769 mis. Note that different scales are used in the graphs of
FIgures 3.10 and 3.11.
ln addition to the dispersion curves, we also do study the behaviour of the
higher order modes near cut-off at the transverse wave velocity. Here we find that the
leaky waves have not dropped down smoothly to the I055less waves. For simplicity, we
show a typical curve for the Semwa mode near cut-off in Figure 3.12. Here it is seen
that the lossless wave approoches the cut-off tangentially from below and the leaky wave
approaches tangentially from ab'oVl. Compari5On of the two waves in Figure 3.12 shows
that this can only be seen by making calculation over a fine set of points.
3.2.2
The Particle Displacements
~
mentioned in Chopter Il, the particle displocement of the Rayleigh modes
hos components U in the direction of wave propagation and U normal to the surface.
3
1
Figure 3.13 illustrotes the amplitude of the particle displacements vs. h (depth - in terms
v
........
{m~
4300
Gold on
v
4100
Fused QU.:Jrtz
t
3700
3300
2900
2500
L3
2100
L
2
1700
1200
o
2K
1_
Vt
LI
4K
6K
8K
c.I
FIGURE
3.8.
VELOelTV DISPERSION
eURVES FOR
lOK
h (fl'"equency x thickness)
LOVE WAVES
VERSUS
c.I
h •
llK
~
v (m/:;~
,
4400
\
Fused Quortz
...
\
"
4000
.
Gold on
V
t
3600
3200
2800
R4
2400
R3
2000
1600
RI
1200
o
2 K
4K
Vt
6K
8K
10 K
Col
FIGURE
3.9.
VELOCITY
DISPERSION
CURVES FOR RAYLEIGH
h (frequency x thickness)
WAVES VERSUS Col h •
11 K
~
..
,-
v (m.;" .~.
2YOO - .
Si on ZnO
Vt
2800
2700
2600
o
0.2
FIGURE
3.10.
0.4
PHASE
•
0.6
VELOC/TY CURVES FOR .;;"t
>
V
t
0.8
1.0
h k (thickness )( wave numbers)
OF RAYLEIGH WAVES.
1.15
~
"',..
v (mA.
2880
Aluminium on Tungsten
2860
2800
V Stoneley
2700
2600
o
2
3
4
h k (thickness x wave numbers)
FIGURE
3.11.
PHASE VELOCITY. CURVES FOR "t
> v
t
OF RAYLEIGH WAVES.
4.5
U'I
Co)
...
.........
.... l
v (rn/s)
4102
v (m/,,)
4100.6
4101
4100.2
4100
4100.
4098
409.98
Gald an
Fused
Quartz
4097
(h)
(a)
4096
.2263
.2265
FIGURE
.2267
h k
3.12.
.2269
THE
(a)
.2271
DISPERSION RELATION OF
Expanded
Scale.
.22662
.22668
h k
.22673
SEZAWA MODES
NEAR eUT-OFF REGION.
(h) Greatly Expanded.
~
.....
.~~
3
Substrate
Layer
First Made,
2
CI>
"'0
2
CI>
u
~
:::>
=
2.0,
Gord on Fused Quartz
Energy Ratio
1.99
v
1446 rn/sec
R
Layer Thickness
Vl
t
h k
0.318
À
CI>
.....~
U
o
>
3
U1
?
- 1
-2
o
-1
X
3
(.n
(.n
h
FIGURE 3.13.
VARIATION OF PARTICLE DISPLACEMENTS WITH DEPTH FOR THE FIRST RAYLEIGH MODE AT hk = 2.0.
56
;.
of wavelengths) for depths up to one loyer thickness. In this figure the relation is for the
cose of a Gold loyer on a Fused Quartz substrate, with the fundamental mode at h k = 2.0
and v =1446 mis. The curves show that the U and U components ore continuous
3
I
R
in the interface, demonstrating that the particle displacements sotisfy the boundary condition requirements. These components in the substrate decreased exponentially os con be
seen from Equation (2.16). The U compone nt in both the loyer and substrate region is
3
positive, while the U compone nt starting in the loyer region starts negatively and then
I
goes positive. The motion con 0150 be
r~presented
in the following way :
Loyer
Substrate
upward is positive,
left hand si de is negative.
From the above diagram, it con be seen that the U compone nt of the displacement in the
3
loyer region hos a symmetric motion in the sense that the free face and interface move in
the sorne direction and it is referred to os the symrnetric mode (MI)' It is 0150 referred
to os the MIl mode [5] by virtue of its similarity to the Mil plate mode.
It is 0150 worth noticing that the displacement amplitude depends upon frequency. Fi~re 3.14 shows the fundomental mode at h k = 0.4, v = 2279
R
comporing wlth Figure 3.13 will prove the above statement.
mis;
..
,-
....
,"
3
C)
u
1
Layer
Substrate
First Made,
~
:>
-8
.~
2
~
Gald on
Energy Ratio
on
;?
h k = 0.4,
Fused
Quartz
0.578
v
2279 rn/sec
s
Layer Thickness= 0.0636 >.
r
C)
....
~
k
--l----
o
- U3
_ _ Ut
-1
-2
L-
,
l
o
-1
X
~
3
h
FIGURE 3.14. VARIATION OF PARTICLE DISPLACEMENTS WITH DEPTH FOR THE FIRST RAYl.:EIGH MODE AT h k = 0.4.
