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Scattering of an obliquely incident Rayleigh wave in an elastic quarterspace
WAVE MOTION 8 (1986) NORTH-HOLLAND
27-41
27
SCATTERING OF AN OBLIQUELY ELASTIC QUARTERSPACE
INCIDENT
RAYLEIGH
WAVE
IN AN
A.K. GAUTESEN Ames Laboratory,
Received
and Department
21 December
of Mathematics,
Iowa State University,
Ames, IA 50011, U.S.A.
1984
We study the three-dimensional problem of scattering of waves in a homogeneous, isotropic, linearly elastic quarter space. We obtain an equation for the Fourier transform of the normal displacements on the free surfaces. For oblique incidence of a Rayleigh surface wave, we numerically solve this equation. Reflection and transmission coefficients are plotted versus angle of incidence. For angles of incidence smaller than a critical angle, we observed that no energy is radiated into the solid by body waves. The farfield scattering patterns are also plotted.
1. Introduction
Wave propagation in a homogeneous, isotropic, linearly elastic quarter space is a canonical problem. In a review paper, Knopoff [7] has discussed the difficulties associated with this problem. Recently Gautesen [4] considered two-dimensional, steady state wave propagation in a quarter space with tractionfree boundary conditions. He separated this problem into one which was symmetric and one which was antisymmetric with respect to the plane that bisects the quarter space. He was then able to obtain uncoupled equations for the Fourier transform of the normal and tangential displacements on the free surfaces. He then numerically solved these equations for incidence of a Rayleigh surface wave, and presented graphs of the complex amplitudes of the Rayleigh waves reflected and transmitted by the comer versus Poisson’s ratio. Subsequently, Gautesen [5] considered the same problem for incidence of a plane wave corresponding to longitudinal motions. In this paper he also derived simple expressions for the farfield scattering patterns (FFSPs) in terms of the Fourier cosine transform of the displacements on the free surfaces. The FFSPs also represent diffraction coefficients in a Geometrical Theory of Diffraction-see e.g. [l]. In this paper we study the three-dimensional scattering of waves in a quarter space. In Section 2, we separate this problem into two problems which have symmetries with respect to the plane that bisects the quarter space. We are then able to obtain a single equation which involves only the Fourier transform of the normal displacements on the free surfaces. This equation is too difficult to solve analytically. We also obtain an equation for each of the tangential displacements in terms of the normal displacement. In Section 3, we analyze our equation which determines the Fourier transform of the normal displacements on the free surfaces for oblique incidence of a Rayleigh surface wave. We solve the resulting equation numerically. In Section 4, we derive expressions for the farfield scattering patterns (FFSPs) in terms of the Fourier transform of the normal displacement on the free surfaces. An interesting consequence of these expressions is that the FFSPs corresponding to longitudinal or vertical polarized transverse motions vanishes on each of the free surfaces, while the normal derivative of the FFSPs corresponding to horizontally polarized 0165-2125/86/%3.50
@ 1986, Elsevier Science Publishers
B.V. (North-Holland)
28
A.K. Gautesen / Rayleigh
transverse
motions
three-dimensional In Section and phase
5 we present
Theory
numerical
transferred
than
to the transmitted coefficient
the amplitude
2. Formulation
results for Poisson’s
versus
a critical
the angle value,
and reflected
vanished and phase
The FFSPs
also represent
diffraction
coefficients
in a
of Diffraction.
and reflection
by the corner
smaller
the reflection plotted
on the free surfaces.
of the transmission
and reflected incidence
vanish
Geometrical
wave in elastic quarterspace
corresponding
of incidence
we found surface
that
to the surface
of the incident all the energy
surface
of the amplitude wave transmitted
wave.
in the incident
For angles
surface
waves. Also, there was an angle of incidence
and the amplitude of the FFSPs
ratio v = $ and $ We give graphs
coefficients
of the transmission
for five angles
coefficient
equalled
of
wave is for which
one. We also
of incidence.
and derivation
We consider three-dimensional, steady state wave propagation in a homogeneous, isotropic, linearly elastic quarter space which occupies the first quadrant of the Cartesian plane (see Fig. 1). We shall denote this region by Z, the boundary of this region by B, and the remaining region in the Cartesian plane by E. Motion is generated by incident waves r?” from line-loads located in region I. These waves satisfy the differential equation (.9?~~“)~=0,
XE E
(2.1)
where (9U)i = rVJ++pw2Ui,
(2.2)
3 = hUJ&rj. + p( Uij + Q).
(2.3)
p is the density of the elastic solid, w is circular frequency of the incident wave, A and p are the Lame constants, and 6, is the Kronecker delta. Here, we have used the convention that a repeated roman subscript implies summation from 1 to 3 and the shorthand notation: usj = au,/ax,
(2.4)
In the sequel, we find it convenient symmetric (I= 2) and antisymmetric
Ufnl(Xi, x2, x3) =where
ui” is recovered
to divide the incident field ui” into two fields ui”‘, I= 1,2 which are (I = 1) with respect to the plane x, = x3. Thus,
1 Ut”(X,, x2, XJ - UEi(X3, x2, x,),
I= 1,
2 { U~(X,,X2,X3)+U~,(X3,X*,X,),
1=2
(2.5)
from ui”’ by
2 uy=
1
Ujn’.
(2.6)
I=1
Note that uinf also satisfies
(2.1) and has the property
@‘(Xi, x2, x3) = (-1)Q!!;_x3,
that
x2, x,).
The total and scattered fields corresponding to ui”’ also have the property defined drop the superscript 1 for simplicity of notation. We emphasize that the equations solved for both I= 1 and I= 2, and the corresponding results added as in (2.6).
(2.7) by (2.7). Hereafter, derived
we
below must be
A.K. Gautesen / Rayleigh wave in elasticquarterspace
The total field u is related u = us=+ “i”, The scattered
(L?U”)i =O,
q-Tnj =
field us’ by
XE I.
field satisfies
and the boundary
to the scattered
29
(2.8)
the differential
equations
XE I
(2.9)
conditions XEB
-Tinj,
where n is the unit interior
(2.10)
normal
to I as shown
in Fig. 1.
We can follow the development in Gautesen [4] to obtain in terms of displacements on the free surfaces B: u,(r)H(I)
= u?(x) -
Tz;k(x-Y)nj(Y)ui(Y)
the following
expression
for the total field
(2.11)
dA,
B
where H(I)
=
1,
XE I,
0,
XE E,
(2.12)
y denotes a point on B, and T$:~ is defined in the Appendix. The importance of (2.11) is that it is valid for all x, not just for x E I. We let x3 approach zero from the lower half-space (i.e. from within the region E) to achieve cc t&x,,
00
X&O) = I
-03
I0
{TE;k(xl - s1, x2-s29 +(-1)‘G:k(X,,
x2-s2,
o-)“i(sl,
s2)
-s~)v~-~(s~,
~2))
ds, ds2,
--oo < x,, x2 < 03 (2.13)
where we have written ui(sI,
s2)
=
out the integral
4ts19
and used the symmetry ui("+9
We remark
s2~ sl)
=
s2,
over each of the free surfaces,
defined (2.14)
O+)
relation (-1)‘y4-i(s1,
that if we had let x3 approach
(2.13). The important property on the semi-infinite plane defined by
(2.15)
S2).
zero from the upper half-space,
we would
also have obtained
of (2.13) is that this equation is valid on the entire (x,, x,)-plane and not just the Fourier transform and its inverse 0 < x1 < co, --co < xZ < 00. We introduce
(2.16)
(2.17)
30
AK.
