- Email: [email protected]
Transient acoustic diffraction in a fluid layer
wnsfE ELSEVIER
Wave Motion 23 (1996) 139-164
MIlTfUll
Transient acoustic diffraction in a fluid layer Kasper F.I. Haak *, Bert Jan Kooij Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Centre for Technical Geoscience, Del@ University of Technology, P.O. Box 5031, 2600 GA De& Netherlands
Received 3 October 1994; revised 29 June 1995
Abstract Acoustic and elastodynamic waves serve as diagnostic tools in the non-destructive search for cracks and holes in material structures. In this article, the diffraction of acoustic waves caused by a semi-infinite screen in a fluid layer is investigated. A
homogeneous half-space models the substructure of the object under investigation. The transient wave problem is then solved by means of an iterative scheme in which successively the Wiener-Hopf technique and the Cagniard-De Hoop method are applied [Kooij and Quak, “Three-dimensional scattering of impulsive acoustic waves by a semi-infinite crack in the plane interface of a half-space and a layer”, J. Math. Phys. 29, 1712-1721 (1988) 1. From the numerical results it turns out that in case of deep embedded semi-infinite screens only the first order diffracted wavefield gives a significant contribution to the geometrical wavefield. Application of these results in the iterative solution procedure yields a very effective solution scheme in which the exact transient wave field in the configuration at hand can be calculated within any finite time window.
1. Introduction
In quantitative non-destructive material exploration, elastodynamic and acoustic waves serve as diagnostic tools to probe the mechanical features of material structures. Examples can be found in examination on cracks in wings of air planes or in concrete of nuclear power stations. The configuration under investigation consists of an elastic layer with a semi-infinite pressure-free void crack in it. An elastic homogeneous half-space represents the substructure of the diffracting crack. Aki and Richards [ 21 show that the elastic wavefield is composed of a dilatational part and an equivoluminal part. In this article we confine ourselves to the equivalent fluid model for dilatational waves in a solid (see De Hoop [ 31) . Hence, the influence of equivoluminal waves is neglected and the configuration is modelled as a fluid layer with a pressure-free semi-infinite void screen in it, together with a fluid half-space substructure. This means that surface wave phenomena like Rayleigh waves are not incorporated in our model. The wave problem is solved by means of a proper combination of the Wiener-Hopf theory and the Cagniard-De Hoop technique. Some numerical results are presented. The diffraction phenomenon has been studied analytically by Sommerfeld [ 41, Copson [ 51 and Noble [ 61. Clemmow [ 71 and Du Cloux [ 81 considered the diffraction by a semi-infinite screen in the single interface of two homogeneous media. The latter presented numerical results of pulsed electromagnetic waves. The problem of acoustic multiple scattering by two parallel half-screens has been considered by Abrahams [ 91, in which the system * Corresponding author. E-mail: k.f.i. [email protected]. 0165-2125/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDIO165-2125(95)00037-2
140
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
of coupled Wiener-Hopf equations is uncoupled by the introduction of an infinite product of poles, each of which contributes to the total wavefield for all ,times ‘ ‘t” without any apparent physical interpretation. Hence, this method yields no exact solution (contribution of a finite number of poles) and is not efficient for the calculation of wavefields in a finite time window. He&man and Van den Berg [ 101 carried out a frequency-domain analysis of diffracted electromagnetic waves by a semi-infinite screen in a layered medium. Here, both multiple edge diffraction as well as repeated reflections are taken into account. However, the conversion of high-bandwidth frequency-domain data to accurate time domain results is no sinecure. Shirai and Felsen [ 1 l] investigated multiple diffraction of electromagnetic waves in an infinite homogeneous medium with a flat strip in it. They developed an iterative technique in which the result of the nth iteration (an n-dimensional integral expression) represents an n-fold half-plane diffraction event caused by the incident wave which diffracts again and again at the opposite edges. In this paper we shall derive an explicit iterative scheme which yields after iV (fixed) iterations an exact time-domain solution within a finite time window. Using the modified Cagniard method Kooij and Quak [l] and Kooij [ 121 obtained closedform expressions in the space-time domain in case of acoustic and elastodynamic waves, respectively. They considered the configuration of a crack located in the plane interface of a uniform layer and a semi-infinite halfspace. In our case the semi-infinite void screen is placed in the fluid layer, rather than on the interface. In this paper the Cagniard-De Hoop technique is used in order to obtain space-time domain results. This method is due to Cagniard [ 131 and has been modified by De Hoop [ 14,151. A description of this method can also be found in Achenbach [ 161, Miklowitz [ 171 and Aki and Richards [ 21.
2. Description of the configuration In the present article we investigate theoretically the diffraction of acoustic waves in a fluid-fluid configuration, as is depicted in Fig. 1. It consists of a fluid layer with a pressure-free surface, and a semi-infinite fluid medium which occupies a half space. In the fluid layer a pressure-free half-plane void screen is present. To specify the geometry in the configuration we employ the coordinatesx,, x2, xs with respect to a fixed, orthogonal, Cartesian frame of reference, with the origin B’ and three mutually perpendicular base vectors ii, iZ, i3 of unit length each. In the indicated order, the base vectors form a right-handed system. The origin d is chosen to be located at the pressure-free surface of the fluid layer right above the tip of the pressure-free half-plane. In accordance with the geophysical convention, i3 points vertically downwards. The base vector ii is chosen such, that the pressure-free half-plane is located at x, > 0 and x3 = a. The plane interface of the fluid layer and the half space is located at x3 = d. The subscript notation for Cartesian vectors is used and the summation convention applies to all repeated subscripts. The position in the configuration is also denoted by the position vectorx =x&r, withx E W3.Furthermore, the symbol t is used for the time coordinate. i2
,
Y' source
in
pressure-free
surface
Q (%I”‘,
zp,
crack
Fig. 1. The geometrical configuration.
23
=
a
z3
=
d
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
141
The media are assumed to be linear, locally reacting, instantaneous reacting, time-invariant, homogeneous and isotropic. The acoustic properties are characterized by its volume density of mass p and its compressibility K, respectively. Then the acoustic wave speed is given by c = (prc) - “‘. We assume that the acoustic field waves are generated by a pulsed acoustic monopole line source, which is present in the fluid layer, parallel to the lateral direction i, and starts to act at the instant t=O. Specifically, the source is located at (xi, x3) = (xi”‘, ~9’). The problem is now reduced to a two-dimensional one, in which the field quantities are x,-invariant, i.e., a2 = 0.
3. Basic equations for the acoustic wave motion The linearized akP
+
@rvk
acoustic fields are governed by the hyperbolic =
%J4u + Ka,P
sk.3.f
=
system of first-order differential
equations
9
(1)
(2)
0,
in whichp and u,, denote the acoustic pressure and the particle velocity, while p and K represent the dynamic medium properties, viz. the volume density of mass and the compressibility, respectively. The symbol Sk,,, represents the Kronecker tensor. Further, the normal component of the dipole line source is characterized by f=f(t>S(x,
-XIS), X3-Xs”‘) .
(3)
Here, f(t) is a causal function of time, i.e. f(t) = 0 when t < 0 and 6(x,, x3) denotes the two-dimensional delta distribution. At the free surface of the fluid layer the boundary condition lim p(x,, x3, s) = 0 , X310
Vx, E W
holds, while at the level of the half-plane lim p(x,, x3la
x3, s) = lim p(x,,
lim P(x,, .qJa
x3, s) = 0
-%?a
Dirac (4)
the conditions
x3, s)
,
Vxl E W ,
(5)
x,>o,
(6)
together with lim +(x1, x3, s) = lim ~(xi, qla xstn apply. Finally, continuity
x3, s) ,
x1
of the pressure and the vertical component
lim p(xI, .x3, s) = lim p(x,, .%Ld x,Td
x3, s)
Vx, E W
(7) of the particle velocity yields (8)
and lim u,(x,, x3, s) = lim u~(x,, x3, s) x3ld
4. Transform-domain
x,Td
Vx, E W .
(9)
acoustic wavefield
In order to exploit the linearity and time-invariance of Eqs. ( 1) and (2), we subject them to a one-sided Laplace transformation with respect to time. The Laplace transform A(x, s) of a quantity A(x, t) is given by
K.F.I. Ha&
142
m
A(x, s)
=
I
t) dt
exp( -st)A(x,
B.J. Kooij/
Wave Motion 23 (1996)
139-164
.
(10)
0
In view of the future application of the Cagniard-De Hoop technique, the parameter s is real and positive and is taken sufficiently large. According to Lerch’s theorem (see Widder [ 181) the solution of the integral equation ( 10) exists, is unique and vanishes in the interval t < 0, thus ensuring causality. In order to exploit the invariance of the medium parameters with respect to the xi-direction, we carry out a spatial Fourier transformation with respect to xi. The spatial Fourier transform pair &Xl,
x3, s)
*
&%X3,
s>
(11)
is given by +m A(% x.3, s) =
I -cc
exp(s~i)&xi,
x3, s) hi
,
(12)
+im
exp( - scrx,)A( (Y,x3, s) da.
(13)
I
-im
Note that scr is an imaginary quantity, hence cry 0. In order to keep the analysis lucid the configuration under (Dc2’),.d investigation is divided in four domains, viz. 0
2 k=l,2
withRe(y)>O,
(14)
y=22 k
(15)
pk ’
respectively, applying the temporal Laplace and spatial Fourier transformation to Eqs. ( 1) and (2) and eliminating u”,results in a coupled system of ordinary first order differential equation, which have exponential functions with positive and negative argument as linearly independent solutions. As a consequence, we have in the domain D(l), D”’ and Dc3’ p”‘(a,
x,, s) =A’,“(a,
s) exp[ -sn(x3-@)I
+At’)(a,
s) exp[ -sn(c--X3)]
,
x,ED(‘)
(16)
and G$:“(cv, x3, S) =Y,{A’,“(a,
S) exp[ --SY1(xX-_)]
-A!?(a,
s) exp[ -sn(F-x3)]},
x,ED(‘) , (17)
(4) Eqs. ( 16) and (17) also apply, in which r E { 1,2,3} and where pimax = min;max{x3(x3ED(‘)}.IndomainD provided that the substitution {yi , Yl } -+ { y2, Y2} is carried out. Further, the coefficient A'_) ( (Y, S) = 0, thus satisfying the physical constraint of causality. From the different equations the source conditions follow with Eq. (3) as lim p( a, x3, s) - Xy~PB( (Y,x3, s) =J’(s) X,IXP
=_?(s) exp(s&‘)
(18)
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
143
Fig. 2. The integration contours L, in the complex w-plane and Lq,_ in the complex q,,,,-plane in case of @,,.
and (19) Together with the boundary conditions (4), (5)) (8) and (9)) which can suitably be transformed into the Fourier domain, they reduce the number of unknown coefficients in Eqs. ( 16) and ( 17) to one, e.g. A’:‘. In order to find the last coefficient, we apply an inverse Fourier integral to the formal spectral solutions. Next, we require the boundary conditions (6) and (7) to satisfy. This is accomplished in the following way. To start with, we introduce the reflection coefficient R, as
y, - y2 Y, -t Y2 .
-
Ro=
(20)
Further, we define GR( (Y,S) and ti( G’<(L~,s) =Ay){
(Y,S) via
1 +R, exp[ -2sy,(d-a)]}
(21)
and 2P(%
s) = Y$’ exp( -~,a)
[exp(~~~~~S’) +exp(
-.vv~~))I
_Ac3j 2YI[1 +
1 - exp( - 2sy,a)
PI
+Ro
exp(
1 -exp(
c22j
-2~41
-2sy,a)
’
Let us finally assume GR( a, S) and ti( CY,S) to be analytical functions in the right half-plane Df3sand in the left half-plane D”, (see Fig. 2)) respectively, with asymptotic behaviour o( 1) as 1a 1-+ ~0in Dt and D”, respectively, then in accordance with Jordan’s lemma and Cauchy’s theorem the boundary conditions (6) and (7) are indeed fulfilled. Eliminating A$? in Eqs. ( 2 1) and ( 22) results in the Wiener-Hopf equation Q’ exp( - sy,a) - GR( (Y,S)
1 + R. exp( - 2sy,d)
1 +R, exp[ -2sy,(d-a)]
=HL(a,
s)
1- exp( - 2sy,a) (23) Yl
’
in which “i
Q'
= i
[
exp( sy,.@))
+ exp( - sylx$“)) ] .