58
The particle displacement for the Sezawa mode is shown in Figure 3.15, at
hk
= 2.0,
v = 2189 m/s, for the same material as above. The U compone nt
3
R
starting at the free surface of the layer region is negative but changes to positive at the
interface. The difference between the U and U components is that the U changes
1
3
1
sign in the substrate region, while the other undergoes this change in the layer region. The
modes can alsa be represented in the following way :
-\--- Layer
I--------~--~--~
...;--- Substrate
From the above figure, it con be seen that the U compone nt of the displacement in the
3
layer has an antisymmetric motion. We cali this the antisymmetric mode
(~),
and it
is also referred to as the plate mode (~1) [5J.
The third mode of the particle displacements is shown in Figure 3,16, which
is simi lar to the fundamental mode, except for the U compone nt , which has two crossing
3
points on the zero amplitude axis, i.e" there are two nodal planes for transverse components' Therefore, the third mode is alsa colled a symmetric mode (M ), and it is
3
alsa referred to as the M mode,
12
Figure 3.17 is the illustration of the fourth mode of the particle displacements, and 15 similar to the Sezawa mode, except that the U component has three
3
crosslng points or nodal planes in the layer. Therefore, the fourth mode is colled an
antisymmetrlc mode (M4), and it is alsa referred to as the
~2
mode,
:
3
r
~
Substrate
G
Second Mode,
U
0
"t:
::t
on
G
]
2
t
G
h k
= 2.0,
Fused Quartz
Energy Ratio
8.81
V
2189 rn/sec
R
layer Thickness
~
u...
Gold on
3.18
À
U3
U
o
1
':'1
-2
o
FIGURE 3.15.
-1
x
3
11
VARIATION OF PART/ClE DISPlACEMENTS WITH DEPTH FOR SEZAWA MODE AT h k = 2.0 •
Ut
-0
L
.... ~
~1'
3
layer
Su bs tra te
4>
Third
u
fv4oode,
.!?
:;
V'II
4>
-0
.2 2
t
4>
4>
h k
=
1.8,
Gold on
Fused Quartz
Energy Ratio
3.29
vR
3239 m/sec
Layer Thickness
0.286 À
ù:.
>~
-----------------------------_U3
o
------=.U
j
1
- 1
-2
2>
o
-1
X3
h
FIGURE 3.16 - VARIATION OF PARTlClE DISPLACEMENTS WITH DEPTH FOR THE THIRD RAYLEIGH MODE AThk = 1.8.
3
Layer
Substrate
cu
u
0
't:
Fourth Mode,
h k = 2.2,
~
-8
V\
.~ 2
~
-<:
cu
Energy Ratio
0.966
v
3953 m/sec
R
Layer Thickness
cu
ù:
Gold on Fused Quartz
0.35
~
U
U
o
3
1
-1
-2
o
FIGURE 3.17.
-1
X3
~
h
VARIATION OF PARJICLE DISPLACEMENTS WITH DEPTH FOR THE FOURTH RAYLEIGH MODE AT hk
=2
62
Figure 3.18 shows the particle displacement for the Love waves, since these
waves are transverse, there is only a U component.
2
The U compone nt of the Love waves of LI and L3 in Figure 3.18 is
2
the same as the U compone nt of the Rayleigh waves of Figures 3.13 and 3.16,
3
respectively.
Therefore, the first and third Love modes are symmetric modes. Similarly,
the U compone nt of L and L4 in Figure 3.18 is the same as the U campane nt of
2
2
3
Figures 3.15 and 3.17, respectively, ,sa that the second and fourth Love modes are 'antisymmetric modes.
ln Figures 3.10 and 3.11, onlyone mode exists for
-it > v of the Rayleigh
t
waves. The parti cie displacement for Figure 3.10 (omitted for simplicity) is the same as
Ml mode. The particle displacement of Figure 3.11 is also the, same as Ml mode for
small wavelengths. AJ the phase velocity approaches the Stoneley wave velocity, the
particle displacement wi Il approach the Stoneley wave particle displacement, this is shawn
in Figures 3.19 and 3.20. These two graphs have the same shape, but represent different
distances from the interface.
3.2.3
The Energy Flow
The results of calculating the ratio of layer to substrate for energy distribution
fOf the Rayleigh waves is shGWn in Figure 3.21. Figure 3.21 iIIustrates how the energy
ratio
~ayer
ta substrate) varies with h k for Gold layer on Fused Quartz. From this
figure, it is seen that only the first mode which the energy ratio is relatively constant and
almost equally divided between layer and substrate up ta film thickness of al~t.
À/4
•
,..
...
~
3
Layer
Substrate
lU
U
~
~
.~
Gold on Fused Quartz
LI
~
2
!
lU
lU
.::
~;;;;;;;~~~~~~~ t~
L3
o
L)
- 1
-2
L
0-
o
FIGURE 3.18.
-
)
X3
h
VAR/ATION OF PARTICLE D/SPLACEMENT W/TH DEPTH FOR THE FIRST FOUR LOVE MODES.
~
...
........
3
Substrate
Layer
cu
u
..E
.2
;=:
t
hk
S
cu
9. 0 m/sec ,
Aluminium an Tungsten
."