Gautesen / Rayleigh
wave in elastic quarterspace
where c2,= (A +2P)lP, ka=wlc,, The Fourier
c+=
(2.18)
PIP,
cu=L,T.
transform
(2.19)
of (2.13) is
2~*Pk”(51,52,0)
= &jfij(51,6)
+ (-l)‘{Bkjcj(yL,
where
25: [ yL = ( 1 py, Y== 2
0
[ -&al/
B=
YT
&552
c=
-bLYT
&a21yL 52a21yL a2
25152
(2.22)
, I
a(&)
&d&)/Y=
2&(K2-5:)/Y=
-25152
51d52)lYT
-x:52/
K =
1 3
YT
-25:
-25152
(K2-12)1’2,
(2.21)
,
K2
2&&rL
-S1a(l)lrT
[
52allYL
25:52/rL
2&YL
1
SldYL
0
K2
A=
(2.20)
52) + Ckjfij(YT, 52))
(2.23)
(2.24) (2.25)
CL/C,,
(2.26)
5=(5:+5:)“‘, U(l)
=
(2.27)
K2-212,
a1 = a(5)
-
YLYT,
(2.28)
a2=a(yL)-25:. Multiplication
(2.29)
of (2.20) by ILK’
yields
R?T/J&, ‘$2)+(-1)‘{~Ld~+Pd~+~THdfH}= where
R is the Rayleigh
function
defined
(2.30)
Wk(&, &)
by
R = a2(l)+412yLyT, dL = (X,Y~,
252~=,
(2.31) (2.32)
a(5)).
(2.33)
~TV=~45:~L~T+~~52)~~l),~5,52~~~5)-~yLyT),--2~25,YL)l~~2-5:)~ dTH =
(&52(4yLyT+
vL=25,[YL&(YL, V -=“=
a([)),
52)+52212(YL,
[5;-4Y=‘)l~l(Y=,
- &a(c) 52)l+a2G3(YL.
52)+52YTfi2(YT,
V -TH = &m2YT&(YT, w/( = RA~‘ill”(~,,
-a’(l)
6) 52,O)
- d52b52(YT,
-45:yLyT,
252YL(YT)*)/(K2-
52)+2&52uY=,
(2.34) (2.35)
52),
52)‘CX1YTfi3(YT,
6%
52), 52)l/YTT
(2.36) (2.37) (2.38)
A.K. Gautesen / Rayleigh wave in elastic quartet-space
31
We remark that (2.30) holds for all real values of & and &-there is no unknown function analytic in the lower half of the complex &-plane. To decouple the system of equations (2.30), we take, with respect to the variable &, the even part of (2.30) when k = 3, the odd part of (2.30) when k = 1, and the even part of (2.30) when k = 2. The result is N&(5,,
&)+ M-51,
5,)1/2+(-l)‘{a(5)a*~~,(rL,6b-45:rLrTG(YT,
= [w,(&, 52) + %(-&r
N&(5,,
6) - &(-CL
52))
5*)1/2,
(2.39)
&)I/2
= -%(-wwT~3(rL,
52)+[[5:yTa(C)+522RlYTllj3(yT, 6m+b1(&,
a-WI(--5*,52)1/2, (2.40)
= -25*(-wwT&(rL,
5*)+5:[YTa(~)-RIyTlV3(YT,5*)/5*~+b*(51,52)+w*(-51,52)1/2. (2.41)
Equations (2.39)-(2.40) represent a decoupled system of equations. We solve (by whatever means) (2.39) for i&. Then with respect to the variable &, (2.40) and (2.41) define the Fourier sine and cosine transform of 6, and i&, respectively, in terms of ti3. Since oa cc (2.42) &(5r, 52) = k’L ui(xI, ~2) exp[ikL(x,& + ~31 dx, dx,, II0 -m this is sufficient to determine o, and v2. For example, (2.43)
3. Analysis for surface wave incidence We consider a Rayleigh surface wave incident on the lower free surface (see Fig. 2) with displacements defined by URi”(xl, x2, 0) = qR’ exp[ikR(-x1 sin &+x2
cos &)]/cos”‘(2&)
(3.1)
where 4 Ri = ((-l)i
sin 4R sin OR,cos & sin OR,cos OR), i = 1, 2,
(3.2)
eR = 2i COSh-'( CT/CR),
(3.3)
kR= w/CR
(3.4)
and CRis the speed of Rayleigh waves. Here, (bR is the angle the incident wave makes with the corner, and the incident wave is normalized so that it has unit complex amplitude. For this problem, all fields have a common factor of exp[ik,x, cos +R], with no other dependence of the spatial variable x2. Hereafter, we omit this common factor. This factor produces a common factor of 2&(e2+ KR cos #R) in the Fourier transform of Vi,where KR =
CL/
CR.
(3.5)
A.K. Gautesen / Rayleigh wave in elastic quarterspace
32
_-
----,
-I*
7
P‘
REFLECTED
/
t”
S.W.
/ /
I
/ /
/
I
/ /
Y
‘h‘
---------
Fig. 1. Geometry
and coordinate
system.
Fig. 2. Incident,
Since V, has no other dependence on &;, we write Q([i) (2.30), as well as (2.39)-(2.41) with &=
-KR
COS
------ --
,$’ 5.
/
/’
and transmitted
in place of Ci(t,, &). Then
surface
waves.
&(&) satisfies
(2.20),
(3.6)
up = 0. Thus,
fiy = Wi = 0. The incident
(3.7)
surface
URre(X1,
0)
7
/
&.
Also in this problem
/
/
INCIDENT >.w.
reflected
/
/
=
and to a transmitted
wave gives rise to a reflected RcqR2 eXp[ik,X,
surface
URtr(O, x3) = T”(qy,
wave on the same free surface
defined
by
Sin &]/coS”‘(2&)
wave defined qy,
surface
qp)
(3.8)
by
exp[ik&
sin &]/cos1”(2&)
where R’ and T’ are the reflection and transmission problem into an antisymmetric (I = 1) and symmetric the surface waves uR on the lower free surface is
(3.9)
coefficients, respectively. When we separate problem (I = 2), we find that the displacement
this of
2uR(x,, 0) = uRi”(xl, 0) + A,uRre(xl, 0)
(3.10)
A, = R’+(-1)‘T”
(3.11)
where
The Fourier
transform
of (3.10) is (3.12)
-2i~R=[qR1/(~1-~R)+ArqRZ/(~1+~R)]/cos1’2(2eR) where & is defined c&=
KR
Sill
by (3.13)
4~.