(24)
Obviously, the denominators of Eqs. (22) and (23) have no zeroes since the Laplace parameter s is real and positive. This means that no surface/interface or pseudo-Rayleigh wave phenomena will appear in our analysis. Since the wave amplitudes A’,“, A?), AT’ and A’,) are functions of either GR or @, e.g.
144
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
-i f 2
{exP[ -syl(2a-x~‘>l +exp(
A(‘) = +
-sy~&))}-GR
exp( -sy,a)
(25)
1- exp( - 2syia)
and -i f - y { 1 +exp[
-2sy,(a-$‘)I}
A”‘=
+GR exp[ -sm(a-x9))]
(26)
,
1 - exp( - 2syiu)
the problem is reduced to the solution of EQ. (23).
5. Iterative solution procedure of the Wiener-Hopf
equation
The original Wiener-Hopf equation is split into a series of less complex Wiener-Hopf equations, each of them corresponding to the wavefield due to k reflections against the interface and the pressure-free half-plane. In doing so we expand GR( CY,s) and @( CY,s) as
{GR,HLl(a, s>= f
{G;, HIk’l(a,s>.
(27)
k=O
Here, the kth term of Eq. (27) is at least of the exponential same exponential order results in Q’exp( -sy,u)
-G:((Y,
s) =@(a,
s)
order exp [ - 2ksy, (d - a) ] . Collecting
1- exp( - 2sy,u) Yl
which is the Wiener-Hopf
the terms of the
’
equation of order zero and k-l
-GF(a,s)--[l-exp(-2sy,a)]
C
p=o =Hi(
(Y,S)
1 - exp( - ;?sy,u)
,
{(-Ro)k-P
exp]
-2(k-p)sy,(d-a)lG,R((Y,
~11
(29)
kal,
Yl
which is the Wiener-Hopf equation of order k. Next, Eqs. (28) and (29) are split into a series of elementary Wiener-Hopf equations of order k, n, each of them corresponding to k reflections against the interface and the pressure-free half-plane and II reflections against the surface of the fluid layer and the half-plane, respectively. Therefore, we expand GF( (Y,s) and Z!& cry,s) as
Here, Gj&( CX,s) and e,,( CX,s) have exponential order exp[ - 2ksy, (dof the same exponential order in Eqs. (28) and (29) results in
a) - (2n + 1) syiu]. Collecting
the terms
(31) where the source terms uk,+ are given by
K.F.I. Husk, B.J. Kooij/
u0.0
Q'
=
exp( -syla)
Wave Motion 23 (1996) 139-164
145
,
(32)
e-1
n>l,
U0.E = Aexp(-2sy,a), Yl
(33)
k-l
uk,O=
c
-
%I( -Ro>k-”T-Q -w--p)sy&i--a)]
, k>
1
(34)
I-‘=0
and uk,n
=
k-l
- C ( -Ro)k-YIG~n-G~:,_lexp(
-
p==o
2~y,a)l exp[ -Xk-p)syl(d-a)l
, k, na 1.
(35)
It is clear (by induction) that GFn( (Y,s) and Z$&(cy, s> are indeed of the exponential order exp [ - 2ksy, (d - a-> - (2n + 1) syra]. Further, it is noted that the evaluation of the wavefield in a finite time window needs only the solution of a finite set of recursive elementary Wiener-Hopf equations. Then, the problem is now broken down to solve Eq. (31). This is accomplished in Appendix A by means of factorization of the kernel, additive decomposition and application of LiouvilIe’s theorem. The solution is given by (36) and
f-&n(~,s>= ti(NQk,(e
$1
(37)
7
in which l/2 yy-(
a)
=
(38)
are analytical functions in DR and DL, respectively. Here, the upper and lower sign correspond with the superscript R and L, respectively. Further, in accordance with Plemelj’s theorem (see Muskhelishvili [ 191) @$;,f-[Qk,nJ
Q:LLb,
(% s)
S>= q$"'"rQk,,l(4
S>,
(Z- %:,:I [Q,,] in which we define the projection
7
~~%~
* E L?k,,z
(39)
(a, s> 3 a-;:,:,
operators (see Hochstadt
[ 201) (40)
and (411
= i [Z+iHlQ,,,(a,
S) .
(42)
146
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
In Eqs. (39)-( 42) Lqk,+is any contour which splits the complex qk,,-plane in a region Df;,,, (on the left) and a region Dik,,$(on the right), and Qk,n is any function which satisfies Holders conditions on L_. Further, Z and H denote the identity and the Hilbert operator, respectively. From Eqs. (32) -( 42) we infer that the norm of the repeated diffraction operators is less than one (contraction operator), due to the presence of the exponential function and the fact that the Laplace parameters is chosen real and positive (see also Appendix C) . Hence, the iterative scheme is convergent. In the present analysis we have
Qk.n(c s)= $W~,,nh
s> .
(43)
In Fig. 2 the integration contour L,,,, is depicted corresponding with the case Q&. In case of Qk, the small semicircle integration path around the integration singularity is replaced with a semi-circle loop on the right of the pole.
6. Spectral-domain
wave constituents
The Fourier-transform-domain pressure field is evaluated by means of the concept of up- and downgoing wave constituents. To this end we substitute d = ma and Eqs. (27) and (30) in the expressions for the coefficients A:, A:, A; and AZ. Here, G& ( cy,S) has exponential order exp [ - 2k( m - 1) a - (2n + 1) a] . Ordering the terms in Eq. (17) in exp[ - (2n+ l)sy,a] then leads to
ip
la=0 l j(m-l)+l &;;b ,
As far as D’ ’ ) is concerned.
V.3.a =
f3 {exp[ -sy 2
(44) j>l.
Here, 1(xF’--x3)]
+exp[
-sy,(xp)+xa)]}
fe,n-da, s)
%:,:b = exp( -ma) I exp( -s-m) +exp(vlx3)
1
?3,;, = exp( -s-w) [exp( -
j ~L(m-l)-l(~~ s) 1 C -t i=l Pl
‘2
syIx3) +exp(sw,)
+ i Y,( -Ro)i-PG~~_i(n~_l)__(LY, i=l p=o
(45)
,
(46)
Pl
S) exp[ -2(i-p)(m-l)sy,a]
>
.
(47)
The generalized ray paths show that the level of the half-plane acts as a secondary (diffraction) source. The vertical particle velocity field in D’*’ (ordered in exp [ - (2n + 1) sy,a] ) is identical to the one in D(l), except if n = 0. In a similar way the spectral field in Dc3’ and Dc4’ can be treated, as well as the acoustic pressure. The mathematical decomposition of the wavefield into generalized ray contributions has a physical sense. Each contribution can be recognized separately in the receiver record if the pulse duration of the source is small compared with the travel time in the multiple reflection/diffraction geometry. In our numerical examples this condition has been satisfied.
7. Analytical continuation of decomposition operators It is anticipated that in accordance with the application of a modified Cagniard-De Hoop technique, as outlined in Section 9, the contours L, and Lq,,o will deform in order to cast the exponential functions in the desired form
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (I996) 139-164
147
exp( - sr), in which T is a real integration variable. The contours L, and LqO,O deform due to the occurrence of exp( - scrxl) and exp( sqo,,.$‘) in the integrand, respectively. Since the modified Cagniard contours Lw,,, . . ., coincide with the imaginary axis we have to distinguish between four different cases, viz. L 40,rr_, case 1:
x,-co { XjS) < 0 ’
case 3:
case 2: and
x,>o -tX$S)< 0 ’
case 4:
During contour deformation singularities in the plane of integration may be encountered, thus requiring analytical continuation of the decomposition operators @$,, and @&,, as indicated in Eq. (39). It is noted here, that domain D,L;:ff denotes the part of the complex qk,,-plane on the left or on the right of the integration contour L,,,,, respectively, including the integration contour L,,,,. Due to the complexity, we confine ourselves to an investigation of the pressure field in D(l) and DC*) of exponential order n < 2m - 1. For practical purposes this turns out to be sufficient in most cases. 7. I. The uertical particle uelocity$eld in D(l) and DC2)(n < m) In order to investigate the wavefield for 1 Q n < m in D(l) and D’*’ it is necessary to investigate The next analysis holds for n > 2. From Eqs. (37) and (39) we have @,?Z-,(%s)
%_L-I(G Application
s)l *
=~((Y)~~,,-,re,,,-,(~,
Repeated application
ZY&_ i (cr, s) .
(48)
of Eqs. (43)) (33), (37) and (39) leads to
s) =~(LY)~~,,_,(exp(-2Syla)dSl,,,_,{cxp(-2sy,a)
s>lIl.
. . . @&[~((.~)Gxo(~~
of Eqs. (46)) (40) and ( 13) leads to an integral-form.
In Fig. 3 the integration
(49)
contours L,, Lqo,O, . . .,
L 40.,,_,have been depicted. Due to the application of a modified Cagniard-De Hoop technique the contours L, and L,,O will deform. In the next subsections case 1 and case 2 are investigated in more detail. 7, I, I, Analytical continuation in case I In case 1 the occurrence of exp( -scrx,)
and exp(sqo,&‘)
in the integrand
will cause a deformation
I+% QO,Ot .,.I Qo,n-1)
a,
q0.0,
.-.,
q0,+
R+, qo,or
---2
>
dane
qoo,,-1)
Fig. 3. The integration contours concerning the integralsL,, Lqop, . . . , L+,,,*_ , before &&mation.
of L, and
148
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
L,,, in the left-half of the complex a-plane and in the right-half of the complex %,O-plane, respectively. analytical continuation of the decomposition operators is required.
Hence, no
7.1.2. Analytical continuation in case 2. In case 2 the occurrence of exp( -S&X,) and exp( sq,,&‘) in the integrand will cause a deformation of L, and L,,, in the right-half of the complex a-plane and the qO,o-plane, respectively. Now, the first decomposition operator in Eq. (49) requires analytical continuation, which according to Eq. (39) results in . . . ~~,,[~((Y)U~,~((Y,S)IJ~
~.,,-1(~,s)=-~(~)~~,,,-,{exp(-2sy,a)~~,,,_,{exp(-2sy,a)
+ tiCa> exp( -Ww)@k,,,-,iexp(
-2wa)@t&31exp(
-2wa)
. . . ~o.o[*(~)Uo,o(a,
s>l I} . (50)
Obviously, analytical continuation of the first decomposition Repeated application of this procedure leads to nZj,,-~(a,s)=--
C
operator in the second term of Eq. (50) is required.
1
exp[-2(n-1-i)sy,a]~(a)~~,i{exp(-2sy,a)
i=l
X%;Y-,{exp( -2syla)...~o,o[~(a)U0,0(a, ~>I11 +exp[ -%n-
l>sy,al~(cw)~~.o[~(a)Uo,o(cu,
x)1 .
(51)
In Fig. 4 the integration contours, concerning the last term of Eq. (5 1) are depicted. Heie, S, and S,,, are the points of intersection with the real a-axis and the real 4 o,O-axis, respectively. Analytical continuation of the decomposition operator CD&,,is required, only if
I%I ’ I~,o,oI.
(52)
If Eq. (52) is satisfied, the last term of Eq. (5 1) is rewritten as
Imb, clo.0)
Fig. 4. The integration contours concerning the integrals L, and Lw.Oafter deformation in case 2.
K.F.I. Haak, B.J. K&j/
In Appendix
B the condition
Wave Motion 23 (1996) 139-164
149
in Eq. (52) will be analysed in more detail.