-0
2
cu
cu
ù:
Energy Ratio
0.0766
v
2769 m/sec
R
Layer Th i ckness
1.43
>.
U
3
U1
o
- 1
-2
o
FIGURE 3.19.
-1
X3
~
n-
PART/CLE DISPLACEMENT OF RAYLEIGH WAVE AS VELOCITY APPROACHES STONELEY VELOCITY.
...
~
3
B
~
t
.E
..,.,:;
Lower Medium
Upper Medium
Tungsten
Aluminium
2
Energy Ratio
0.099
v
2769 m/sec
CI>
CI>
ù:
s
Depth
o
0.55
À
b======----+--====~:::::::>-=========
U3
U)
- 1
-2
_)
o
FIGURE 3.20.
STONELEY WAVE
PARTICLE
DISPLACEMENTS.
x3
h
0-
U.
....
r-
16
I
Gold on Fused Quartz
0
.~
14
a:::
>-.
0)
~
u...J
12
lU
'ê
.Ê
::::>
V"I
10
2
Q;
>-.
0
->
8
R3
6
4
2
o
o
FIGURE
2
3.21.
VARIATION OF THE ENERGY RATIO WITH
h k
3.
4
h k (Thlckness x wave numbers).
OF THE RAYLEIGH WAVES AT v
> t •
t
v
4.6
~
67
,.
Th energy of the higher order modes is entirely concentrated on the substrate near cut-off
velocity of substrate and progressively transits from the substrate to the layer surface with
increasing the layer thickness.
The Sezawa mode energy increases steadily and more
sharply than the first mode. There is a peak exhibited around h k = 2.5, and then the
mode falls off slowly and then rises again after h k =4.0. The third and higher arder
modes follow roughly the same pattern, their peaks being progressively shifted to the right.
The energy calculation described in Section 2.5 is only va!id for the lossless
modes. Since the leaky modes involve a radiating component into the solid causing attenuation of the wave, no energy flow calculation for this complex situation was attempted.
The graph for the Love wave energy ratio against h k for the first four modes
of the some materials as above is plotted in Figure 3.22. The ratio increases with large
values of h k in 011 four cases. In the higher order modes, however, the increase rate
of energy ratio becomes less rapid. The value of h k at zero energy· ratio corresponds to
the cut-off of the transverse wave velocity of the substrate for the first mode. It is also
interesting ta note that, in Figure 3.22, the energy ratio curves versus h k curves for
higher order modes starting at h k = l, 2, 3, ...... and 50 on. This is due ta the
effect of the dispersion relation for Love wave (Equation (2.36) for the particular materiol
2
used. Forexample, when y=y ,Equation (2.36) becomes ton [h k J(v t/y 2 -1)]=0,
2
t
t
2
or h k Jtt 1Yt - 1) = nI, n = 0, l, 2, ••••• •
Yt of the Fused Quartz = 4100
mis.
Y of the Gold Quartz
mis .
t
= 1239
The h k value con be obtained from the above equation by substituting the velocities, and
2
it is found that
J Yt 1 vt2 - 1 is close to
1 50
that h k
coincidence and not the general case for 011 materials.
IW
n. Hence, it is just 0
;
'.
~
18
Gold on
.2
Fused
Quartz
16
Ô
co:
~
!
14
~
-s
~
1
12
.2
......
.....~1O
8
6
4
2
o
0
FIGURE 3.22.
VARIATION
2
3
h k (Thickness x W'ove numbers)
OF THE ENERGY RATIO WITH h k OF THE" LOVE WAVES AT v
>
t
4.6
4
...
v
t
&;
•
69
f
Figures 3.23 and 3.24 are far the case where
vt > vt of the Rayleigh
woves. The energy ratio for Si on ZnO rises faster for AI on Tungsten, and then drops
rapidly to cut-off. The energy ratio for AI on Tungsten tends to approach that of Stoneley
waves as thickness increases.
".,
"!
0.16
Aluminium on Tungsten
0.14
o
.~
00:
li;
0 . 12
!
u..I
.!!
~ 0.10
.B
~
S?
i-0 •08
-'
0.06
0.04
0.02
o
o
2
3
4
.
4.6
h k (Thickness x wove numbers)
FIGURE 3.23.
VARIATION
OF THE
ENERGY RATIO WITH h k
OF THE
RAYLEIGH WAVES AT ~
> v
t
•
C:t
il""
~
o
!
li;
!
U.I
1!
Si on ZnO
j~
2
1
0.1
0.08
0.06
0.04
0.02
Near
Cut-off
o
LV______
~
______
o
~~
______
~
0.2
FIGURE
3.24.
VARIATION
______
~
0.4
OF Tlie
______
~
______
~
______
0.6
ENERGY RATIO WITH h k
~
______
~
________
0.8
~--
__
~
1.0
h k (Thickness x wave numbers)
OF THE RAYLEIGH WAVES AT
~
vt
> v
t
•
72
CHAPTER IV
EXPERIMENTAL APPARATUS, TECHNIQUES AND RESULTS
4.1
Introduction
There are several methods of exciting elastic surface waves with transducers.