In this problem motions are generated by the incoming (2.39) that is numerically tractable, we write &([I) = ii;+
&‘$(&)/[2
c0s1’*(2&)]
Rayleigh
wave. To obtain
an alternate
form of
(3.14)
AX
33
Gauiesen / Rayleigh wave in elastic quarterspace
where $‘( 5,) is analytic in the upper half of the complex 5, -plane as well as in a neighborhood of & = f &. The poles at 6, = f& in &(&), which yield the Rayleigh surface waves, are contained in ii!. Upon substitution of (3.13) into (2.39), we obtain the result RM(Sr)+
w%yL) -~~?Y~Y~$‘(Y~)] = i[O-(5,) - A@(&)1 ~~(-51)1/2+(-1)'{a(5)
(3.15)
where (3.16)
9*(5,)=(-l)‘{-4T:rLrTl(SR*yT)+~(3)~2/(51fyL)}+5RR/(52R--5:).
To determine A, from (3.15), we evaluate this equation at t1 = &. Note that at ,$I= &, R vanishes and $‘(=t&) is bounded. The result is O+(SR)A, = Q-(SR)+i(-I)‘{a(KR)[~(~~)
-25;lfi3'(rid -~&Y~Y%‘(Y:))
(3.17)
where 7;; = i[ K; - c’,/ c;]“‘,
(3.18)
(Y= L, T.
When we substitute for A, from (3.17) into (3.15), we obtain an equation where the only unknown is i$‘. This equation is solved numerically by exactly the same procedure as described in Gautesen [4]. Then, we compute the reflection and transmission coefficients from (3.11) and (3.17).
4. Farlield scattering patterns In this Section we compare the farfield scattering patterns (FFSPs) which for this problem are also diffraction coefficients-see, e.g. [l]. From (2.11), it follows that in the fartield, (4.1)
where r(cos 8, sin e) = (x,, x3), E”(r) = exp[i{k,r(&/c2, 4L= (6, -5*, ~9,
(4.2)
- 5:)“‘+P/4}]/[8pkLT(cZL/c~
fil = eL= (1 -W*
4 n=(-rT,0,51)/(K2-5:)“2, QTH=
(5152,
K*-
t2YT)/[‘dK2-
&“*I,
(4.3)
~0s 0,
(4.4)
&=&‘(K*-@‘*COS 4%
- .c$~‘~]“*,
61=
(4.5)
8,
(4.6)
tT,
iK2~“(e)=2YL[~~~~(-~~)-~2~2(-~~)1+[~(~,)-2~:14(-~)
+(-1)‘{25,[yL~,(-yL)-52U2(-YL)I+[a(yL)-25:1~~,(-yL)}, i(K2-~~)“2~n(~)
=[~:-~(~~)l~,(-~~)-~~[~2~2(-~~)-2YT~~(-~~)l -(-1)‘{5~-a(rT)~,(-YT)-YT[52~2(-yT)-25,63(-YT)1},
iK(K2-#‘*DTH(@=
(4.7)
&‘5L,
(4.8)
&=ifT,
rT[25152~l(-51)+a(5)~2(-51)+252rT4(-51)l +(-1)‘{51[252rT~l(-rT)+u(52)~22(-YT)+25152~~3(-YT)1},
61=
tT*
(4.9)
34
A.K. Gauiesen / Rayleigh waue in elastic quarterspace
We now derive we subtract
expressions
for the FFSPs
the scalar product
_iK2DL(8)/2
in terms of &([,)
alone.
of (2.30) with (25,yL, 2&yL, a(l))/R
= 2YL[51%(51)-t
52G(51)1 +[a(51)
We begin
with D”(0).
From
(4.7)
to achieve
-x:lGY&)
+(-1)‘{25,[rLV’;(YL)+52Uq(YL)I+r,(YL)-25:1~-P(yL)},
(4.10)
5,=5L
where Z(5)
=
vr(5) =
Lvi(5) + fii(-5)1/2,’
(4.11)
[G(5)+ 8i(-5)l/2.
(4.12)
From (2.40) and (2.41) we find (4.13)
2~L[5,~~(5,)+52~5(5,)1=-[~(5*)-25:1~;(51)-(-l)f[~(rL)-25:l~~(rL)
where we have substituted for &(yT) from (2.39). Then substitution into (4.10) from (4.13) and from (4.13) with 5, replaced iKZ~L(~)I4=[~(51)-25:l~~(51)+(-l)’[~(rL)-25:l~S(rL),
Next we deal with DTv(0). [:, -5,c2, -2 yT5,)/ R to achieve
From
(4.8)
by yL yields (4.14)
51=5L.
we
add
the
scalar
product
of
(2.30)
with
(a(&,)-
--i(K2-5:)1’2~TV(~)/2=[5:-~(51)l~~(5,)+5152~~(~l)+225*YT~~(S,) -(-1)‘{[5:-u(rT)lV~(yT)+YTS2ZIS(YT)+25,YTB~(yT)},
tI=tT.
(4.15) From (2.40) and (2.41), we find (4.16)
r5:-~(51)l~~(51)+5152~S(5,)=25,rT[-~4(5,)+(-1)’~3(YT)I
where we have substituted (K2-&)“2DTV(e)
for &(yL) from (2.39). Upon =
-8i[,rT[i$(~r)
substitution
(4.17)
5, = &.
-(-l)‘v;(yT)],
Finally, to obtain an alternate expression for D’“(O), yT(25,5,, -u(t2), 2-yTt2)/R from (4.9), and then substitute
from (4.16) into (4.15), we obtain
we subtract the scalar product from (2.39)-(2.14) to achieve
of (2.30) with
(4.18)
~(~~-~~)“~~‘~(~)=-8i~~~(~~)~~~(~,)+(-1)’~~~~(~~)~. We observe
from (4.17) that DTv( 0) = 0, at 0 = 0, 90”. This is in agreement
of Gautesen [6], which also imply that D”(e) should vanish obvious from (4.14). Thus, we consider in lieu of (4.14): eq. (4.14)-
F([,)
eq. (2.39)-(-l)‘F(yL){eq.
with the asymptotic
on the free surfaces.
results
That this is true is not
(4.19)
(2.39)(,1=,1}
where F(5,) The result
=
&/[a(5)(1-
(4.20)
‘91.
is iK*(l-&0”(8)/4=
r”{‘y”U(l)[ti;(&)-
U3(51)]-45:52yTtije(5,)/U(~)+451(yL)3yTZ13(yT)/U2}
+(-1)‘5,~51a2[~;(rL)-~,(rL)1-4(rL)2(1-5:)rT~S(rL)l~2
+4yLS:yT~-3(YT)Ia(5)}.
5, = CL
(4.21)
AX.
35
Gautesen / Rayleigh watre in elastic quarterspace
where fL
[ K2_ (#]‘/2.
Now from (4.21), we find that DL( 6) = 0 at 0 = 0, 90”. Also, we observe
from (4.18) that aDTH(B)/M
at e = 0,900. We remark that all expressions
of plane waves corresponding
either longitudinal
or transverse
for FFSPs motions,
remain
provided
valid for incidence that the definition
(3.6) of t2 is changed
= 0, to
appropriately.
5. Results and discussion When the incident Rayleigh surface-wave ray intersects the corner (1) reflects a surface-wave ray on the same free surface, (2) transmits a surface-wave ray to the other free surface, and (3) diffracts The inner
two cones of body-wave
cone of rays corresponds I$~ = arc cos[
The
K~
cos
& defined
r#+= In the farfield
XC
rays.
to longitudinal
motions
and has a half angle 4L defined
by
q&]
(5.1)
outer cone of rays corresponds
a half angle
it
to vertically
and horizontally
polarized
transverse
motions,
and has
by
COS[(KdK)
COS
I$~].