7.2. The vertical particle velocityjeld
in D”’ and 0”’ (n < 2m - 1)
In order to investigate the wavefield for m < n < 2m - 1 in D (r) and D@) (due to a reflection against the interface) it is necessary to investigate
@&Lb%s> -Ro@,-.,(w s>exp[ YI(~)
-2sy,(m-
1)al
.
A similar investigation as presented in the previous section can be carried out. This results in a series of operator expressions which may need analytical continuation in case 1 and case 2, depending on the integral deformations caused by the modified Cagniard technique. Consistent application of Bq. (39) yields expressions like Eqs. (5 1) and (53).
8. Classification
of generalized rays
Substitution of Eq. (40) in the relevant expressions of the previous section results in the integral-form equivalents of the decomposition operator expressions. Application of Eqs. (46) and ( 13) yields the Laplace-transform-domain integral expressions, concerning the vertical particle velocity for the two different cases. Due to the analytical continuation of the decomposition operators in the previous section, no integral singularities with respect to the integration variable (Yand qO,Oduring the contour deformation occur. The Laplace-transform-domain vertical particle velocity in D (‘) and D(‘) for 1~ n < m is arranged according to ~exp{-s[y,(2na+x,+x:‘))+rw(n,-nl”)]}
da
(55)
and
Xexp[ -~{~~(m)[a+x$S91
-q.o,~Pll
exp -2s i yl(qOJa [ i=l
xexp[-s(y,(~)[(2n-2k-l)a+x,)+cwc,}l
dqo,,...dq,,da,
1 (56)
in which Ak=
rI(~>fi(40,0> P1(90,0-40.1)...(qo,k---qqo,k)(qO,k-~)
.
(57)
So far, the generalized ray path consists of the modified Cagniard interpretation of the successive exponential functions in Eqs. (55) and (56). Each geometrical and diffraction contribution in Eqs. (55) and (56), respectively,
150
K.F.I. Haak, B.J. Kooij/
consists of four generalized is in case 1 given by -(n)
-
u3
4n,n-
- V3;d
Wave Motion 23 (1996) 139-164
ray paths, viz. kxj and +x2). In this way, the wavefield in D(l) and D(*) for 1
x,
1) -f
xp
(58)
< 0
and in case 2 is given by (59) In case 2 also the geometrical fS)
wavefield contributes
if the condition
(cf. Appendix B)
a+x$“’
I I EL<
(2n-
Xl
(60)
l)afx,
is satisfied. The above analysis holds for n > 2. For n = 0, 1 some diffraction phenomena do not contribute yet, while the condition for the geometrical field to exist differs from Eq. (60). Now the contribution due to one reflection against the interface is investigated. Substitution of Eq. (40) in the relevant operator expressions results in the integral-form equivalents of the decomposition operator expressions. Application of Eqs. (47) and ( 13) yields the Laplace-transform-domainintegral expressions, concerning the vertical particle velocity for the two different cases. Due to the analytical continuation of the decomposition operators in the previous section, no integral singularities with respect to the integration variable cr and qo,o during the contour deformation are encountered. The integral expressions are arranged in a geometrical contribution and a diffraction contribution, viz. corresponding to generalized rays which do not and which do diffract at the edge, respectively. The diffracted contribution is decomposed as rays in which reflection against the interface before (2 types), during ( 1 type) and after (2 types) the repeated diffraction phenomenon occurs. Again, each geometrical and diffraction contribution consists of four generalized ray paths, viz. +x, and +x, (‘). We now infer that in case 2 also the geometrical wavefield contributes if the condition (2m-
l)a+x$“’
(2n-2m+l)afX3
XiS) < Iy
(2m+
l)a+xP’
I < (2n-2m-l)afx,
(61)
is satisfied. This restriction is easily found with the aid of classical ray theory. The above analysis holds for n > m + 2. For n = m, m + 1 some diffraction types do not contribute yet, while the condition for the geometrical field to exist, differs from Eq. (61).
9. Space-time-domain
acoustic wave field
In order to apply the standard Cagniard technique with respect to each integration variable (see De Hoop [ 141 and Kooij [ 121) , the integrals with more than three integration variables are in Appendix C rewritten as principle value integrals. The integrand contains the s-dependence only through the exponential function exp( -ST,) in which 7,=D,,Yr(u)
and u stands for each of the integration which is given by
Here,
(62)
+ux, variables
(Y,qo,o, . . ., qo,m.IQ. (62) represents
a branch of a hyperbola,
K. F.I. Haak, B.J. Kooij / Wave Motion 23 (1996) 139-164
R,= (Xz+Dz)1’2,
.L=x,
Re(R,,) >O,
(65)
xp
f
(66)
)
Uf= [%,I,
..*3
40,ml
and D, is some vertical propagation T,, =
(64)
,
x,,, = x,=0
151
(67) distance. Further,
5
(68)
Cl
From Eqs. (63)) (65)-( 67) it is clear, that only the contours L, and L,,, deform, away from the imaginary Due to the occurrence of the vertical slowness yi in the integrand, branch cuts along IRe(u
iT
Im(u)
=0
(69)
in the complex u-plane are introduced. introduced along IRe(u
ij
Im(u)
axis.
If the reflection factor Ro( u) occurs in the integrand,
also branch cuts are
=O.
(70)
We assume, that the position of the source and the receiver are chosen such, that the hyperbolas do not intersect the branch cut. Hence, no head waves occur. In this way the principle value integrals can be evaluated as
Xex~?( --ST,) exp(-S~O,O) fi ew[-2~pw(qo,p)~ldqo,o.-.dqo,,da p=l
=
(2)
T, To,,,To,,- I
To,.2 To.1To,o
(71) Here, B nm+l(~,
%,o, --.r qo,nr) =~~(‘dBm,m(~,
qo,o, . . . . qo,m) +~(~*)Bm,m(a*,
qo,o, a.., qo,m)
(72)
=~~(40,P,,,t-~t~,
40,0,
. . . . 40,b
. . . . qO,m)+‘Y~tq~t)Bm,t--l((Y,
40,0,
. . ..
d,b . . .. q0,A , l
(73)
152
K.F.I. Haak, B.J. K&j/
Wave Motion 23 (1996) 139-164
and k&0( a> 40,OY. . . >a,,,> = YI(cIO,OM~(~~ qo,ov..-, c~o,m) + ~~(d&J&z(~~ do, . ..v qo,nJ . In Eq. (72) Schwarz’s reflection principle
(Titchmarsh
[ 211) has been used. Further, &,,+
B,,,, + I ( 7m ~o,o,. . . > ~o,m)= &,m + I ( (~7qo,o, . . ., %,A 9
(74) 1 is given by (75)
where the integration variables [r,, ro,o, . . ., rO,J are related to the variables [a, qo,o, . . ., qo,,,] via Eq. (62). According to Lerch’s theorem, application of the shift-rule of the Laplace transform to Eq. (71) leads to
(76) in which U(t) denotes the Heaviside step function. In Eq. (76) we have used the differention rule of the Laplace transform. Further, the region of integration D is on account of the causal source function q(t) given by
.r,>T, D = ~a, ro,o,. . .,
7o.m I
T~,~~T~,~, ZE [O, . . .. ml ~7LY+C;1=070,16t. 1
It can be verified that the region of integration
ro,oE [To,o,. D=
~-a,ro,o,. . .,
70,nzI
D can be characterized
(77)
by
t-CK=~ro,k-r,l 70,~ E [ToJ, . . . . t-C~=t+lro,k-r,-C~~;=bTO,k] ZfZ [ 1, . . . . m-l] ..I
ro,nzE [To,,, . . . . t-r,CE,‘T,,k 7, E [T,, . . . . t-C~zoTO,k] .
,
1
(78)
Concerning the arrival time tm = C;LoTo,l + T, the approximation tarr= (2nalcr) holds in D(l) and D(*) if n is large enough and the offset is small enough. A similar reasoning applies for the transient wavefield in Dc3). Consequently, the space-time-domain expressions are exact within some finite time window. The upper limit of this window is determined by the minimal arrival time of the (n + 1) st wave constituent.
10. Numerical results In this section we pay our attention to the numerical implementation. In the geometrical configuration (cf. Fig. 1) the distance between the half-plane and the free surface equals a = 1.O m, while the thickness of the fluid layer is given by d=4.0 m. Further, the volume source density of force is located at (xl’), xi’)) = ( - 1.0, 0.7) m. The
K.F.I. Hmk. B.J. Kooij/ Table 1 The acoustic properties
Wave Motion 23 (1996) 139-164
153
of the configuration p, = 2.4 x p2= 1.0x K,=5.0x tc2=4.4x c1 =2.9x c,= 1.5x
lo3 103 IO-” lo-‘0 103 103
kg/m3 kg/m3 Pa-’
Pa-’ m/s m/s
dynamic acoustic properties of the media at hand are chosen as indicated in Table 1, thus simulating the practical situation of a concrete wall which isolates a nuclear reactor. Since c, > c2 it is clear that no head wave contributions occur (see Appendix B) . We have numerically evaluated the Green function concerning the particle velocity field of some generalized ray path contributions. These Green’s functions are to be convolved with the second derivative with respect to time of the source signaturef( t) (cf. Eq. (3) ) , which is in our case a four-point optimum Blackman pulse of unit amplitude, given by
s(t) =Q(t)
=
-~~(~~cos(~
r)
0
when
--co
when
O
when
-tp
(79)
The Blackman pulse approximates the classical Ricker wavelet, which is often employed in seismics (De Hoop and Van der Hijden [22] ), quite well. Here, the constants b, and the pulse duration tp are given by bo=0.35869, bl = -0.48829, b,=0.14128, b3= -0.01168 and tp= 80 ps. We have evaluated the geometric, first order and second order diffracted fields in various points of observation. In Fig. 5 a typical Green function pertaining to a first order diffracted contribution is shown. In Figs. 6,7 and 8 the convolved versions of a geometrical wavefield and the corresponding first and second order diffracted wavefields (n = 2) are depicted, all at the receiver location (xi, x3) = (2.0, 0.5) m. The contributions in Figs. 9-11 are due to one reflection against the interface (n = 12). Further, the convolved first order and second order diffracted wavefield are shown in Figs. 12 and 13 for the observation points (xi, x3) = (0.5, 0.5) m. The geometrical field in this case (n = 2) does not exist. We have also
-5. 1.75
1.80
1.85
1.90
t Fig. 5. The Green function pertaining
[ml.
1.95 x 10-3
to a first order diffracted field (vertical ray path 4a -x3 +.$’
at the observation
point (xl, x3) = (2.0,0.5)
154
K.F.I. Hauk, B.J. Kooij/ Wave Motion 23 (1996) 139-164
geometrical
contribution
(n=Z)
t Fig. 6. The convolved
contribution
pertaining
to a geometrical
x 10-3
contribution
(vertical
ray path 4a-x3+$))
first order diffraction
1.75
1.80
1.85
t 1.95
1.90
at the observation
point
at the observation
point
i 2.00
t
Fig. 7. The convolved
contribution
pertaining
to a first order diffracted
x 10-3
field (vertical
ray path 4a-x3
+xp’)
[ml.
second order diffraction
2.10
2.15
2.20
2.25
(n=2)
2.30
tie Fig, 8. The convolved Cxlr x3) = (2.0,0.5)
point
(x1=2)
-1000
(Xl, x3) = (2.CO.5)
at the observation
[ml.