The earlier transducers for ultrasonic surface-wave excitation usually consisted of a wedge
arrangement [24] which converted a homogeneous plane wave supported by the wedge into
a surface wove-olong the interface region under the weclge. The surface wave generated
then propagated outon to the free surface. More recently, interdigital comb structures
[2] deposited and photo-etched onto piezoelectric as weil as nonpiezoelectric substrates
have made it possible to generate and propagate surface waves in giga-hertz frequency
range.
ln this chapter, an ultra50nic ganiometer used to measure the fundamental
Rayleigh and Sezawa modes will be described. This device utilizes a single transducer
and a reflector composed of two plane surfaces arranged perpendicular to each other. The
transducer and reflector are to be immersed in a distilled water bath. In addition, the
layer thickness measurement and 50mple cleaning procedure will al50 be discussed. Finally,
the accuracy of the experimental results are evaluated.
4.2
UltJosonic Reflectivity at Liquid-Solid Interface
The reflection of ultrasonic energy at a liquid-solid interface con often be
used as a rapidtechnique for evaluating certain material choracteristics. Reflection measurements can provide information on ultrasonic velocities, impedance mismatch, surface roughness
et cetera [25].
73
When an ultrasonicbeam is incident upon a liquid-solid Interface as shown in
Figure [4.1], the geometrical relationship that exists between incident and refracted
portions of the ultrasonic beam are given by Snell's Law :
sin 9
i
sin 9
l
sin 9
S
(4.1)
VU =VL2 ='752
• • • • 1 .....
where
V
L1
is the compressional wave velocity in Liquid,
Vl2
is the longitudinal wave velocity in solid,
VS2
is the shear wave velocity in solid.
LlQUID
l
'
V
L2
V
S2
SOUD
FIGURE 4.1. REFlECTION AND REFRACTION AT UQUID-SOLID INTERFACE.
,
When VL2 > VS2 > VlI' the refraction angles are olways larger thon the ongle of in'
j
cidence and it is possible to produce criticol refroction, i.e. the condition where the ongle
of refraction equals 90 degrees. When the angle of incidence is greater than thot necessory
74
to produce crltical refraction of the shear wave, it may be possible to excite a surface
wave along the boundary. Equation (4.1) can also be used to express the angle of incidence for maximum excitation of surface waves. Since the angle of refraction is equal
to 90 degrees, we get
sin 9.
1
1
=
VU T
(4.2)
where v is the velocity of the Rayleigh wove.
R
However, disregarding the possible existence of surface waves and referring to
Figure 4.1 we may obtain an expression for ultrasonic reflectivity as a function of the
angle of incidence. The formai derivation of the reflectivity considers only plane waves.
Ergin [26] shows that the amplitude ratio of the reflected to the incidence ultrasonic wave
at a liquid-solid boundary as a function of the angle of incidence is given by :
cos 9
R
ï
where
=
A'=
12
- A cos 9 (1 - B)
L1
+ A cos 9 (1 - B)
12
L1
cos 9
32
~1
(4.3)
vL2
\1
A typical plot (25] of R/1 as function of 9 is shown in Figure 4.2, which was
U
obtained by substituting the values for densities and velocities for aluminium and water,
where the solid line and open circles represent theoretical and experimental results, respective Iy. The a{P'eement between theory and experiment is very good except at angles of
incidence neor 31 degrees, i.e., when the angle of incidence is less than that which produces tatal reflection.
The anomalous ref~ection results obtained at the critical conditions
......
o
"'0
o
80
cu
Û
-!!
o
~
co:
0
o
ëcu
u
~ 60
~
!
Open Circles
Experimental
Solid-Line
Calculated Results of
Results
Mayer's.
~
40
20
o
12
FIGURE 4.2.
16
20
24
Angle of Incidence
REFLECTED ENERGY VERSUS ANGLE OF
28
(Degrees)
32
36
INCIDENCE FOR WATER ALUMINIUM
40
41.5
'-1
c..n
BOUNDARY.
76
are undoubtedly associated with surface wave excitation (11). Recently, Oiachok and
Mayer (27) experimentally showed that when a bounded beam is incident at the Rayleigh
angle at liquid"50lid interface, the beam is displaced along the interface before it is reradiated into the water. Schoch [27] derived an expression for this displacement
Â
= 2~S 2 [
'1ï
1/2
a (a - b) J
2
1 +6 b (1 - c) - 2 b (3 - 2 c)
b (b - 1)
where ~ is the wavelength in the liquid a = (
S2
c = (v /v )2,
L2
(4.4)
b- c
vS2
2
vS2
/ \1) , b = (
2
/ v)
R
,
subscripts 1 and 2 refer to liquid and salid, respectively.
The experiment performed by Oiachak indicates that the velocity with which
the ultrasariic signal is displaced along the solid is the Rayleigh wave velocity. We have
used the measurement value of the longitudinal wave velocities, the shear wave velocities,
and the densities of two media to obtain the beam displacement (â). For nickel immersed
in water, we then predict a displacement of 60 ~ which corresponds to a displacement of
3.36 cm at 5 MHz. For aluminium immersed in water 1 we predict a displacement of 22).
which corresponds to a displacement of 1.32 cm at 5 MHz. 8ecause the displacement of
the wave is significantly large with respect to the width of the beam from the transducer
(approximately 1.25 cm diameter). it wi Il cause a partial loss of energy at the, Rayleigh
angle of Figure 4.2. In other words, Rollins (25] experiment cqnnat receive 011 the
reflected signai from the specimen is shown in Figure 4.3 , and a sharp dip oppears in the
graph. Since the displacement (6) is frequency dependent the sharpness of the dip wi Il
also be frequency dependent. This probably accounts for the frequency dependent observed
by Rollin.