(5.2)
these waves are, of course,
defined
by (4.1). When the angle of incidence
satisfies (5.3)
qh,<&-=arcCOS(K/Kd
both cones consist of non-propagating rays. Thus all of the energy in the incident surface wave is transferred to the reflected and transmitted surface waves, and we must have the following identities: IAll = JA2) = IR”* T”I = 1,
In order for the reflection
&<
and transmission
E=[~RC]2+(Tc]2]1’2=1,
4;.
(5.4) coefficients
&
to satisfy
(5.4), we must have (5.5)
and Phase{ T”} - Phase{ R’} = (2n + 1)90”,
c#+ < 4;
(5.6)
where n is an integer. We remark that this result has been observed analytically by Freund [3] (for a surface wave obliquely incident on a semi-infinite crack) and by Angel and Achenbach [2] (for a surface wave obliquely incident on an edge crack). In Fig. 3, we plotted IT’I, (R’I and E (as defined by (5.5)) versus the angle of incidence 4R for Poisson’s ratio Y = t. We used 90 equally spaced values of 4 R. We note that E = 1 for C&C &= 23.2”, and that there is a value of &= 13” where ( T’I = 1 and IR’I = 0. In Fig. 4, we plotted the phase of T’, the phase of R”, and the difference of these two phases. We note that for &< r#& the phase difference is either 90” or 270” as required by (5.6). The 180” jump in the phase of R” occurs at the value of #Jo where R’=O. The phase of the transmission coefficients is fairly constant, varying between 266 and 286”.
36
A. K. Gautesen / Rayleigh
wave in elastic quarterspace
For Y = f, we have plotted in Figs. 5 and 6 the same quantities as in Figs. 3 and 4, respectively. For this value of V, &= 21.2”. Again, these curves satisfy (5.5) and (5.6). Also R” = 0 for 4K = 15”. Except in a neighborhood of & = &, the phases the same as for Y = i. For v = a, we have plotted
of the transmission
in Figs. 7 and 8 the amplitude
and reflection and phase,
versus 0 for & = 90, 85, 80, 75, 70”. For v = f, we have plotted When the incident angle C#IKis smaller than
coefficients
for v =! are nearly
respectively,
of DL( 0) from (4.21)
the same quantities
in Figs. 9 and
d’,=arccos(l/~K) the longitudinal
10.
(5.7)
body wave-rays
are non-propagating.
Figs. 7 and 9, we note that the amplitude
For v = a, 4: = 62.2” and for v = f, 4: = 57.9”. From
of DL( 0) for Y = a is somewhat
DL( 0) for v = f at corresponding angles of incidence. third quadrant for all angles of incidence.
For both
larger than
the amplitude
of
v = $ and f the phase of DL( f3) lies in the
For & = 90”, Gautesen [4] plotted the amplitude of ]D”( 0)l versus 13for both v = $ and f. He used a formula for D”(0) which involved only the tangential displacements on the free surfaces. He solved for these displacements numerically. Our expression (4.21) for DL( 0) involves only the normal displacements on the free surfaces. When we compared our curves with the corresponding ones of Gautesen [4] we found no significant differences. We add that the phases agreed as well. For v = $, we have plotted in Figs. 11 and 12 the amplitude and phase, respectively, of DTV( (3) versus 6 for & = 90,75,60,45,30”. For v = $, we have plotted the same quantities in Figs. 13 and 14. We observe that at corresponding angles of incidence, the amplitude of DTv( 0) is somewhat larger for v = f than for v = a. Also, the phases of DTv( 0) for v = a and f are nearly the same at corresponding angles of incidence. For v =$, we have plotted in Figs. 15 and 16 the amplitude and phase of D=“(e) versus 0 for &=90, 75, 60, 45, 30”. For v = 3, we have plotted the same quantities in Figs. 17 and 18. Again, at corresponding angles of incidence, the amplitude of DTH( f3) is somewhat larger for v = 4 than for v = $. Also, for 0 > 15”, the phases of DTH( 19) for v = i and $ are nearly the same at corresponding angles of incidence.
Appendix In this Appendix,
we give the fundamental
stress tensor and its Fourier
transform.
This tensor is defined
by
where
k, and K are given by (2.19) and (2.25), and (A.2)
r = 1x1, H(k,r) The Fourier
= exp[ik,r]/(4Tr).
transform
of (A.l)
(A.3) is
(A.4) where
ye is defined (C, E" =
Pz”, C)
by (2.24) and (2.25), = (-&,
exp[ik,y”]x,(],
-e2, Y" sgdxd), LY= L, T.
Q = L, T,
(A.5) (A.6)
A. K. Gautesen / Rayleigh wave in elastic quarterspace I
,
I
I
:
I
kl
8
i
\I ,i
I
I
:
37
I
I
i
\
I
\_
I
I
\
c I I
I II
1
3anlIldWV
[3301 3SVHd
30ll111dWV
(330) 3SVHd
: I 1
A.K. Gautesen / Rayleigh
38
wave in elastic quarterspace
.’
!,‘, 1v
:
/.......
,’
3CJlLIldWt- dSJ+1
m
1
8
,:“ 1: i
:
ii
l
3SVHd - dS&J-1
3 N
3ClillIldWV - dSj+1
0
N
0 N
3SVHd - dS.U-1
A.K. Gautesen / Rayleigh wave in elastic quarterspace
-s-
U-J
m
N
30flLIldWV- dSJ+hl
u-l
d
m
N
30fILIldWV- dSA-Al
3SVHd - dSj+hl
0
39
A.K. Gauresen f Rayleigh waoe in elastic quarterspace
40
-1
I :
30flLIldWV- dS4-HL
m
N
3CinlIldWV- dSL!-HI
3SVHd - dSj+Hl
I
A. K. Gautesen / Rayleigh wave in elastic quarterspace
41
Acknowledgment This work was supported Energy
Research,
(in part) by the Applied
U.S. Department
of Energy,
under
Mathematical Contract
Sciences
subprogram
of the Office of
No. W-7405-82.
References [I] J.D. Achenbach, A.K. Gautesen and H. McMaken, Ray Methods for Waves in Hasric Solids with Applications to Scattering by Cracks, Pitman, Boston (1982). “Reflection and transmission of obliquely incident Rayleigh waves by a surface breaking [2] Y.C. Angel and J.D. Achenbach, crack”, .I. Acousr. Sot. Amer. 75, 313-319 (1984). [3] L.B. Freund, “The oblique reflection of a Rayleigh wave from a crack tip”, Internat. J. Solids Struct. 7, 1199-1205 (1971). [4] A.K. Gautesen, “Scattering of a Rayleigh wave by an elastic quarter space”, .I. Appl. Mech., accepted for publication. [5] A.K. Gautesen, “Scattering of a plane longitudinal wave by an elastic quarter space”, Wave Motion 7, 557-568 (1985). [6] A.K. Gautesen, “On matched asymptotic expansions for two dimensional elastodynamic diffraction by cracks”, Wave Motion I, 127-140 (1979). [7] L. Knopoff, “Elastic wave propagation in a wedge” , in: J. Miklowitz, ed., Wave Propagation in Solids, ASME, New York (1969) 3-42.