(x1, $1 = (2JIO.5)
contribution
[ml.
pertaining
to a second order diffracted
2.35
2.40
x1r3 field (vertical
ray path 4a-x,+x:“‘)
K.F.I. Hnak, B.J. Kooij I Wave Motion 23 (1996) 139-164
geometrical
contribution
(n=lZ)
/
-2.0
5.45
,_
8.50
8.55
8.60
t Fig. 9, The convolved (xl, x3) = (2.2,0.5)
contribution
pertaining
to a geometrical
155
8.65 XPO-3
contribution
(vertical
ray path 24a -x3 -t xp))
at the observation
point
at the observation
point
at the observation
point
[ml.
first order
diffraction
(x=12)
i 8.45
8.50
8.55
8.60
t
Fig. 10. The convolved contribution (x,, .x,1 = (2.2,OS) [ml.
pertaining
to a first order diffracted
_60 i
i
field (vertical
second order di&action
(n=lZ)
i;
i
i
8.65 x 10-3
?
ray path 24a-x3
+xy’)
; I
8.425 8.450 8.475 8.500 8.525 8.550 8.575 8.600 X10 -3
time
Fig.
11.The convolved contribution
(n,, -Q) = (2.2,0.5)
[ml.
pertaining
to a second order diffracted
field (vertical
ray path 24~ -x3 +xp’)
K. F.I. Haak, B.J. Kooij / Wave Motion 23 (1996) 139-164
156
Table 2 The maximum amplitudes
of the convolved
contributions
evaluated at different points of observation
Vertical generalized ray path geom 1st diff 2nd diff geom I st diff 2nd diff
4a-x,+x!“’
4a-x,
Maximum absolute amplitude ug [m/s]
x3 [ml 2.0
0.5
2.88 x 1O-3 (100%) 8.17x lo+ (28%) 1.60x 1O-6 (0.06%)
0.5
0.5
-
+xp)
8.12x 2.80x
1O-4 (100%) 1O-5 (3%)
geom 1st diff 2nd diff
24a-x,+x!“’
2.2
0.5
1.28x 1O-3 (100%) 6.24x 1O-4 (49%) 5.00x 1o-5 (4%)
geom 1st diff 2nd diff
x3-xp
2.6
2.0
8.78X 1o-4 (100%) 2.78 X lo+ (32%) 1.60x lo+ (0.2%)
2.0
2.0
-
geom I st diff 2nd diff
2a+x,-x:S’ -
(rel)
1.25 x lo-=’ (100%) 5.23x lo+ (4%)
xg +x:s’ 2a +x, +x:S)
computed some contributions for receiver locations below the half-plane. The results are depicted in Figs. 14-18. In all figures the corresponding generalized ray path is shown as well. In Table 2 the maximum values of the convolved contributions are indicated, together with the vertical generalized ray path. Obviously, the second order diffracted wavefield turns out to be only a fraction of the geometrical contribution. Physically, this is caused by the scattering nature of the diffracting edge. Hence, the wavefield due to a multiple diffraction phenomenon is attenuated considerably. Consequently, for practical purposes, the second and higher diffracted wave fields need not be computed. This will decrease the computation time of the total field within a finite window considerably. One can argue on the same physical grounds that the second order diffracted field compared with the first order diffracted field will decrease when the receiver location moves away from the edge, or more precisely, when the angle between the half-plane and the last part of the ray path decreases. This expectation is confirmed by the numerical experiments (see Table 2).
first order diffraction
(n=Z)
~
.._ _
-------------
.
-1000/ 1.55
1.60
1.65
1.70 t
Fig. 12. The convolved (x1, ns) = (C).5,0.5)
[ml.
contribution
pertaining
to a first order diffracted
1.75 x10-3
field (vertical
ray path 4u--~a+#‘)
at the observation
point
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
157
second order diffraction (a%?)
_30
1.600 1.625
1.650
1.675
1.700
1.725
1.750
time
Fig. 13. The convolved (x,. xx) = (0.5,0.5)
contribution
pertaining
to a second order diffracted
1.775 x 10-3
field (vertical
ray path 4a-xx
+np’)
at the observation
point
[ml.
geometrical contribution (n=O)
1.15
1.20
1.25
1.30
t Fig. 14. The convolved (2.0,2.0) [ml.
contribution
pertaining
to a geometrical
1.35 x 10-3
contribution
(vertical ray path x,+x 9’) at the observation
point (x,, x3) =
first order diffraction (n=O)
1.15
1.25
1.20 t
Fig. 15. The convolved (2.0,2.0) [ml.
contribution
pertaining
to a first order diffracted
1.30
1.35 x10-3
field (vertical ray path x,--x p’) at the observation
point (x1, x3) =
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
158
second order di%action
(n=Z)
time
Fig. 16. The convolved (x,, x3) = (2.0.2.0)
contribution
pertaining
to a second order diffracted
x10-3
field (vertical
di%xtion
Fig. 17. The convolved (2.0,2.0) [ml.
contribution
pertaining
to a first order diffracted
2.15
field (vertical ray path x,+x :“‘) at the observation
2.20
2.25
2.30
convolved [ml.
contribution
pertaining
to a second order diffracted
point (x,, x3) =
(n=Z)
2.35
2.40
time
The
point
x 10-3
second order diffraction
18.
at the observation
(~2)
t
(x,, x3) = (2.0,2.0)
-xp’)
lml.
firstorder
fig.
ray path 2n+x,
2.45 x 10-3
field (vertical
ray path 2a+x,
+n:“‘)
at the observation
point
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-144
159
11. Conclusion The strategy of solving the original Wiener-Hopf equation iteratively enables us to solve the scattering problem exactly within any finite time window. After classifying the different wave constituents closed-form space-timedomain expressions for the pressure field are obtained by means of a modified version of the Cagniard-De Hoop technique. Due to the complexity and to the methodological character of this study we have confined ourselves to the wavefield due to at most one reflection with the interface. For practical purposes this turns out to be sufficient in most cases. The results of the numerical implementation are interesting. The amplitude of the second order diffraction wavefield turns out to be only a fraction of the geometrical contribution. Physically, this is caused by the scattering nature of the diffracting edge of the pressure-free half-plane. Mathematically, this can be explained by the observation that the diffraction operators have a norm less than one. It seems to be a difficult task to estimate the norm of the contraction operators more precisely in order to predict the maximum amplitudes of higher order diffracted wave components. But the numerical results show that, for practical purposes, in case of deep embedded semi-infinite screens only the first order diffracted wavefields have to be computed, thus avoiding expensive, time-consuming evaluations of multiple integrals.
Appendix A Standard solution procedure The elementary
of the elementary
Wiener-Hopf
Wiener-Hopf
equation
equation
(A-1) is solved by factorizing n(a)
=fi(a)S(a)
the vertical slowness yi ( (Y) as (A.21
9
in which #(a) and $( (Y) are analytical functions in D”, and D”,, respectively as 1a 1-+ CCin their regions of regularity. By the additive decomposition
Qiwz(a, $1 =T%W~,,=Q~,<~~
s> +Q%w
in which QE( LY,s) and Qkn( (Y,S) are analytical Eq. (A.2) changes into -
Y&j
@G(CG s)
-Q&SW
(see Eq. (38) ), and behave as o( 1)
~1 , functions
s> = - Y%W&(W
(A.31 in D”, and in D”, respectively
s> +Q:&,
~1 .
(see Eqs. (39)-(
42) ) ,
(A.41
The left hand side of Eq. (A.4) is an analytical function in D”, with asymptotic behaviour o( 1) as 1aI -+ ~0in Dk, while the right hand side is an analytical function in DE with asymptotic behaviour o( &‘*) as 1a I --f 00 in DE. Consequently, they form an entire function which according to Liouville’s theorem (Tit&marsh [ 211) vanishes. As a result we have
(A.3
160
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
Appendix B Conditions for intersection
of the modijied Cagniard contours with the real axis
In this appendix the conditions to obtain a geometrical field contribution are investigated. Starting with the integral form expressions, the points of intersection of the modified Cagniard contours on the one hand and the real q,,,-axis or a-axis, respectively, on the other hand are determined. In the last section a condition for the presence of headwaves is derived. B.1. General geometrical
condition for the geometrical$eld
We consider the integral form expression
u^,tx,, x39 s> =-
A,( a, qo,o)
-qo,~~s’l~ exp{-s[yl(~)D,+~ll~ X expt - s[ 35 ( 40,0) Dqo,o
dq,,,da,
(B.1)
in which Ao=
~WYwlo,o) Pl(40.0 - a> .
U3.2)
The modified Cagniard method now aims at parametrizing
T,=Y,t~m,+~x,
U3.3)
3
in which u E [a, qo,o] and rU is real and positive. Eq. (B.3) represents a branch of a hyperbola, XL,
U f-- ~2
1,
rL,fi $$ ($-Tz)“*, I,
ru a T, .
which is given by (B-4)
Here, R,= (X~+D~)“2,
Re(R,)
>O,
(B.5)
-G=x, >
v3.6)
x,,, = - xy
(B.7)
and D,, is some vertical propagation T,, =
3. Cl
distance. Further,
(B.8)
Due to the occurrence of a singularity in the integrand of Eq. (B.l) it is necessary to investigate the points of these points are given intersection S, and S,,, with the real (Yand qo,o-axis, respectively. From Eqs. (B.4)-(B.8) by
K.F.I. Hank, B.J. Kooij/ Wave Motion 23 (1996) 139-164
XL‘
161
1
&I=(x;+D;)l’*
(B.9)
;’
It can be verified that the condition (B.lO)
I&I 5 I%0,0I is satisfied if
WI
5 1x1I 40.0 D, D
(B.ll)
and vice versa. Now we are in a position to transpose the conditions concerning real axis in conditions concerning the location of the source and the receiver. B.2. The vertical partical velocity$eld In case 2 the vertical propagation D,=
(2n-
l)a+x,
the points of intersection
with the
in D”’ and 0”’ (n < m) distances D, and D,,,, are given by
,
(B.12) (B.13)
DqO,O=a+~~S) . Consequently,
the condition
(cf. Eq. (52) )
ILI > I~,o,oI
(B.14)
transposes in
IXP I < Ix1I
LZ+J$’
B.3. General geometrical
(B.15)
’
(2n-l)a+x,
condition for a headwaue contribution of the geometrical field
Let us consider the integral form expression
fi33(%x3 s) =
W-h
2(2ni)
~2):
of a geometrical
field
exp[-s(Y~D~+LYX,)I da,
(B.16)
Lt in which the integration path L, coincides with the imaginary a-axis. Using the modified Cagniard method the integration contour L, deforms into the complex a-plane. The presence of the double valued functions 3/, and x in the integrand causes branch cuts along Br,=
]Re(a)]
> $,
(B.17)
k=1,2.
When the integration contour intersects a branch cut a head wave contribution Hijden [ 221) _In Section B. 1 we found the intersection point to be
arises (see De Hoop and Van der
(B.18) from which we conclude that L, never intersects the branch cut Br,. The branch cut Br, is intersected
when
162
K.F.I. Haak, B.J. Kooij/
s,>
Wave Motion 23 (1996) 139-164
1.
(B.19)
c2
Hence, a head-wave
contribution
of the geometrical
field only occurs when c2 > ci and (B.20)
Appendix C The conversion
into principle
value integrals
In this appendix Eq. ( 56) is evaluated in terms of principal value integrals. This can be accomplished by application of Cauchy’s theorem to the original integration contour, resulting in a principle value integral and half a residue contribution. However, a lucid investigation in this way is no sinecure. Therefore we rewrite the integral-form expression in the equivalent operator form. Then, the relevant decomposition operators are evaluated in terms of identity and Hilbert operators. We start with the operator-form equivalent of one term on the right hand side of Eq. (56)) viz. Int = qa[B,( a) 1, in which ?Pa= -g
I ev( -SKI>
exp( -wa)
[exp( -wxd
L,
+exp(sy,x,)
Y?(a)
I -B,(a) Pl
da
cc.11
and
cc.21 Here, the argument z is given by z=exp(
-2sy,a)~~,~[~(c~)Uo,o(cy,
s)l .