77
,
'
FIGURE 4.3.
LOSS OF REFLECTION SIGNAL SV RECEIVING TRANSDUCER
RESULTING FROM LARGE BEAM DISPLACEMENTS
4.3
(~).
Measurement Using A Right-Angle Reflector
Aschematic dia~am of the ultrasonic goniometer using right-anglereflector
is shown in Figure 4.4. The transducer rotates about an axis which is collinear with the
interaction of two orthogonal surfaces, i.e., the rernovable somple and the fixed reflector.
When the transducer is properly aligned, the centre line of the ultrasonic beam is perpendicular to the axis of rotation. With this geometrical arrangement, an incident beam of
,>
plane waves will produce a reflected beam which returns to the original sending transducer.
'.'
The removable sample can be easily positioned by placing it in intimate contact with a
reference flot, which in turn is perpendicular ta the fixed reflecting surface. A block
78
REMOVABLE
SAMPLE
1
1
/
/
/
1
/
V
FIGURE 4.4.
SCHEMATIC DIAGRAM OF ULTRASONIC GONIOMETER
USING RIGHT -ANGLE REFLECTOR.
79
<-
diagram of the 9Oriiometer and associated signal-processing equipment is shown in Figure 4.5.
When the system is in use, the waveform translator is positioned to accept only the first pulse
echo from the right-angle reflector. The repetitive pulse is integrated and converted to a
OC signal that serves as the input to an X-Y recorder. With the bottom part of the 9Onio·
meter immersed in distilled water or other suitable coupling liquid, the angular position of
the transducer is moved bya motor drive, and the amplitude of the reflected signal is thus
automatically recorded as a function of the angle of incidence on the X-Y recorder. The
frequency can be changed by changing the coil of the R.F. generator. The amplitude of
the reflected signal decays with the frequency change due to absorption and scattering.
AtYpical chart containing the amplitude variations from a water-aluminium
boundary is shown in Figure 4.6. Starting at the left of the chart, the various features can
be explained as follows: The sharp maximum (A) occurs when the transducer axis is per0
pendicular to the sample and th us represents an incident angle of 0
•
The amplitude of
this maximum would be approximately twice as great if·the entire beam could strike the
0
sample, but at 0 , one-half of the ultrasonic beam is blocked by the end of the standard
reflector. The slowly increasing amplitude observed near (8) is produced as the rightangle reflector starts to function and accepts more and more of the incident beam. The
maximum at (C) occurs at the angle of incidence corresponding to critical refraction of
longitudinal waves in aluminium. This angle can be used together with Snell's law to calculate the velocity of longitudinal waves in the solid. At higher angles of incidence, the
ampl itude drops as much of the energy which is coupled into aluminium. At (0), the amplitude reaches another maximum, related to critical refraction of shear waves in aluminium,
,.
i
and then drops to a relatively sharp minimum at (E). The minimum at (E) is a surfacewave effect. The angular position of the minimum is used together with SnelPs law to
80
ATTENUATOR
1--4---J
RF
GENERATOR
UNIT
GONIOMETER
PULSE
GENERATOR
TRIGGER
WIDE
BAND
AMPLIFIER
X-Y
CHART
RECORDER
WAVEFORM
~-...OSCILLOSCOPu---:w TRANSLATOR
FIGURE 4.5. BLOCK DIAGRAM OF SIGNAL-PROCESSING EQUIPMENT.
,
~
-~
.
~
.
C
D'
'.
.À
C'
ë
.!!
u
~
~
A'
.~
.
û
;:
~
o o.
~
A
t
E'
E
00
FIGURE 4.6.
X - Y RECOROING OF THE REa..ECTEO SIGNAL ._FROM AN ALUMINIUM SPECIMEN.
00
82
calculate the phase velocity of the related surface wave. Since the moveable sample
and fixed reflecting surface are orthogonal, a given transducer position yields complementory angles of incidence as measured with respect to the two walls. We see that the
features labelled AI
N
El are related to the fixed stainless steel reflector in the same
way that the iJnprimed features are related to the aluminium sample. Between (E) and
(El), both surfaces are totally reflecting.
4.4 Other Preparations and Measurements
To make the wave-guide layer, one must obtain a perfectly polished plane
on the substrate and cleaning procedures are of primary importance in order to produce pure
films with good adhesion to the substrate. It has been found necessary to use a rigorous
cleaning procedure for 011 highly polished substrates to prevent the deposited films from
peeling. We use a chemical cleaning method for this as outlined below.
Removal of the impurities by chromic ocid, and scrubbing the
surface with a cotton swab.
Rinse in running distilled woter to remove the acid.
Boiling in methanal (or isopropyl alcohol) to remove the water.
Boiling in trichlorethylene for about ten minutes to remove
the grease.
Boiling in methanol again.
83
,.
1
Then holding the sampi es immediately transferred to the vacuum evaporator or cathodic
sputtering system waiting for deposition.