27-41
27
SCATTERING OF AN OBLIQUELY ELASTIC QUARTERSPACE
INCIDENT
RAYLEIGH
WAVE
IN AN
A.K. GAUTESEN Ames Laboratory,
Received
and Department
21 December
of Mathematics,
Iowa State University,
Ames, IA 50011, U.S.A.
1984
We study the three-dimensional problem of scattering of waves in a homogeneous, isotropic, linearly elastic quarter space. We obtain an equation for the Fourier transform of the normal displacements on the free surfaces. For oblique incidence of a Rayleigh surface wave, we numerically solve this equation. Reflection and transmission coefficients are plotted versus angle of incidence. For angles of incidence smaller than a critical angle, we observed that no energy is radiated into the solid by body waves. The farfield scattering patterns are also plotted.
1. Introduction
Wave propagation in a homogeneous, isotropic, linearly elastic quarter space is a canonical problem. In a review paper, Knopoff [7] has discussed the difficulties associated with this problem. Recently Gautesen [4] considered two-dimensional, steady state wave propagation in a quarter space with tractionfree boundary conditions. He separated this problem into one which was symmetric and one which was antisymmetric with respect to the plane that bisects the quarter space. He was then able to obtain uncoupled equations for the Fourier transform of the normal and tangential displacements on the free surfaces. He then numerically solved these equations for incidence of a Rayleigh surface wave, and presented graphs of the complex amplitudes of the Rayleigh waves reflected and transmitted by the comer versus Poisson’s ratio. Subsequently, Gautesen [5] considered the same problem for incidence of a plane wave corresponding to longitudinal motions. In this paper he also derived simple expressions for the farfield scattering patterns (FFSPs) in terms of the Fourier cosine transform of the displacements on the free surfaces. The FFSPs also represent diffraction coefficients in a Geometrical Theory of Diffraction-see e.g. [l]. In this paper we study the three-dimensional scattering of waves in a quarter space. In Section 2, we separate this problem into two problems which have symmetries with respect to the plane that bisects the quarter space. We are then able to obtain a single equation which involves only the Fourier transform of the normal displacements on the free surfaces. This equation is too difficult to solve analytically. We also obtain an equation for each of the tangential displacements in terms of the normal displacement. In Section 3, we analyze our equation which determines the Fourier transform of the normal displacements on the free surfaces for oblique incidence of a Rayleigh surface wave. We solve the resulting equation numerically. In Section 4, we derive expressions for the farfield scattering patterns (FFSPs) in terms of the Fourier transform of the normal displacement on the free surfaces. An interesting consequence of these expressions is that the FFSPs corresponding to longitudinal or vertical polarized transverse motions vanishes on each of the free surfaces, while the normal derivative of the FFSPs corresponding to horizontally polarized 0165-2125/86/%3.50
@ 1986, Elsevier Science Publishers
B.V. (North-Holland)
28
A.K. Gautesen / Rayleigh
transverse
motions
three-dimensional In Section and phase
5 we present
Theory
numerical
transferred
than
to the transmitted coefficient
the amplitude
2. Formulation
results for Poisson’s
versus
a critical
the angle value,
and reflected
vanished and phase
The FFSPs
also represent
diffraction
coefficients
in a
of Diffraction.
and reflection
by the corner
smaller
the reflection plotted
on the free surfaces.
of the transmission
and reflected incidence
vanish
Geometrical
wave in elastic quarterspace
corresponding
of incidence
we found surface
that
to the surface
of the incident all the energy
surface
of the amplitude wave transmitted
wave.
in the incident
For angles
surface
waves. Also, there was an angle of incidence
and the amplitude of the FFSPs
ratio v = $ and $ We give graphs
coefficients
of the transmission
for five angles
coefficient
equalled
of
wave is for which
one. We also
of incidence.
and derivation
We consider three-dimensional, steady state wave propagation in a homogeneous, isotropic, linearly elastic quarter space which occupies the first quadrant of the Cartesian plane (see Fig. 1). We shall denote this region by Z, the boundary of this region by B, and the remaining region in the Cartesian plane by E. Motion is generated by incident waves r?” from line-loads located in region I. These waves satisfy the differential equation (.9?~~“)~=0,
XE E
(2.1)
where (9U)i = rVJ++pw2Ui,
(2.2)
3 = hUJ&rj. + p( Uij + Q).
(2.3)
p is the density of the elastic solid, w is circular frequency of the incident wave, A and p are the Lame constants, and 6, is the Kronecker delta. Here, we have used the convention that a repeated roman subscript implies summation from 1 to 3 and the shorthand notation: usj = au,/ax,
(2.4)
In the sequel, we find it convenient symmetric (I= 2) and antisymmetric
Ufnl(Xi, x2, x3) =where
ui” is recovered
to divide the incident field ui” into two fields ui”‘, I= 1,2 which are (I = 1) with respect to the plane x, = x3. Thus,
1 Ut”(X,, x2, XJ - UEi(X3, x2, x,),
I= 1,
2 { U~(X,,X2,X3)+U~,(X3,X*,X,),
1=2
(2.5)
from ui”’ by
2 uy=
1
Ujn’.
(2.6)
I=1
Note that uinf also satisfies
(2.1) and has the property
@‘(Xi, x2, x3) = (-1)Q!!;_x3,
that
x2, x,).
The total and scattered fields corresponding to ui”’ also have the property defined drop the superscript 1 for simplicity of notation. We emphasize that the equations solved for both I= 1 and I= 2, and the corresponding results added as in (2.6).
(2.7) by (2.7). Hereafter, derived
we
below must be
A.K. Gautesen / Rayleigh wave in elasticquarterspace
The total field u is related u = us=+ “i”, The scattered
(L?U”)i =O,
q-Tnj =
field us’ by
XE I.
field satisfies
and the boundary
to the scattered
29
(2.8)
the differential
equations
XE I
(2.9)
conditions XEB
-Tinj,
where n is the unit interior
(2.10)
normal
to I as shown
in Fig. 1.
We can follow the development in Gautesen [4] to obtain in terms of displacements on the free surfaces B: u,(r)H(I)
= u?(x) -
Tz;k(x-Y)nj(Y)ui(Y)
the following
expression
for the total field
(2.11)
dA,
B
where H(I)
=
1,
XE I,
0,
XE E,
(2.12)
y denotes a point on B, and T$:~ is defined in the Appendix. The importance of (2.11) is that it is valid for all x, not just for x E I. We let x3 approach zero from the lower half-space (i.e. from within the region E) to achieve cc t&x,,
00
X&O) = I
-03
I0
{TE;k(xl - s1, x2-s29 +(-1)‘G:k(X,,
x2-s2,
o-)“i(sl,
s2)
-s~)v~-~(s~,
~2))
ds, ds2,
--oo < x,, x2 < 03 (2.13)
where we have written ui(sI,
s2)
=
out the integral
4ts19
and used the symmetry ui("+9
We remark
s2~ sl)
=
s2,
over each of the free surfaces,
defined (2.14)
O+)
relation (-1)‘y4-i(s1,
that if we had let x3 approach
(2.13). The important property on the semi-infinite plane defined by
(2.15)
S2).
zero from the upper half-space,
we would
also have obtained
of (2.13) is that this equation is valid on the entire (x,, x,)-plane and not just the Fourier transform and its inverse 0 < x1 < co, --co < xZ < 00. We introduce
(2.16)
(2.17)
30
AK.