CC.31
If k E [ 0, 11, no integration singularities arise. The following investigation holds for k > 2. In order to keep the analysis lucid, we omit the integration variables qo,k- i, . . ., qo,l and the argument z in the notation. Next, we apply Eq. (42) to the multiple operator, resulting in k-l [exp( -2~y,a)diL,~]~--l= in which we have introduced
0 $
[J+Plk-‘,
(C.4)
the operators J and P as
J= exp( - 2sy,a)l,
cc.3
P=exp(
(C.6)
-2sy,a)iH.
Here, I and H denote the identity and the Hilbert operator, respectively. The norm of the multiple operator in Eq. (C.4) is less than one (contraction operator), due to the presence of the exponential function and the fact that the Laplace parameter s is chosen real and positive. Hence the repeated diffraction operator is a contraction operator and the scheme is convergent. It is noted that Eq. (C.4) does not represent a binomial series, since the operators J and P do not commute. Equation (C.4) is now evaluated as
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
163
k-l
0
[exp( - 2syra) @vH] k-1 = i
Y-l+
-1-
cc.71 Here, the summation m
E [2, . . . . k] E [l, . . . . k+l-m] E [l, . . . . k-m+p-CfZjkJ =k-C;:;‘k,.
4 kP k i nr Consequently, B,(a)
variables range according to
Eq. (C.2) is now evaluated as
= &
nz,k,?km-,
$>I I
~~.,{exp(-2ksyla)~~,,[~(a)U,,,(cy,
@&,,,,{exp( c2ik-’
-2Lw~WL
+
(C-8)
PE [2, . . . . m-l]
,
l
X~ex~(-2k,,-~~~~~)...~~Iex~(-2k~~~~~)~~,~~~(~)~~,~(~,~)llll. Finally, substitution of the integral-form to the integral form
*sf Int=(2)k(27ri)3 .,,k,,?k,..-,
+
X fi
&IY~,
equivalent
(C.9)
of the operators and application
%,o) exp( - 2ksyl (%,r)a)
of Eqs. (C. 1) and (C.2) leads
dq,.,dq,,,da
L Lqa.1ko.0
1j f -“iri)z+“’
(2)k+1
exp[ -2k pw (40,&l
.a.
L,
LqO,m
L~~,~_,
$o,o.. .dqo,,da
f
j
Am.f(a~40,ol
L‘l0.I Lga,o
(C.10)
9
p=l in which the summation f(a,
%,o) =exp]
variables range according to Eq. (C.8) andA f~, qo,o) is given by -~[n(~o,o)(~+x~))
-~o,dc~“‘l
-s[n(a!)(a*~~)
+a~~]}
.
(C.11)
References [ 11 B.J. Kooij and D. Quak, “Three-dimensional scattering of impulsive acoustic waves by a semi-infinite crack in the plane interface of a half-space and a layer”, J. Math. Phys. 29, 1712-1721 (1988). [2] K. Aki and P.G. Richards, Quantifatiue Seismology, Theory and Methods, W.H. Freeman, San Fransisco ( 1980). [3] A.T. De Hoop, Handbook ofradiation and scattering ofwaues, Section 10.13, Academic Press, London, New York (1995) to appear. [4] A. Sommerfeld, “Mathematische Theorie der Diffraction”, Marh. Ann. 47,317-374 (1896). [5] E.T. Copson, “An integral equation method of solving plane diffraction problems”, Proc. R. Sot. (London) A 186,100-l 18 (1946). 161 B. Noble, Mefhods based on the Wiener-Hopf technique, Pergamon, New York (1958). [ 71 P.C. Clemmow, “Radio propagation over a flat earth across a boundary separating two different media”, Phil. Trans. R. Sot. (London) A 246, l-55 ( 1953). Also in: M. Born and E. Wolf, Principles of Optics, Pergamon, New York (1959).
164
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
[ 81 R. Du Cloux, “Pulsed electromagnetic radiation from a line source in the presence of a semi-infinite screen in the plane interface of two different media”, Waoe Motion 6,459-476 (1984). [9] I.D. Abrahams, “Scattering of sound by two parallel semi-infinite screens”, Waue Motion 9,289-300 (1987). [IO] W.G. Heitman and P.M. van den Berg, “Diffraction of electromagnetic waves by a semi-infinite screen in a layered medium”, Can. J. Ph.?% 53,1305-1317 (1975). [ 111 H. Shirai and L.B. Felsen, “Spectral method for multiple edge diffraction by a flat strip”, Wave Motion 8,499-524 (1986). [ 121 B.J. Kooij, “Scattering of point-source generated transient elastodynamic waves by a semi-infinite void crack in a half-space”, in: Elastic waue propagation, Eds. M.F. McCarthy and M.A. Hayes, Elsevier, Amsterdam (1989), pp. 217-222. [ 131 L. Cagniard, R@ections et R&fractions des Ondes Seismiques Progressives, Gauthiers-Villars, Paris (1939, in French). Translated and revised version: Reflection and Refraction of Progressive Seismic Waves, Eds. E.A. Flinn and C.H. Dix, McGraw-Hill, New York ( 1962). [ 141 A.T. De Hoop, “A modification of Cagniard’s method for solving seismic pulse problems”, Appl. Sci. Res. B 8,349-356 ( 1960). [ 151 A.T. De Hoop, “Theoretical determination of the surface motion of a uniform elastic half-space produced by a dilatational, impulsive, point source”, Proc. Colloque International du CNRS (Marseille) Ill, 21-32 (1961). [ 161 J.D. Achenbach, Wave Propagation in Elastic Solids, North Holland, Amsterdam (1973). [ 171 J. Miklowitz, The Theory of Elastic Waves and Waveguides, North Holland, Amsterdam (1978). [ 181 D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ (1946). [ 191 N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff, Groningen (1953). [20] H. Hochstadt, Integral Equations, Pure &Applied Mathematics, Wiley, New York ( 1973). [ 211 E.C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford (1932). “Generation of Acoustic Waves by an Impulsive Line Source in a Fluid/Solid Configuration [22] A.T.DeHoopandJ.H.M.T.vanderHijden, with a Plane Boundary”, J. Acoustic Sot. of America 74,333-342 ( 1983).
Wave Motion 23 (1996) 139-164
MIlTfUll
Transient acoustic diffraction in a fluid layer Kasper F.I. Haak *, Bert Jan Kooij Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Centre for Technical Geoscience, Del@ University of Technology, P.O. Box 5031, 2600 GA De& Netherlands
Received 3 October 1994; revised 29 June 1995
Abstract Acoustic and elastodynamic waves serve as diagnostic tools in the non-destructive search for cracks and holes in material structures. In this article, the diffraction of acoustic waves caused by a semi-infinite screen in a fluid layer is investigated. A
homogeneous half-space models the substructure of the object under investigation. The transient wave problem is then solved by means of an iterative scheme in which successively the Wiener-Hopf technique and the Cagniard-De Hoop method are applied [Kooij and Quak, “Three-dimensional scattering of impulsive acoustic waves by a semi-infinite crack in the plane interface of a half-space and a layer”, J. Math. Phys. 29, 1712-1721 (1988) 1. From the numerical results it turns out that in case of deep embedded semi-infinite screens only the first order diffracted wavefield gives a significant contribution to the geometrical wavefield. Application of these results in the iterative solution procedure yields a very effective solution scheme in which the exact transient wave field in the configuration at hand can be calculated within any finite time window.
1. Introduction
In quantitative non-destructive material exploration, elastodynamic and acoustic waves serve as diagnostic tools to probe the mechanical features of material structures. Examples can be found in examination on cracks in wings of air planes or in concrete of nuclear power stations. The configuration under investigation consists of an elastic layer with a semi-infinite pressure-free void crack in it. An elastic homogeneous half-space represents the substructure of the diffracting crack. Aki and Richards [ 21 show that the elastic wavefield is composed of a dilatational part and an equivoluminal part. In this article we confine ourselves to the equivalent fluid model for dilatational waves in a solid (see De Hoop [ 31) . Hence, the influence of equivoluminal waves is neglected and the configuration is modelled as a fluid layer with a pressure-free semi-infinite void screen in it, together with a fluid half-space substructure. This means that surface wave phenomena like Rayleigh waves are not incorporated in our model. The wave problem is solved by means of a proper combination of the Wiener-Hopf theory and the Cagniard-De Hoop technique. Some numerical results are presented. The diffraction phenomenon has been studied analytically by Sommerfeld [ 41, Copson [ 51 and Noble [ 61. Clemmow [ 71 and Du Cloux [ 81 considered the diffraction by a semi-infinite screen in the single interface of two homogeneous media. The latter presented numerical results of pulsed electromagnetic waves. The problem of acoustic multiple scattering by two parallel half-screens has been considered by Abrahams [ 91, in which the system * Corresponding author. E-mail: k.f.i. [email protected]. 0165-2125/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDIO165-2125(95)00037-2
140
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
of coupled Wiener-Hopf equations is uncoupled by the introduction of an infinite product of poles, each of which contributes to the total wavefield for all ,times ‘ ‘t” without any apparent physical interpretation. Hence, this method yields no exact solution (contribution of a finite number of poles) and is not efficient for the calculation of wavefields in a finite time window. He&man and Van den Berg [ 101 carried out a frequency-domain analysis of diffracted electromagnetic waves by a semi-infinite screen in a layered medium. Here, both multiple edge diffraction as well as repeated reflections are taken into account. However, the conversion of high-bandwidth frequency-domain data to accurate time domain results is no sinecure. Shirai and Felsen [ 1 l] investigated multiple diffraction of electromagnetic waves in an infinite homogeneous medium with a flat strip in it. They developed an iterative technique in which the result of the nth iteration (an n-dimensional integral expression) represents an n-fold half-plane diffraction event caused by the incident wave which diffracts again and again at the opposite edges. In this paper we shall derive an explicit iterative scheme which yields after iV (fixed) iterations an exact time-domain solution within a finite time window. Using the modified Cagniard method Kooij and Quak [l] and Kooij [ 121 obtained closedform expressions in the space-time domain in case of acoustic and elastodynamic waves, respectively. They considered the configuration of a crack located in the plane interface of a uniform layer and a semi-infinite halfspace. In our case the semi-infinite void screen is placed in the fluid layer, rather than on the interface. In this paper the Cagniard-De Hoop technique is used in order to obtain space-time domain results. This method is due to Cagniard [ 131 and has been modified by De Hoop [ 14,151. A description of this method can also be found in Achenbach [ 161, Miklowitz [ 171 and Aki and Richards [ 21.
2. Description of the configuration In the present article we investigate theoretically the diffraction of acoustic waves in a fluid-fluid configuration, as is depicted in Fig. 1. It consists of a fluid layer with a pressure-free surface, and a semi-infinite fluid medium which occupies a half space. In the fluid layer a pressure-free half-plane void screen is present. To specify the geometry in the configuration we employ the coordinatesx,, x2, xs with respect to a fixed, orthogonal, Cartesian frame of reference, with the origin B’ and three mutually perpendicular base vectors ii, iZ, i3 of unit length each. In the indicated order, the base vectors form a right-handed system. The origin d is chosen to be located at the pressure-free surface of the fluid layer right above the tip of the pressure-free half-plane. In accordance with the geophysical convention, i3 points vertically downwards. The base vector ii is chosen such, that the pressure-free half-plane is located at x, > 0 and x3 = a. The plane interface of the fluid layer and the half space is located at x3 = d. The subscript notation for Cartesian vectors is used and the summation convention applies to all repeated subscripts. The position in the configuration is also denoted by the position vectorx =x&r, withx E W3.Furthermore, the symbol t is used for the time coordinate. i2
,
Y' source
in
pressure-free
surface
Q (%I”‘,
zp,
crack
Fig. 1. The geometrical configuration.
23
=
a
z3
=
d
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
141
The media are assumed to be linear, locally reacting, instantaneous reacting, time-invariant, homogeneous and isotropic. The acoustic properties are characterized by its volume density of mass p and its compressibility K, respectively. Then the acoustic wave speed is given by c = (prc) - “‘. We assume that the acoustic field waves are generated by a pulsed acoustic monopole line source, which is present in the fluid layer, parallel to the lateral direction i, and starts to act at the instant t=O. Specifically, the source is located at (xi, x3) = (xi”‘, ~9’). The problem is now reduced to a two-dimensional one, in which the field quantities are x,-invariant, i.e., a2 = 0.