The film deposition is performed on an Edward Mode 1 : E 12 E vacuum
evaporator. This unit is easy to operate and it serves as a most convenient piece of apparatus for this simple evaporation pro cess •
The layer thickness is measured by using a commercial Sioan Instruments
DTM - 3 type thickness monitor, which is an instrument capable of monitoring and control/ing the film thickness during the deposition process. The thickness monitor has two quartz
crystal oscil/ators, one of which is the reference oscil/ator while the other acts as the monitor oscil/ator. The ground surface of the monitor quartz plate is exposed to the vacuum in
the evaporator and is held at the same temperature as the substrate. As the films condense
on the quartz crystal, its fundamental resonant frequency reduces in direct proportion to
the mass of the deposition film. The signais generated by the reference and monitor 05cillators are mixed and the resulting difference frequency can be estimated according to the
relation
Af
where
Af
T
=~
(4.5)
=
frequency difference
=
the bulk density of the deposited material (gm / cm 3)
=
thickness (Angstroms)
(cycles /sec)
Hence, by reading out the change in frequency of the monitor osci Ilator, the thickness of
the deposited film can be rneasured. Since the film density is nat always equal to bYlk
density during the evaporation, the above formula is on approximation.
84
There are optical methods which can also be used for film thickness measurement. We have used a commercial WATON interference objective ta measure the film
thickness. The instrument is quite straightforward. It is necessary ta observe a boundary
of the film and the substrate, sa that the hei"t of the substrate surface may be compared
with the film surface. If the film is more than two microns thick, separate white light
fringe patterns can be observed on the top and bock surfaces of the film.
The above method is more accurate than the Sloon thickness monitor because
the latter is temperature dependent. Error is introduced if the sensor head temperature
a
exceeds 120 C •
4.5
Error and Precision of the Results
The theoretical analysis in the preceding chapters has been vl!rified by using
,
Q
right-angle reflector technique ta measure the phase velocity. A 112" diameter lithium
sulphate transducer which was immersed in distilled water was used as the source and detector
element. The measurernents were carried out ot frequencies lying between 5 and 50 MHz,
on layers of up ta 20 ~ thickness. The phase velocity dispersion measured with various
material combinat ion are shown in Figures 4.7, 4.8, 4.9and 4.10. The crosses represent experirnent data obtained from reflectivity measurernents, and colculated using
Equation (4.2). The salid curves are theoretically calculoted using the rneasured density
and stiffness constants for the substrate materials.
The oc cu roc y of the meosured product h x k (thickness x wove numbers) is
limited by :-
MEASUREMENT POINT
SIL VER ON ALUMINIUM
'"
4200
~ 1-1'
t
3800
'"
0-
POSSIBLE
T MAXIMUM
ERROR IN v
L MEASUREMENT
MAXIMUM POSSIBLE ERROR
MEASUREMENT
3400
u
V
~
IN h k
t
..s.
>- 3000
1-
g
CALCULATED MODE 2
--'
.....
>2 600
.....
on
-<
if
2200
CALCULATED MODE 1
Vt
1800
1400
-----
o
FIGURE 4.7.
3
2
THICKNESS X WAVE VECTOR
VELOCITY DISPERSION
4
(h k)
CURVES FOR THE RAYLEIGH AND SEZAWA MODES VERSUS h k •
00
c.n
,..
4600
SILVER ON NICKEL
MEASUREMENT POINT
+
4200
3800
TMAXIMUM POSSIBLE
ERROR IN v
L MEASUREMENT
-4
1-
MAXIMUM POSSIBLE ERROR IN h k
MEASUREMENT
'"J 3400
~
E
V
~3000
8.....
t
CALCULATED MODE 2
-1
>
..... 2600
V'I
<
if
2200
CALCULATED MODE
1800
V
1
t
1400
o
2
3
THICKNESS X WAVE VECTOR
FIGURE 4.8.
VELOCITY
DISPERSION CURVES FOR THE RAYLEIGH
4
(Il k)
AND SEZAWA MODES VERSUS h k •
co
0-
.......
TIN ON ALUMINIUM
MEASUREMENT POINT
+
~
~
>-
!:::
1-
~~~,<
r
1
MAXIMUM POSSIBLE
ERROR IN v
1. MEASUREMENT
-1
MAXIMUM POSSIBLE ERROR IN h k
MEASUREMENT
_"_vt
OPTIMIZE MODE 2
30
~
CALCULATED MODE 2
-J
LU
>
LU
V\
«
:x:
a...
OPTIMIZE MODE 1
22
CALCULATED MODE
18001
Vt
140P,L,____________________
o
~--------------------~--------------------;_--------------------~
2
3
4
THICKNESS X WAVE VEcrOR (hk)
FIGURE 4.9.
VELOCITY DISPERSION CURVES FOR THE RAYLEIGH AND SEZAWA MODES VERSUS h k •
~
4600
+
TIN
ON
NICKEL
4200
MEASUREMENT
+
3800
-
t--
Ü 3400
~
POINT
T
MAXIMUM POSSI BLE
ERROR IN v
1
MEASUREMENT
--f
MAXIMUM POSSIBLE ERROR IN
MEASUREMENT
vt
...s.