Gautesen / Rayleigh
wave in elastic quarterspace
where c2,= (A +2P)lP, ka=wlc,, The Fourier
c+=
(2.18)
PIP,
cu=L,T.
transform
(2.19)
of (2.13) is
2~*Pk”(51,52,0)
= &jfij(51,6)
+ (-l)‘{Bkjcj(yL,
where
25: [ yL = ( 1 py, Y== 2
0
[ -&al/
B=
YT
&552
c=
-bLYT
&a21yL 52a21yL a2
25152
(2.22)
, I
a(&)
&d&)/Y=
2&(K2-5:)/Y=
-25152
51d52)lYT
-x:52/
K =
1 3
YT
-25:
-25152
(K2-12)1’2,
(2.21)
,
K2
2&&rL
-S1a(l)lrT
[
52allYL
25:52/rL
2&YL
1
SldYL
0
K2
A=
(2.20)
52) + Ckjfij(YT, 52))
(2.23)
(2.24) (2.25)
CL/C,,
(2.26)
5=(5:+5:)“‘, U(l)
=
(2.27)
K2-212,
a1 = a(5)
-
YLYT,
(2.28)
a2=a(yL)-25:. Multiplication
(2.29)
of (2.20) by ILK’
yields
R?T/J&, ‘$2)+(-1)‘{~Ld~+Pd~+~THdfH}= where
R is the Rayleigh
function
defined
(2.30)
Wk(&, &)
by
R = a2(l)+412yLyT, dL = (X,Y~,
252~=,
(2.31) (2.32)
a(5)).
(2.33)
~TV=~45:~L~T+~~52)~~l),~5,52~~~5)-~yLyT),--2~25,YL)l~~2-5:)~ dTH =
(&52(4yLyT+
vL=25,[YL&(YL, V -=“=
a([)),
52)+52212(YL,
[5;-4Y=‘)l~l(Y=,
- &a(c) 52)l+a2G3(YL.
52)+52YTfi2(YT,
V -TH = &m2YT&(YT, w/( = RA~‘ill”(~,,
-a’(l)
6) 52,O)
- d52b52(YT,
-45:yLyT,
252YL(YT)*)/(K2-
52)+2&52uY=,
(2.34) (2.35)
52),
52)‘CX1YTfi3(YT,
6%
52), 52)l/YTT
(2.36) (2.37) (2.38)
A.K. Gautesen / Rayleigh wave in elastic quartet-space
31
We remark that (2.30) holds for all real values of & and &-there is no unknown function analytic in the lower half of the complex &-plane. To decouple the system of equations (2.30), we take, with respect to the variable &, the even part of (2.30) when k = 3, the odd part of (2.30) when k = 1, and the even part of (2.30) when k = 2. The result is N&(5,,
&)+ M-51,
5,)1/2+(-l)‘{a(5)a*~~,(rL,6b-45:rLrTG(YT,
= [w,(&, 52) + %(-&r
N&(5,,
6) - &(-CL
52))
5*)1/2,
(2.39)
&)I/2
= -%(-wwT~3(rL,
52)+[[5:yTa(C)+522RlYTllj3(yT, 6m+b1(&,
a-WI(--5*,52)1/2, (2.40)
= -25*(-wwT&(rL,
5*)+5:[YTa(~)-RIyTlV3(YT,5*)/5*~+b*(51,52)+w*(-51,52)1/2. (2.41)
Equations (2.39)-(2.40) represent a decoupled system of equations. We solve (by whatever means) (2.39) for i&. Then with respect to the variable &, (2.40) and (2.41) define the Fourier sine and cosine transform of 6, and i&, respectively, in terms of ti3. Since oa cc (2.42) &(5r, 52) = k’L ui(xI, ~2) exp[ikL(x,& + ~31 dx, dx,, II0 -m this is sufficient to determine o, and v2. For example, (2.43)
3. Analysis for surface wave incidence We consider a Rayleigh surface wave incident on the lower free surface (see Fig. 2) with displacements defined by URi”(xl, x2, 0) = qR’ exp[ikR(-x1 sin &+x2
cos &)]/cos”‘(2&)
(3.1)
where 4 Ri = ((-l)i
sin 4R sin OR,cos & sin OR,cos OR), i = 1, 2,
(3.2)
eR = 2i COSh-'( CT/CR),
(3.3)
kR= w/CR
(3.4)
and CRis the speed of Rayleigh waves. Here, (bR is the angle the incident wave makes with the corner, and the incident wave is normalized so that it has unit complex amplitude. For this problem, all fields have a common factor of exp[ik,x, cos +R], with no other dependence of the spatial variable x2. Hereafter, we omit this common factor. This factor produces a common factor of 2&(e2+ KR cos #R) in the Fourier transform of Vi,where KR =
CL/
CR.
(3.5)
A.K. Gautesen / Rayleigh wave in elastic quarterspace
32
_-
----,
-I*
7
P‘
REFLECTED
/
t”
S.W.
/ /
I
/ /
/
I
/ /
Y
‘h‘
---------
Fig. 1. Geometry
and coordinate
system.
Fig. 2. Incident,
Since V, has no other dependence on &;, we write Q([i) (2.30), as well as (2.39)-(2.41) with &=
-KR
COS
------ --
,$’ 5.
/
/’
and transmitted
in place of Ci(t,, &). Then
surface
waves.
&(&) satisfies
(2.20),
(3.6)
up = 0. Thus,
fiy = Wi = 0. The incident
(3.7)
surface
URre(X1,
0)
7
/
&.
Also in this problem
/
/
INCIDENT >.w.
reflected
/
/
=
and to a transmitted
wave gives rise to a reflected RcqR2 eXp[ik,X,
surface
URtr(O, x3) = T”(qy,
wave on the same free surface
defined
by
Sin &]/coS”‘(2&)
wave defined qy,
surface
qp)
(3.8)
by
exp[ik&
sin &]/cos1”(2&)
where R’ and T’ are the reflection and transmission problem into an antisymmetric (I = 1) and symmetric the surface waves uR on the lower free surface is
(3.9)
coefficients, respectively. When we separate problem (I = 2), we find that the displacement
this of
2uR(x,, 0) = uRi”(xl, 0) + A,uRre(xl, 0)
(3.10)
A, = R’+(-1)‘T”
(3.11)
where
The Fourier
transform
of (3.10) is (3.12)
-2i~R=[qR1/(~1-~R)+ArqRZ/(~1+~R)]/cos1’2(2eR) where & is defined c&=
KR
Sill
by (3.13)
4~.
In this problem motions are generated by the incoming (2.39) that is numerically tractable, we write &([I) = ii;+
&‘$(&)/[2
c0s1’*(2&)]
Rayleigh
wave. To obtain
an alternate
form of
(3.14)
AX
33
Gauiesen / Rayleigh wave in elastic quarterspace
where $‘( 5,) is analytic in the upper half of the complex 5, -plane as well as in a neighborhood of & = f &. The poles at 6, = f& in &(&), which yield the Rayleigh surface waves, are contained in ii!. Upon substitution of (3.13) into (2.39), we obtain the result RM(Sr)+
w%yL) -~~?Y~Y~$‘(Y~)] = i[O-(5,) - A@(&)1 ~~(-51)1/2+(-1)'{a(5)
(3.15)
where (3.16)
9*(5,)=(-l)‘{-4T:rLrTl(SR*yT)+~(3)~2/(51fyL)}+5RR/(52R--5:).