3. Basic equations for the acoustic wave motion The linearized akP
+
@rvk
acoustic fields are governed by the hyperbolic =
%J4u + Ka,P
sk.3.f
=
system of first-order differential
equations
9
(1)
(2)
0,
in whichp and u,, denote the acoustic pressure and the particle velocity, while p and K represent the dynamic medium properties, viz. the volume density of mass and the compressibility, respectively. The symbol Sk,,, represents the Kronecker tensor. Further, the normal component of the dipole line source is characterized by f=f(t>S(x,
-XIS), X3-Xs”‘) .
(3)
Here, f(t) is a causal function of time, i.e. f(t) = 0 when t < 0 and 6(x,, x3) denotes the two-dimensional delta distribution. At the free surface of the fluid layer the boundary condition lim p(x,, x3, s) = 0 , X310
Vx, E W
holds, while at the level of the half-plane lim p(x,, x3la
x3, s) = lim p(x,,
lim P(x,, .qJa
x3, s) = 0
-%?a
Dirac (4)
the conditions
x3, s)
,
Vxl E W ,
(5)
x,>o,
(6)
together with lim +(x1, x3, s) = lim ~(xi, qla xstn apply. Finally, continuity
x3, s) ,
x1
of the pressure and the vertical component
lim p(xI, .x3, s) = lim p(x,, .%Ld x,Td
x3, s)
Vx, E W
(7) of the particle velocity yields (8)
and lim u,(x,, x3, s) = lim u~(x,, x3, s) x3ld
4. Transform-domain
x,Td
Vx, E W .
(9)
acoustic wavefield
In order to exploit the linearity and time-invariance of Eqs. ( 1) and (2), we subject them to a one-sided Laplace transformation with respect to time. The Laplace transform A(x, s) of a quantity A(x, t) is given by
K.F.I. Ha&
142
m
A(x, s)
=
I
t) dt
exp( -st)A(x,
B.J. Kooij/
Wave Motion 23 (1996)
139-164
.
(10)
0
In view of the future application of the Cagniard-De Hoop technique, the parameter s is real and positive and is taken sufficiently large. According to Lerch’s theorem (see Widder [ 181) the solution of the integral equation ( 10) exists, is unique and vanishes in the interval t < 0, thus ensuring causality. In order to exploit the invariance of the medium parameters with respect to the xi-direction, we carry out a spatial Fourier transformation with respect to xi. The spatial Fourier transform pair &Xl,
x3, s)
*
&%X3,
s>
(11)
is given by +m A(% x.3, s) =
I -cc
exp(s~i)&xi,
x3, s) hi
,
(12)
+im
exp( - scrx,)A( (Y,x3, s) da.
(13)
I
-im
Note that scr is an imaginary quantity, hence cry 0. In order to keep the analysis lucid the configuration under (Dc2’),.
2 k=l,2
withRe(y)>O,
(14)
y=22 k
(15)
pk ’
respectively, applying the temporal Laplace and spatial Fourier transformation to Eqs. ( 1) and (2) and eliminating u”,results in a coupled system of ordinary first order differential equation, which have exponential functions with positive and negative argument as linearly independent solutions. As a consequence, we have in the domain D(l), D”’ and Dc3’ p”‘(a,
x,, s) =A’,“(a,
s) exp[ -sn(x3-@)I
+At’)(a,
s) exp[ -sn(c--X3)]
,
x,ED(‘)
(16)
and G$:“(cv, x3, S) =Y,{A’,“(a,
S) exp[ --SY1(xX-_)]
-A!?(a,
s) exp[ -sn(F-x3)]},
x,ED(‘) , (17)
(4) Eqs. ( 16) and (17) also apply, in which r E { 1,2,3} and where pimax = min;max{x3(x3ED(‘)}.IndomainD provided that the substitution {yi , Yl } -+ { y2, Y2} is carried out. Further, the coefficient A'_) ( (Y, S) = 0, thus satisfying the physical constraint of causality. From the different equations the source conditions follow with Eq. (3) as lim p( a, x3, s) - Xy~PB( (Y,x3, s) =J’(s) X,IXP
=_?(s) exp(s&‘)
(18)
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
143
Fig. 2. The integration contours L, in the complex w-plane and Lq,_ in the complex q,,,,-plane in case of @,,.
and (19) Together with the boundary conditions (4), (5)) (8) and (9)) which can suitably be transformed into the Fourier domain, they reduce the number of unknown coefficients in Eqs. ( 16) and ( 17) to one, e.g. A’:‘. In order to find the last coefficient, we apply an inverse Fourier integral to the formal spectral solutions. Next, we require the boundary conditions (6) and (7) to satisfy. This is accomplished in the following way. To start with, we introduce the reflection coefficient R, as
y, - y2 Y, -t Y2 .
-
Ro=
(20)
Further, we define GR( (Y,S) and ti( G’<(L~,s) =Ay){
(Y,S) via
1 +R, exp[ -2sy,(d-a)]}
(21)
and 2P(%
s) = Y$’ exp( -~,a)
[exp(~~~~~S’) +exp(
-.vv~~))I
_Ac3j 2YI[1 +
1 - exp( - 2sy,a)
PI
+Ro
exp(
1 -exp(
c22j
-2~41
-2sy,a)
’
Let us finally assume GR( a, S) and ti( CY,S) to be analytical functions in the right half-plane Df3sand in the left half-plane D”, (see Fig. 2)) respectively, with asymptotic behaviour o( 1) as 1a 1-+ ~0in Dt and D”, respectively, then in accordance with Jordan’s lemma and Cauchy’s theorem the boundary conditions (6) and (7) are indeed fulfilled. Eliminating A$? in Eqs. ( 2 1) and ( 22) results in the Wiener-Hopf equation Q’ exp( - sy,a) - GR( (Y,S)
1 + R. exp( - 2sy,d)
1 +R, exp[ -2sy,(d-a)]
=HL(a,
s)
1- exp( - 2sy,a) (23) Yl
’
in which “i
Q'
= i
[
exp( sy,.@))
+ exp( - sylx$“)) ] .
(24)
Obviously, the denominators of Eqs. (22) and (23) have no zeroes since the Laplace parameter s is real and positive. This means that no surface/interface or pseudo-Rayleigh wave phenomena will appear in our analysis. Since the wave amplitudes A’,“, A?), AT’ and A’,) are functions of either GR or @, e.g.
144
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
-i f 2
{exP[ -syl(2a-x~‘>l +exp(
A(‘) = +
-sy~&))}-GR
exp( -sy,a)
(25)
1- exp( - 2syia)
and -i f - y { 1 +exp[
-2sy,(a-$‘)I}
A”‘=
+GR exp[ -sm(a-x9))]
(26)
,
1 - exp( - 2syiu)
the problem is reduced to the solution of EQ. (23).
5. Iterative solution procedure of the Wiener-Hopf
equation
The original Wiener-Hopf equation is split into a series of less complex Wiener-Hopf equations, each of them corresponding to the wavefield due to k reflections against the interface and the pressure-free half-plane. In doing so we expand GR( CY,s) and @( CY,s) as
{GR,HLl(a, s>= f
{G;, HIk’l(a,s>.
(27)
k=O
Here, the kth term of Eq. (27) is at least of the exponential same exponential order results in Q’exp( -sy,u)
-G:((Y,
s) =@(a,
s)
order exp [ - 2ksy, (d - a) ] . Collecting
1- exp( - 2sy,u) Yl
which is the Wiener-Hopf
the terms of the
’
equation of order zero and k-l
-GF(a,s)--[l-exp(-2sy,a)]
C
p=o =Hi(
(Y,S)
1 - exp( - ;?sy,u)
,
{(-Ro)k-P
exp]
-2(k-p)sy,(d-a)lG,R((Y,
~11
(29)
kal,
Yl
which is the Wiener-Hopf equation of order k. Next, Eqs. (28) and (29) are split into a series of elementary Wiener-Hopf equations of order k, n, each of them corresponding to k reflections against the interface and the pressure-free half-plane and II reflections against the surface of the fluid layer and the half-plane, respectively. Therefore, we expand GF( (Y,s) and Z!& cry,s) as
Here, Gj&( CX,s) and e,,( CX,s) have exponential order exp[ - 2ksy, (dof the same exponential order in Eqs. (28) and (29) results in
a) - (2n + 1) syiu]. Collecting
the terms
(31) where the source terms uk,+ are given by
K.F.I. Husk, B.J. Kooij/
u0.0
Q'
=
exp( -syla)
Wave Motion 23 (1996) 139-164
145
,
(32)
e-1
n>l,
U0.E = Aexp(-2sy,a), Yl
(33)
k-l
uk,O=
c
-
%I( -Ro>k-”T-Q -w--p)sy&i--a)]
, k>
1
(34)
I-‘=0
and uk,n
=
k-l
- C ( -Ro)k-YIG~n-G~:,_lexp(
-
p==o
2~y,a)l exp[ -Xk-p)syl(d-a)l
, k, na 1.
(35)
It is clear (by induction) that GFn( (Y,s) and Z$&(cy, s> are indeed of the exponential order exp [ - 2ksy, (d - a-> - (2n + 1) syra]. Further, it is noted that the evaluation of the wavefield in a finite time window needs only the solution of a finite set of recursive elementary Wiener-Hopf equations. Then, the problem is now broken down to solve Eq. (31). This is accomplished in Appendix A by means of factorization of the kernel, additive decomposition and application of LiouvilIe’s theorem. The solution is given by (36) and
f-&n(~,s>= ti(NQk,(e
$1
(37)
7
in which l/2 yy-(
a)
=
(38)
are analytical functions in DR and DL, respectively. Here, the upper and lower sign correspond with the superscript R and L, respectively. Further, in accordance with Plemelj’s theorem (see Muskhelishvili [ 191) @$;,f-[Qk,nJ
Q:LLb,
(% s)
S>= q$"'"rQk,,l(4
S>,
(Z- %:,:I [Q,,] in which we define the projection
7
~~%~
* E L?k,,z
(39)
(a, s> 3 a-;:,:,
operators (see Hochstadt
[ 201) (40)
and (411
= i [Z+iHlQ,,,(a,
S) .
(42)
146
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
In Eqs. (39)-( 42) Lqk,+is any contour which splits the complex qk,,-plane in a region Df;,,, (on the left) and a region Dik,,$(on the right), and Qk,n is any function which satisfies Holders conditions on L_. Further, Z and H denote the identity and the Hilbert operator, respectively. From Eqs. (32) -( 42) we infer that the norm of the repeated diffraction operators is less than one (contraction operator), due to the presence of the exponential function and the fact that the Laplace parameters is chosen real and positive (see also Appendix C) . Hence, the iterative scheme is convergent. In the present analysis we have
Qk.n(c s)= $W~,,nh
s> .
(43)
In Fig. 2 the integration contour L,,,, is depicted corresponding with the case Q&. In case of Qk, the small semicircle integration path around the integration singularity is replaced with a semi-circle loop on the right of the pole.
6. Spectral-domain
wave constituents
The Fourier-transform-domain pressure field is evaluated by means of the concept of up- and downgoing wave constituents. To this end we substitute d = ma and Eqs. (27) and (30) in the expressions for the coefficients A:, A:, A; and AZ. Here, G& ( cy,S) has exponential order exp [ - 2k( m - 1) a - (2n + 1) a] . Ordering the terms in Eq. (17) in exp[ - (2n+ l)sy,a] then leads to
ip
la=0 l
As far as D’ ’ ) is concerned.
V.3.a =
f3 {exp[ -sy 2
(44) j>l.