~3000
h k
OPTIMIZE MODE 2
o
o
CALCULATED MODE 2
~
~
.... 2600
~
J(
iE
"',
i
OPTIMIZE MODE
1
CALCULATED MODE
2200
1800
---------
"t
----
1400
o
2
THICKNESS X WAVE VECTOR
FIGURE 4.10.
3
4
(h k)
VELOCITY DISPERSION CURVES FOR THE RAYLEIGH AND SEZAWA MODES VERSUS h k.
CD
CD
89
The maximum uncertainty in the measurement of frequency is
:!: 3.7 % ,
and is just due to an uncettainty in resetting the
generator frequency scale.
-
Film thickness measurement errors, which are within :!: 1.5 % •
This yields a maximum possible error of :!: 5.2 % in the h x k measurement. The longitudinal velocity of water versus temperature is shown in Figure 4.11. From the data, an
error of :!: 1.5 % can be estimated in the phase velocity calculation on the basis of observed temperature changes of 8°C during the experiment.
The maximum possible error is shown in the figure by error bars. For
simpl icity, error bars are shown on only a few of the experimental points.
ln Figures 4.7 and 4.8, the agreement between experimental and theoretical dispersion curves is good for the fundamental Rayleigh mode, and fairly good for the
Sezawa mode and Rayleigh leaky mode.
Figure 4.9 shows that the experimental and theoretical values are in good
agreement for the fundame nta 1Rayleigh mode and are within 3 % for the Sezawa mode
and Rayleigh leaky mode. Since the shear wave velocity of the actual tin film is not
measurable using our technique, the stiffness constants of the layer are taken from Anderson
[23 J. The difference between bulk waves and film values may be a source of inoccuracy.
If we use the measured density of the tin film and optimize the stiffness values, it is possible
to get a fairly good a~eement with measured curves. This is shown by the broken line of
Figure 4.9. The difference between these two values are shown in the following table.
90
1.560
1.540
1.520
1.500
0
N
:c
f·
.
1.480
u
5/
~
"'-
E
E
1.460
1.440
1.420
1.400
0°
10°
20°
'SJ0
TEMPERATURE
40°
50°
600
Oc
FIGURE 4.11. THE LONGITUDINAL VELOCITY OF WATER VERSUS TEMPERATURE.
91
ANDERSON VALUE
J(gm /cm 3)
.7.32
OPTIMIZE VALUE
.. 7.26
% DIFFERENCE
0.83
(Measured Value)
2 11
C (N1m x 10 )
ll
0.748
0.700
6.4
0.278
0.314
12.8
1791
1630
9
3198
3104
2.9
Figure 4.10 shows that for bath the fundamental Rayleigh and Sezawa modes
the experimental results and theory are within 2 %. However using the same method as
above, we can obtain a fairly good agreement with measured curves. This is shawn by the
broken line of Figure 4.10 .
The most significant difference between these experimental results and the
usual theoretical predictions is the total absence of a cut-off at the substrate transverse
velocity. The measured velocity increases sharply above v for ail the material comt
binations we have tested indicating the excitation of a leaky wave.
92
CHAPT ER V
SUMMARY AND CONCLUSIONS
T~e
theory of elastic waves guided by layers has been summarized and the
results of calculations using a general numerical technique have been presented for
Rayleigh, Love and Stoneley waves. The velocity dispersion for the two lowest Rayleigh
modes was measured for a number of material combinations using a right-angle corner
reflector technique.
ln Chapter 1/, the theoryof layer waves has been developed in some detail,
particularly with regard to the higher order modes, the leaky Rayleigh wave, and the
energy flow, since these aspects have not been previously studied quantitatively.
The computing technique and the calculated results are presented in Chapter
"1
primarily in the form of a series of curves which essentially characterize the layer
waves. On the basis of the the ory and calculations the following canclusions are given :
(1)
Age ne ra 1, rapid and reliable numerical search technique
enabled a more detailed study of the properties of Rayleigh,
Love and Stone ley waves •
m
For Rayleigh waves, only one mode of propagation of layer
waves exists if the velocity of shear waves in the layer is
higher thon the velocity of shear waves in the substrate. This
made is a low pass mode.
(3)
For bath Rayleigh waves and Love waves :
93
~
.•
,.
Q)
The number of modes increases with decreasing wavelength or increasing layer thickness or frequency if the
velocity of shear waves in the layer is less than the
velocity of shear waves in the substrate. Ali of these
modes are dispersive ; the first mode is an ail pass mode,
ail the others are high pass. The velocity of ail the
high order modes asymptotes to the layer shear velocity
and the first mode asymptotes to the layer Rayleigh velocity •
Qi)
Two kinds of particle displacement motions ar~ found, they differ
in their nature, giving rise to symmetric and antisymmetric
modes of a free-plate modified by the contact of the lower
surface.
Oii)
Except for the first Rayleigh mode, the group velocity will
approach the shear wave velocity of the layer as the layer
thickness or the frequency increases for the case where the
velocity of shear wave in the layer is less than the velocity
of shear waves in the substrate.
(4)
The Rayleigh leaky modes exist in ail the higher order modes.
There is no leaky wave existing in Love modes.
ln Chapter IVan experimental apparatus and a technique has been described
for measuring velocity dispersion of layer waves and the results of a number of meosurements
94
carried out on four combinations of materials are compared with theoretical calculations.