To determine A, from (3.15), we evaluate this equation at t1 = &. Note that at ,$I= &, R vanishes and $‘(=t&) is bounded. The result is O+(SR)A, = Q-(SR)+i(-I)‘{a(KR)[~(~~)
-25;lfi3'(rid -~&Y~Y%‘(Y:))
(3.17)
where 7;; = i[ K; - c’,/ c;]“‘,
(3.18)
(Y= L, T.
When we substitute for A, from (3.17) into (3.15), we obtain an equation where the only unknown is i$‘. This equation is solved numerically by exactly the same procedure as described in Gautesen [4]. Then, we compute the reflection and transmission coefficients from (3.11) and (3.17).
4. Farlield scattering patterns In this Section we compare the farfield scattering patterns (FFSPs) which for this problem are also diffraction coefficients-see, e.g. [l]. From (2.11), it follows that in the fartield, (4.1)
where r(cos 8, sin e) = (x,, x3), E”(r) = exp[i{k,r(&/c2, 4L= (6, -5*, ~9,
(4.2)
- 5:)“‘+P/4}]/[8pkLT(cZL/c~
fil = eL= (1 -W*
4 n=(-rT,0,51)/(K2-5:)“2, QTH=
(5152,
K*-
t2YT)/[‘dK2-
&“*I,
(4.3)
~0s 0,
(4.4)
&=&‘(K*-@‘*COS 4%
- .c$~‘~]“*,
61=
(4.5)
8,
(4.6)
tT,
iK2~“(e)=2YL[~~~~(-~~)-~2~2(-~~)1+[~(~,)-2~:14(-~)
+(-1)‘{25,[yL~,(-yL)-52U2(-YL)I+[a(yL)-25:1~~,(-yL)}, i(K2-~~)“2~n(~)
=[~:-~(~~)l~,(-~~)-~~[~2~2(-~~)-2YT~~(-~~)l -(-1)‘{5~-a(rT)~,(-YT)-YT[52~2(-yT)-25,63(-YT)1},
iK(K2-#‘*DTH(@=
(4.7)
&‘5L,
(4.8)
&=ifT,
rT[25152~l(-51)+a(5)~2(-51)+252rT4(-51)l +(-1)‘{51[252rT~l(-rT)+u(52)~22(-YT)+25152~~3(-YT)1},
61=
tT*
(4.9)
34
A.K. Gauiesen / Rayleigh waue in elastic quarterspace
We now derive we subtract
expressions
for the FFSPs
the scalar product
_iK2DL(8)/2
in terms of &([,)
alone.
of (2.30) with (25,yL, 2&yL, a(l))/R
= 2YL[51%(51)-t
52G(51)1 +[a(51)
We begin
with D”(0).
From
(4.7)
to achieve
-x:lGY&)
+(-1)‘{25,[rLV’;(YL)+52Uq(YL)I+r,(YL)-25:1~-P(yL)},
(4.10)
5,=5L
where Z(5)
=
vr(5) =
Lvi(5) + fii(-5)1/2,’
(4.11)
[G(5)+ 8i(-5)l/2.
(4.12)
From (2.40) and (2.41) we find (4.13)
2~L[5,~~(5,)+52~5(5,)1=-[~(5*)-25:1~;(51)-(-l)f[~(rL)-25:l~~(rL)
where we have substituted for &(yT) from (2.39). Then substitution into (4.10) from (4.13) and from (4.13) with 5, replaced iKZ~L(~)I4=[~(51)-25:l~~(51)+(-l)’[~(rL)-25:l~S(rL),
Next we deal with DTv(0). [:, -5,c2, -2 yT5,)/ R to achieve
From
(4.8)
by yL yields (4.14)
51=5L.
we
add
the
scalar
product
of
(2.30)
with
(a(&,)-
--i(K2-5:)1’2~TV(~)/2=[5:-~(51)l~~(5,)+5152~~(~l)+225*YT~~(S,) -(-1)‘{[5:-u(rT)lV~(yT)+YTS2ZIS(YT)+25,YTB~(yT)},
tI=tT.
(4.15) From (2.40) and (2.41), we find (4.16)
r5:-~(51)l~~(51)+5152~S(5,)=25,rT[-~4(5,)+(-1)’~3(YT)I
where we have substituted (K2-&)“2DTV(e)
for &(yL) from (2.39). Upon =
-8i[,rT[i$(~r)
substitution
(4.17)
5, = &.
-(-l)‘v;(yT)],
Finally, to obtain an alternate expression for D’“(O), yT(25,5,, -u(t2), 2-yTt2)/R from (4.9), and then substitute
from (4.16) into (4.15), we obtain
we subtract the scalar product from (2.39)-(2.14) to achieve
of (2.30) with
(4.18)
~(~~-~~)“~~‘~(~)=-8i~~~(~~)~~~(~,)+(-1)’~~~~(~~)~. We observe
from (4.17) that DTv( 0) = 0, at 0 = 0, 90”. This is in agreement
of Gautesen [6], which also imply that D”(e) should vanish obvious from (4.14). Thus, we consider in lieu of (4.14): eq. (4.14)-
F([,)
eq. (2.39)-(-l)‘F(yL){eq.
with the asymptotic
on the free surfaces.
results
That this is true is not
(4.19)
(2.39)(,1=,1}
where F(5,) The result
=
&/[a(5)(1-
(4.20)
‘91.
is iK*(l-&0”(8)/4=
r”{‘y”U(l)[ti;(&)-
U3(51)]-45:52yTtije(5,)/U(~)+451(yL)3yTZ13(yT)/U2}
+(-1)‘5,~51a2[~;(rL)-~,(rL)1-4(rL)2(1-5:)rT~S(rL)l~2
+4yLS:yT~-3(YT)Ia(5)}.
5, = CL
(4.21)
AX.
35
Gautesen / Rayleigh watre in elastic quarterspace
where fL
[ K2_ (#]‘/2.
Now from (4.21), we find that DL( 6) = 0 at 0 = 0, 90”. Also, we observe
from (4.18) that aDTH(B)/M
at e = 0,900. We remark that all expressions
of plane waves corresponding
either longitudinal
or transverse
for FFSPs motions,
remain
provided
valid for incidence that the definition
(3.6) of t2 is changed
= 0, to
appropriately.
5. Results and discussion When the incident Rayleigh surface-wave ray intersects the corner (1) reflects a surface-wave ray on the same free surface, (2) transmits a surface-wave ray to the other free surface, and (3) diffracts The inner
two cones of body-wave
cone of rays corresponds I$~ = arc cos[
The
K~
cos
& defined
r#+= In the farfield
XC
rays.
to longitudinal
motions
and has a half angle 4L defined
by
q&]
(5.1)
outer cone of rays corresponds
a half angle
it
to vertically
and horizontally
polarized
transverse
motions,
and has
by
COS[(KdK)
COS
I$~].