Here, 1(xF’--x3)]
+exp[
-sy,(xp)+xa)]}
fe,n-da, s)
%:,:b = exp( -ma) I exp( -s-m) +exp(vlx3)
1
?3,;, = exp( -s-w) [exp( -
j ~L(m-l)-l(~~ s) 1 C -t i=l Pl
‘2
syIx3) +exp(sw,)
+ i Y,( -Ro)i-PG~~_i(n~_l)__(LY, i=l p=o
(45)
,
(46)
Pl
S) exp[ -2(i-p)(m-l)sy,a]
>
.
(47)
The generalized ray paths show that the level of the half-plane acts as a secondary (diffraction) source. The vertical particle velocity field in D’*’ (ordered in exp [ - (2n + 1) sy,a] ) is identical to the one in D(l), except if n = 0. In a similar way the spectral field in Dc3’ and Dc4’ can be treated, as well as the acoustic pressure. The mathematical decomposition of the wavefield into generalized ray contributions has a physical sense. Each contribution can be recognized separately in the receiver record if the pulse duration of the source is small compared with the travel time in the multiple reflection/diffraction geometry. In our numerical examples this condition has been satisfied.
7. Analytical continuation of decomposition operators It is anticipated that in accordance with the application of a modified Cagniard-De Hoop technique, as outlined in Section 9, the contours L, and Lq,,o will deform in order to cast the exponential functions in the desired form
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (I996) 139-164
147
exp( - sr), in which T is a real integration variable. The contours L, and LqO,O deform due to the occurrence of exp( - scrxl) and exp( sqo,,.$‘) in the integrand, respectively. Since the modified Cagniard contours Lw,,, . . ., coincide with the imaginary axis we have to distinguish between four different cases, viz. L 40,rr_, case 1:
x,-co { XjS) < 0 ’
case 3:
case 2: and
x,>o -tX$S)< 0 ’
case 4:
During contour deformation singularities in the plane of integration may be encountered, thus requiring analytical continuation of the decomposition operators @$,, and @&,, as indicated in Eq. (39). It is noted here, that domain D,L;:ff denotes the part of the complex qk,,-plane on the left or on the right of the integration contour L,,,,, respectively, including the integration contour L,,,,. Due to the complexity, we confine ourselves to an investigation of the pressure field in D(l) and DC*) of exponential order n < 2m - 1. For practical purposes this turns out to be sufficient in most cases. 7. I. The uertical particle uelocity$eld in D(l) and DC2)(n < m) In order to investigate the wavefield for 1 Q n < m in D(l) and D’*’ it is necessary to investigate The next analysis holds for n > 2. From Eqs. (37) and (39) we have @,?Z-,(%s)
%_L-I(G Application
s)l *
=~((Y)~~,,-,re,,,-,(~,
Repeated application
ZY&_ i (cr, s) .
(48)
of Eqs. (43)) (33), (37) and (39) leads to
s) =~(LY)~~,,_,(exp(-2Syla)dSl,,,_,{cxp(-2sy,a)
s>lIl.
. . . @&[~((.~)Gxo(~~
of Eqs. (46)) (40) and ( 13) leads to an integral-form.
In Fig. 3 the integration
(49)
contours L,, Lqo,O, . . .,
L 40.,,_,have been depicted. Due to the application of a modified Cagniard-De Hoop technique the contours L, and L,,O will deform. In the next subsections case 1 and case 2 are investigated in more detail. 7, I, I, Analytical continuation in case I In case 1 the occurrence of exp( -scrx,)
and exp(sqo,&‘)
in the integrand
will cause a deformation
I+% QO,Ot .,.I Qo,n-1)
a,
q0.0,
.-.,
q0,+
R+, qo,or
---2
>
dane
qoo,,-1)
Fig. 3. The integration contours concerning the integralsL,, Lqop, . . . , L+,,,*_ , before &&mation.
of L, and
148
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
L,,, in the left-half of the complex a-plane and in the right-half of the complex %,O-plane, respectively. analytical continuation of the decomposition operators is required.
Hence, no
7.1.2. Analytical continuation in case 2. In case 2 the occurrence of exp( -S&X,) and exp( sq,,&‘) in the integrand will cause a deformation of L, and L,,, in the right-half of the complex a-plane and the qO,o-plane, respectively. Now, the first decomposition operator in Eq. (49) requires analytical continuation, which according to Eq. (39) results in . . . ~~,,[~((Y)U~,~((Y,S)IJ~
~.,,-1(~,s)=-~(~)~~,,,-,{exp(-2sy,a)~~,,,_,{exp(-2sy,a)
+ tiCa> exp( -Ww)@k,,,-,iexp(
-2wa)@t&31exp(
-2wa)
. . . ~o.o[*(~)Uo,o(a,
s>l I} . (50)
Obviously, analytical continuation of the first decomposition Repeated application of this procedure leads to nZj,,-~(a,s)=--
C
operator in the second term of Eq. (50) is required.
1
exp[-2(n-1-i)sy,a]~(a)~~,i{exp(-2sy,a)
i=l
X%;Y-,{exp( -2syla)...~o,o[~(a)U0,0(a, ~>I11 +exp[ -%n-
l>sy,al~(cw)~~.o[~(a)Uo,o(cu,
x)1 .
(51)
In Fig. 4 the integration contours, concerning the last term of Eq. (5 1) are depicted. Heie, S, and S,,, are the points of intersection with the real a-axis and the real 4 o,O-axis, respectively. Analytical continuation of the decomposition operator CD&,,is required, only if
I%I ’ I~,o,oI.
(52)
If Eq. (52) is satisfied, the last term of Eq. (5 1) is rewritten as
Imb, clo.0)
Fig. 4. The integration contours concerning the integrals L, and Lw.Oafter deformation in case 2.
K.F.I. Haak, B.J. K&j/
In Appendix
B the condition
Wave Motion 23 (1996) 139-164
149
in Eq. (52) will be analysed in more detail.
7.2. The vertical particle velocityjeld
in D”’ and 0”’ (n < 2m - 1)
In order to investigate the wavefield for m < n < 2m - 1 in D (r) and D@) (due to a reflection against the interface) it is necessary to investigate
@&Lb%s> -Ro@,-.,(w s>exp[ YI(~)
-2sy,(m-
1)al
.
A similar investigation as presented in the previous section can be carried out. This results in a series of operator expressions which may need analytical continuation in case 1 and case 2, depending on the integral deformations caused by the modified Cagniard technique. Consistent application of Bq. (39) yields expressions like Eqs. (5 1) and (53).
8. Classification
of generalized rays
Substitution of Eq. (40) in the relevant expressions of the previous section results in the integral-form equivalents of the decomposition operator expressions. Application of Eqs. (46) and ( 13) yields the Laplace-transform-domain integral expressions, concerning the vertical particle velocity for the two different cases. Due to the analytical continuation of the decomposition operators in the previous section, no integral singularities with respect to the integration variable (Yand qO,Oduring the contour deformation occur. The Laplace-transform-domain vertical particle velocity in D (‘) and D(‘) for 1~ n < m is arranged according to ~exp{-s[y,(2na+x,+x:‘))+rw(n,-nl”)]}
da
(55)
and
Xexp[ -~{~~(m)[a+x$S91
-q.o,~Pll
exp -2s i yl(qOJa [ i=l
xexp[-s(y,(~)[(2n-2k-l)a+x,)+cwc,}l
dqo,,...dq,,da,
1 (56)
in which Ak=
rI(~>fi(40,0> P1(90,0-40.1)...(qo,k---qqo,k)(qO,k-~)
.
(57)
So far, the generalized ray path consists of the modified Cagniard interpretation of the successive exponential functions in Eqs. (55) and (56). Each geometrical and diffraction contribution in Eqs. (55) and (56), respectively,
150
K.F.I. Haak, B.J. Kooij/
consists of four generalized is in case 1 given by -(n)
-
u3
4n,n-
- V3;d
Wave Motion 23 (1996) 139-164
ray paths, viz. kxj and +x2). In this way, the wavefield in D(l) and D(*) for 1
x,
1) -f
xp
(58)
< 0
and in case 2 is given by (59) In case 2 also the geometrical fS)
wavefield contributes
if the condition
(cf. Appendix B)
a+x$“’
I I EL<
(2n-
Xl
(60)
l)afx,
is satisfied. The above analysis holds for n > 2. For n = 0, 1 some diffraction phenomena do not contribute yet, while the condition for the geometrical field to exist differs from Eq. (60). Now the contribution due to one reflection against the interface is investigated. Substitution of Eq. (40) in the relevant operator expressions results in the integral-form equivalents of the decomposition operator expressions. Application of Eqs. (47) and ( 13) yields the Laplace-transform-domainintegral expressions, concerning the vertical particle velocity for the two different cases. Due to the analytical continuation of the decomposition operators in the previous section, no integral singularities with respect to the integration variable cr and qo,o during the contour deformation are encountered. The integral expressions are arranged in a geometrical contribution and a diffraction contribution, viz. corresponding to generalized rays which do not and which do diffract at the edge, respectively. The diffracted contribution is decomposed as rays in which reflection against the interface before (2 types), during ( 1 type) and after (2 types) the repeated diffraction phenomenon occurs. Again, each geometrical and diffraction contribution consists of four generalized ray paths, viz. +x, and +x, (‘). We now infer that in case 2 also the geometrical wavefield contributes if the condition (2m-
l)a+x$“’
(2n-2m+l)afX3
XiS) < Iy
(2m+
l)a+xP’
I < (2n-2m-l)afx,
(61)
is satisfied. This restriction is easily found with the aid of classical ray theory. The above analysis holds for n > m + 2. For n = m, m + 1 some diffraction types do not contribute yet, while the condition for the geometrical field to exist, differs from Eq. (61).
9. Space-time-domain
acoustic wave field
In order to apply the standard Cagniard technique with respect to each integration variable (see De Hoop [ 141 and Kooij [ 121) , the integrals with more than three integration variables are in Appendix C rewritten as principle value integrals. The integrand contains the s-dependence only through the exponential function exp( -ST,) in which 7,=D,,Yr(u)
and u stands for each of the integration which is given by
Here,
(62)
+ux, variables
(Y,qo,o, . . ., qo,m.IQ. (62) represents
a branch of a hyperbola,
K. F.I. Haak, B.J. Kooij / Wave Motion 23 (1996) 139-164
R,= (Xz+Dz)1’2,
.L=x,
Re(R,,) >O,
(65)
xp
f
(66)
)
Uf= [%,I,
..*3
40,ml
and D, is some vertical propagation T,, =
(64)
,
x,,, = x,=0
151
(67) distance. Further,
5
(68)
Cl
From Eqs. (63)) (65)-( 67) it is clear, that only the contours L, and L,,, deform, away from the imaginary Due to the occurrence of the vertical slowness yi in the integrand, branch cuts along IRe(u
iT
Im(u)
=0
(69)
in the complex u-plane are introduced. introduced along IRe(u
ij
Im(u)
axis.
If the reflection factor Ro( u) occurs in the integrand,
also branch cuts are
=O.