Good agreement was found. It has also been shown that the bulk values and film values
obtained are different. The stiffness values were optimized with respect to the film
values. These optimal stiffness values were used for computing the theoretical curves
which are in good agreement with experimental curves.
95
REFERENCES
[1]
Achenbach, J.O. and Epsten, H.I., "Dynamic Interaction of a Layer and
a Half-Space", Proceedings of the American Society of Civil Engineers,
Vol. EM5, p. 27, 1967.
[2]
Morgan, D. P., "Dispersive Delay Unes Using Acoustic Surface Waves",
Ph.D. Thesis, July 1969. University College London.
[3]
Sezawa, K. and Kanai, K., "The M2 Seismic Waves", Bull, Earthquake
Research Institute, Vol. 13, p. 471, 1935.
[4]
Farnell, G.W., "Properties of Elastic Surface Waves", Physical Acoustics,
Moson, W.P. and Thurston, R.N., Vol. VIA, Chapter III, Academic
Press, N.Y., 1970.
[5]
Tolstoy, 1. and Usdin, E., "Dispersive Properties of Stratified Elastic and
Liquid Media: ARay Theory", Geophysics, Vol. 18, p. 844, 1953.
[6]
Stoneley, R., "Elastic Waves at the Surface of Separation of Two Solids",
Proceedings of the Royal Society, Vol. A106, p. 416, 1924.
[7]
Scholte, J.G., "The Range of Existence of Rayleigh and Stoneley Waves",
Monthly Notice, R. Astr. Society of Geophysics Supplement, Vol. 5,
p. 120, 1947.
[8]
Schmidt, R.V. and Voltmer, F.W., "Piezoelectric Elastic Surface Waves
in Anisotropic Layered Media", IEEE Transactions on Microwave Theory
and Techniques, Vol. MIT - 17, No. Il, 1969.
[9]
Lim, T.C. and Farnell, G.W., "Search for Forbidden Directions of Elastic
Surface Wave Propagation in Anisotropic Crystals", Journal of
Physics, Vol. 39, p. 4319, 1968.
~plied
96
[10]
Adkins, L.R. and Hughes, A. J., "Elastic Surface Waves Guide by
Thin Films: Gold on Fused Quartz", IEEE Transactions on Microwave Theory and Techniques, Vol. MIT-17, No. 1J, 1969.
[11]
Rollins, F. R. Jr., "Ultrasonic Examination of Liquid-Solid Boundaries Using
a Right-Angle Reflector Technique ", Journal of the Acoustical Society
of America, Vol. 44, p. 431,1968.
[12J
Rollins, F.R. Jr., Lim,
Le.
and Farnell, G.W., "Ultrasonic Reflectivity
and Surface-Wave Phenomena on Surface of Copper Single Crystals",
Applied Physics Letters, Vol. 12, p. 236, 1968.
[13J
Adler, E.L., Farnell, G.W. and Sun, I.H., "Velocity Dispersion of Elastic
Waves Guide by Thin Films", Paper E-9, 1969 IEEE Ultrasonics
Symposi um, St. Louis, September 1969.
[14]
Mattews, H. and Van de Vaart, H., "Observation of Love Wave Propagation
at UHF Frequencies", Applied Physics Letters, Vol. 14, No. 5, p. 171,
1967.
[J5]
Nye, J.F., "Physical Properties of Crystals", Clarendon Press, Oxford, 1957.
[J6]
Tiersten, H.F., "On Elastic Surface Waves Guided by Thin Film", Journal
of Applied Physics, Vol. 40, p.
[l7J
no, 1969.
Tournois, D. and Lardat, C., "Love Wave-Dispersive Delay Lines for Wideband Pulse
Compr~ssion",
IEEE Transactions on Sonics and Ultrasonics,
Vol. SU-16, No.'3, July 1969.
[18]
Ewing, W.M., Jardentzky, W.S. and Press, F., "Elastic Waves in Layered
Media", McGraw-Hill, New York, 1957.
[19]
~en, T.E., "Surface Wave Phenomene in Ultrasonic", Prol1ess in Applied
Materials Reseorch, R' 71,1965.
[20]
Adler, E. L., "To be publ ished Il •
[21]
Lim, T.C., "Elastie Surface Wave Propagating in Anisotropie Crystals",
Ph.D. Thesis, July, 1968. MeGill University.
[22]
Wilde, D.J., "Optimum Seeking Methods", Prentice Hall, New Jersey,
1964.
[23]
Anderson,O.L., "Physieal Acousties", Vol. 3B, Chapter Il, edited by
Mason, W.P., Academie Press, New York, 1965.
[24]
Vietorov, A.V., "Rayleigh and Lamb Waves", Plenum Press, New York,
1967.
[25J
Rollins, F.R. Jr., "Critical Ultrasonie Refleetivity - A Neglected Tooi for
Material Evaluation", Material Evaluation XXIV, p. 683, 1966.
[26 J
Ergin, K., "Energy Ratio of the Seismic Waves Reflected and Refracted at
a Rock-Water Boundary", Bull. of Seism. Society of America,
Vol. 42, p. 349, 1952.
[27 J
Diachok, 0.1. and ~yer, W. G. , liVe loeity Measurement of Lateral
Beam Displacement upon Reflection ", IEEE Transactions of Sonies
and Ultrasonics, Oetober, 1969.