(5.2)
these waves are, of course,
defined
by (4.1). When the angle of incidence
satisfies (5.3)
qh,<&-=arcCOS(K/Kd
both cones consist of non-propagating rays. Thus all of the energy in the incident surface wave is transferred to the reflected and transmitted surface waves, and we must have the following identities: IAll = JA2) = IR”* T”I = 1,
In order for the reflection
&<
and transmission
E=[~RC]2+(Tc]2]1’2=1,
4;.
(5.4) coefficients
&
to satisfy
(5.4), we must have (5.5)
and Phase{ T”} - Phase{ R’} = (2n + 1)90”,
c#+ < 4;
(5.6)
where n is an integer. We remark that this result has been observed analytically by Freund [3] (for a surface wave obliquely incident on a semi-infinite crack) and by Angel and Achenbach [2] (for a surface wave obliquely incident on an edge crack). In Fig. 3, we plotted IT’I, (R’I and E (as defined by (5.5)) versus the angle of incidence 4R for Poisson’s ratio Y = t. We used 90 equally spaced values of 4 R. We note that E = 1 for C&C &= 23.2”, and that there is a value of &= 13” where ( T’I = 1 and IR’I = 0. In Fig. 4, we plotted the phase of T’, the phase of R”, and the difference of these two phases. We note that for &< r#& the phase difference is either 90” or 270” as required by (5.6). The 180” jump in the phase of R” occurs at the value of #Jo where R’=O. The phase of the transmission coefficients is fairly constant, varying between 266 and 286”.
36
A. K. Gautesen / Rayleigh
wave in elastic quarterspace
For Y = f, we have plotted in Figs. 5 and 6 the same quantities as in Figs. 3 and 4, respectively. For this value of V, &= 21.2”. Again, these curves satisfy (5.5) and (5.6). Also R” = 0 for 4K = 15”. Except in a neighborhood of & = &, the phases the same as for Y = i. For v = a, we have plotted
of the transmission
in Figs. 7 and 8 the amplitude
and reflection and phase,
versus 0 for & = 90, 85, 80, 75, 70”. For v = f, we have plotted When the incident angle C#IKis smaller than
coefficients
for v =! are nearly
respectively,
of DL( 0) from (4.21)
the same quantities
in Figs. 9 and
d’,=arccos(l/~K) the longitudinal
10.
(5.7)
body wave-rays
are non-propagating.
Figs. 7 and 9, we note that the amplitude
For v = a, 4: = 62.2” and for v = f, 4: = 57.9”. From
of DL( 0) for Y = a is somewhat
DL( 0) for v = f at corresponding angles of incidence. third quadrant for all angles of incidence.
For both
larger than
the amplitude
of
v = $ and f the phase of DL( f3) lies in the
For & = 90”, Gautesen [4] plotted the amplitude of ]D”( 0)l versus 13for both v = $ and f. He used a formula for D”(0) which involved only the tangential displacements on the free surfaces. He solved for these displacements numerically. Our expression (4.21) for DL( 0) involves only the normal displacements on the free surfaces. When we compared our curves with the corresponding ones of Gautesen [4] we found no significant differences. We add that the phases agreed as well. For v = $, we have plotted in Figs. 11 and 12 the amplitude and phase, respectively, of DTV( (3) versus 6 for & = 90,75,60,45,30”. For v = $, we have plotted the same quantities in Figs. 13 and 14. We observe that at corresponding angles of incidence, the amplitude of DTv( 0) is somewhat larger for v = f than for v = a. Also, the phases of DTv( 0) for v = a and f are nearly the same at corresponding angles of incidence. For v =$, we have plotted in Figs. 15 and 16 the amplitude and phase of D=“(e) versus 0 for &=90, 75, 60, 45, 30”. For v = 3, we have plotted the same quantities in Figs. 17 and 18. Again, at corresponding angles of incidence, the amplitude of DTH( f3) is somewhat larger for v = 4 than for v = $. Also, for 0 > 15”, the phases of DTH( 19) for v = i and $ are nearly the same at corresponding angles of incidence.
Appendix In this Appendix,
we give the fundamental
stress tensor and its Fourier
transform.
This tensor is defined
by
where
k, and K are given by (2.19) and (2.25), and (A.2)
r = 1x1, H(k,r) The Fourier
= exp[ik,r]/(4Tr).
transform
of (A.l)
(A.3) is
(A.4) where
ye is defined (C, E" =
Pz”, C)
by (2.24) and (2.25), = (-&,
exp[ik,y”]x,(],
-e2, Y" sgdxd), LY= L, T.
Q = L, T,
(A.5) (A.6)
A. K. Gautesen / Rayleigh wave in elastic quarterspace I
,
I
I
:
I
kl
8
i
\I ,i
I
I
:
37
I
I
i
\
I
\_
I
I
\
c I I
I II
1
3anlIldWV
[3301 3SVHd
30ll111dWV
(330) 3SVHd
: I 1
A.K. Gautesen / Rayleigh
38
wave in elastic quarterspace
.’
!,‘, 1v
:
/.......
,’
3CJlLIldWt- dSJ+1
m
1
8
,:“ 1: i
:
ii
l
3SVHd - dS&J-1
3 N
3ClillIldWV - dSj+1
0
N
0 N
3SVHd - dS.U-1
A.K. Gautesen / Rayleigh wave in elastic quarterspace
-s-
U-J
m
N
30flLIldWV- dSJ+hl
u-l
d
m
N
30fILIldWV- dSA-Al
3SVHd - dSj+hl
0
39
A.K. Gauresen f Rayleigh waoe in elastic quarterspace
40
-1
I :
30flLIldWV- dS4-HL
m
N
3CinlIldWV- dSL!-HI
3SVHd - dSj+Hl
I
A. K. Gautesen / Rayleigh wave in elastic quarterspace
41
Acknowledgment This work was supported Energy
Research,
(in part) by the Applied
U.S. Department
of Energy,
under
Mathematical Contract
Sciences
subprogram
of the Office of
No. W-7405-82.
References [I] J.D. Achenbach, A.K. Gautesen and H. McMaken, Ray Methods for Waves in Hasric Solids with Applications to Scattering by Cracks, Pitman, Boston (1982). “Reflection and transmission of obliquely incident Rayleigh waves by a surface breaking [2] Y.C. Angel and J.D. Achenbach, crack”, .I. Acousr. Sot. Amer. 75, 313-319 (1984). [3] L.B. Freund, “The oblique reflection of a Rayleigh wave from a crack tip”, Internat. J. Solids Struct. 7, 1199-1205 (1971). [4] A.K. Gautesen, “Scattering of a Rayleigh wave by an elastic quarter space”, .I. Appl. Mech., accepted for publication. [5] A.K. Gautesen, “Scattering of a plane longitudinal wave by an elastic quarter space”, Wave Motion 7, 557-568 (1985). [6] A.K. Gautesen, “On matched asymptotic expansions for two dimensional elastodynamic diffraction by cracks”, Wave Motion I, 127-140 (1979). [7] L. Knopoff, “Elastic wave propagation in a wedge” , in: J. Miklowitz, ed., Wave Propagation in Solids, ASME, New York (1969) 3-42.