(70)
We assume, that the position of the source and the receiver are chosen such, that the hyperbolas do not intersect the branch cut. Hence, no head waves occur. In this way the principle value integrals can be evaluated as
Xex~?( --ST,) exp(-S~O,O) fi ew[-2~pw(qo,p)~ldqo,o.-.dqo,,da p=l
=
(2)
T, To,,,To,,- I
To,.2 To.1To,o
(71) Here, B nm+l(~,
%,o, --.r qo,nr) =~~(‘dBm,m(~,
qo,o, . . . . qo,m) +~(~*)Bm,m(a*,
qo,o, a.., qo,m)
(72)
=~~(40,P,,,t-~t~,
40,0,
. . . . 40,b
. . . . qO,m)+‘Y~tq~t)Bm,t--l((Y,
40,0,
. . ..
d,b . . .. q0,A , l
(73)
152
K.F.I. Haak, B.J. K&j/
Wave Motion 23 (1996) 139-164
and k&0( a> 40,OY. . . >a,,,> = YI(cIO,OM~(~~ qo,ov..-, c~o,m) + ~~(d&J&z(~~ do, . ..v qo,nJ . In Eq. (72) Schwarz’s reflection principle
(Titchmarsh
[ 211) has been used. Further, &,,+
B,,,, + I ( 7m ~o,o,. . . > ~o,m)= &,m + I ( (~7qo,o, . . ., %,A 9
(74) 1 is given by (75)
where the integration variables [r,, ro,o, . . ., rO,J are related to the variables [a, qo,o, . . ., qo,,,] via Eq. (62). According to Lerch’s theorem, application of the shift-rule of the Laplace transform to Eq. (71) leads to
(76) in which U(t) denotes the Heaviside step function. In Eq. (76) we have used the differention rule of the Laplace transform. Further, the region of integration D is on account of the causal source function q(t) given by
.r,>T, D = ~a, ro,o,. . .,
7o.m I
T~,~~T~,~, ZE [O, . . .. ml ~7LY+C;1=070,16t. 1
It can be verified that the region of integration
ro,oE [To,o,. D=
~-a,ro,o,. . .,
70,nzI
D can be characterized
(77)
by
t-CK=~ro,k-r,l 70,~ E [ToJ, . . . . t-C~=t+lro,k-r,-C~~;=bTO,k] ZfZ [ 1, . . . . m-l] ..I
ro,nzE [To,,, . . . . t-r,CE,‘T,,k 7, E [T,, . . . . t-C~zoTO,k] .
,
1
(78)
Concerning the arrival time tm = C;LoTo,l + T, the approximation tarr= (2nalcr) holds in D(l) and D(*) if n is large enough and the offset is small enough. A similar reasoning applies for the transient wavefield in Dc3). Consequently, the space-time-domain expressions are exact within some finite time window. The upper limit of this window is determined by the minimal arrival time of the (n + 1) st wave constituent.
10. Numerical results In this section we pay our attention to the numerical implementation. In the geometrical configuration (cf. Fig. 1) the distance between the half-plane and the free surface equals a = 1.O m, while the thickness of the fluid layer is given by d=4.0 m. Further, the volume source density of force is located at (xl’), xi’)) = ( - 1.0, 0.7) m. The
K.F.I. Hmk. B.J. Kooij/ Table 1 The acoustic properties
Wave Motion 23 (1996) 139-164
153
of the configuration p, = 2.4 x p2= 1.0x K,=5.0x tc2=4.4x c1 =2.9x c,= 1.5x
lo3 103 IO-” lo-‘0 103 103
kg/m3 kg/m3 Pa-’
Pa-’ m/s m/s
dynamic acoustic properties of the media at hand are chosen as indicated in Table 1, thus simulating the practical situation of a concrete wall which isolates a nuclear reactor. Since c, > c2 it is clear that no head wave contributions occur (see Appendix B) . We have numerically evaluated the Green function concerning the particle velocity field of some generalized ray path contributions. These Green’s functions are to be convolved with the second derivative with respect to time of the source signaturef( t) (cf. Eq. (3) ) , which is in our case a four-point optimum Blackman pulse of unit amplitude, given by
s(t) =Q(t)
=
-~~(~~cos(~
r)
0
when
--co
when
O
when
-tp
(79)
The Blackman pulse approximates the classical Ricker wavelet, which is often employed in seismics (De Hoop and Van der Hijden [22] ), quite well. Here, the constants b, and the pulse duration tp are given by bo=0.35869, bl = -0.48829, b,=0.14128, b3= -0.01168 and tp= 80 ps. We have evaluated the geometric, first order and second order diffracted fields in various points of observation. In Fig. 5 a typical Green function pertaining to a first order diffracted contribution is shown. In Figs. 6,7 and 8 the convolved versions of a geometrical wavefield and the corresponding first and second order diffracted wavefields (n = 2) are depicted, all at the receiver location (xi, x3) = (2.0, 0.5) m. The contributions in Figs. 9-11 are due to one reflection against the interface (n = 12). Further, the convolved first order and second order diffracted wavefield are shown in Figs. 12 and 13 for the observation points (xi, x3) = (0.5, 0.5) m. The geometrical field in this case (n = 2) does not exist. We have also
-5. 1.75
1.80
1.85
1.90
t Fig. 5. The Green function pertaining
[ml.
1.95 x 10-3
to a first order diffracted field (vertical ray path 4a -x3 +.$’
at the observation
point (xl, x3) = (2.0,0.5)
154
K.F.I. Hauk, B.J. Kooij/ Wave Motion 23 (1996) 139-164
geometrical
contribution
(n=Z)
t Fig. 6. The convolved
contribution
pertaining
to a geometrical
x 10-3
contribution
(vertical
ray path 4a-x3+$))
first order diffraction
1.75
1.80
1.85
t 1.95
1.90
at the observation
point
at the observation
point
i 2.00
t
Fig. 7. The convolved
contribution
pertaining
to a first order diffracted
x 10-3
field (vertical
ray path 4a-x3
+xp’)
[ml.
second order diffraction
2.10
2.15
2.20
2.25
(n=2)
2.30
tie Fig, 8. The convolved Cxlr x3) = (2.0,0.5)
point
(x1=2)
-1000
(Xl, x3) = (2.CO.5)
at the observation
[ml.
(x1, $1 = (2JIO.5)
contribution
[ml.
pertaining
to a second order diffracted
2.35
2.40
x1r3 field (vertical
ray path 4a-x,+x:“‘)
K.F.I. Hnak, B.J. Kooij I Wave Motion 23 (1996) 139-164
geometrical
contribution
(n=lZ)
/
-2.0
5.45
,_
8.50
8.55
8.60
t Fig. 9, The convolved (xl, x3) = (2.2,0.5)
contribution
pertaining
to a geometrical
155
8.65 XPO-3
contribution
(vertical
ray path 24a -x3 -t xp))
at the observation
point
at the observation
point
at the observation
point
[ml.
first order
diffraction
(x=12)
i 8.45
8.50
8.55
8.60
t
Fig. 10. The convolved contribution (x,, .x,1 = (2.2,OS) [ml.
pertaining
to a first order diffracted
_60 i
i
field (vertical
second order di&action
(n=lZ)
i;
i
i
8.65 x 10-3
?
ray path 24a-x3
+xy’)
; I
8.425 8.450 8.475 8.500 8.525 8.550 8.575 8.600 X10 -3
time
Fig.
11.The convolved contribution
(n,, -Q) = (2.2,0.5)
[ml.
pertaining
to a second order diffracted
field (vertical
ray path 24~ -x3 +xp’)
K. F.I. Haak, B.J. Kooij / Wave Motion 23 (1996) 139-164
156
Table 2 The maximum amplitudes
of the convolved
contributions
evaluated at different points of observation
Vertical generalized ray path geom 1st diff 2nd diff geom I st diff 2nd diff
4a-x,+x!“’
4a-x,
Maximum absolute amplitude ug [m/s]
x3 [ml 2.0
0.5
2.88 x 1O-3 (100%) 8.17x lo+ (28%) 1.60x 1O-6 (0.06%)
0.5
0.5
-
+xp)
8.12x 2.80x
1O-4 (100%) 1O-5 (3%)
geom 1st diff 2nd diff
24a-x,+x!“’
2.2
0.5
1.28x 1O-3 (100%) 6.24x 1O-4 (49%) 5.00x 1o-5 (4%)
geom 1st diff 2nd diff
x3-xp
2.6
2.0
8.78X 1o-4 (100%) 2.78 X lo+ (32%) 1.60x lo+ (0.2%)
2.0
2.0
-
geom I st diff 2nd diff
2a+x,-x:S’ -
(rel)
1.25 x lo-=’ (100%) 5.23x lo+ (4%)
xg +x:s’ 2a +x, +x:S)
computed some contributions for receiver locations below the half-plane. The results are depicted in Figs. 14-18. In all figures the corresponding generalized ray path is shown as well. In Table 2 the maximum values of the convolved contributions are indicated, together with the vertical generalized ray path. Obviously, the second order diffracted wavefield turns out to be only a fraction of the geometrical contribution. Physically, this is caused by the scattering nature of the diffracting edge. Hence, the wavefield due to a multiple diffraction phenomenon is attenuated considerably. Consequently, for practical purposes, the second and higher diffracted wave fields need not be computed. This will decrease the computation time of the total field within a finite window considerably. One can argue on the same physical grounds that the second order diffracted field compared with the first order diffracted field will decrease when the receiver location moves away from the edge, or more precisely, when the angle between the half-plane and the last part of the ray path decreases. This expectation is confirmed by the numerical experiments (see Table 2).
first order diffraction
(n=Z)
~
.._ _
-------------
.
-1000/ 1.55
1.60
1.65
1.70 t
Fig. 12. The convolved (x1, ns) = (C).5,0.5)
[ml.
contribution
pertaining
to a first order diffracted
1.75 x10-3
field (vertical
ray path 4u--~a+#‘)
at the observation
point
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
157
second order diffraction (a%?)
_30
1.600 1.625
1.650
1.675
1.700
1.725
1.750
time
Fig. 13. The convolved (x,. xx) = (0.5,0.5)
contribution
pertaining
to a second order diffracted
1.775 x 10-3
field (vertical
ray path 4a-xx
+np’)
at the observation
point
[ml.
geometrical contribution (n=O)
1.15
1.20
1.25
1.30
t Fig. 14. The convolved (2.0,2.0) [ml.
contribution
pertaining
to a geometrical
1.35 x 10-3
contribution
(vertical ray path x,+x 9’) at the observation
point (x,, x3) =
first order diffraction (n=O)
1.15
1.25
1.20 t
Fig. 15. The convolved (2.0,2.0) [ml.
contribution
pertaining
to a first order diffracted
1.30
1.35 x10-3
field (vertical ray path x,--x p’) at the observation
point (x1, x3) =
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
158
second order di%action
(n=Z)
time
Fig. 16. The convolved (x,, x3) = (2.0.2.0)
contribution
pertaining
to a second order diffracted
x10-3
field (vertical
di%xtion
Fig. 17. The convolved (2.0,2.0) [ml.
contribution
pertaining
to a first order diffracted
2.15
field (vertical ray path x,+x :“‘) at the observation
2.20
2.25
2.30
convolved [ml.
contribution
pertaining
to a second order diffracted
point (x,, x3) =
(n=Z)
2.35
2.40
time
The
point
x 10-3
second order diffraction
18.
at the observation
(~2)
t
(x,, x3) = (2.0,2.0)
-xp’)
lml.
firstorder
fig.
ray path 2n+x,
2.45 x 10-3
field (vertical
ray path 2a+x,
+n:“‘)
at the observation
point
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-144
159
11. Conclusion The strategy of solving the original Wiener-Hopf equation iteratively enables us to solve the scattering problem exactly within any finite time window. After classifying the different wave constituents closed-form space-timedomain expressions for the pressure field are obtained by means of a modified version of the Cagniard-De Hoop technique. Due to the complexity and to the methodological character of this study we have confined ourselves to the wavefield due to at most one reflection with the interface. For practical purposes this turns out to be sufficient in most cases. The results of the numerical implementation are interesting. The amplitude of the second order diffraction wavefield turns out to be only a fraction of the geometrical contribution. Physically, this is caused by the scattering nature of the diffracting edge of the pressure-free half-plane. Mathematically, this can be explained by the observation that the diffraction operators have a norm less than one. It seems to be a difficult task to estimate the norm of the contraction operators more precisely in order to predict the maximum amplitudes of higher order diffracted wave components. But the numerical results show that, for practical purposes, in case of deep embedded semi-infinite screens only the first order diffracted wavefields have to be computed, thus avoiding expensive, time-consuming evaluations of multiple integrals.
Appendix A Standard solution procedure The elementary
of the elementary
Wiener-Hopf
Wiener-Hopf
equation
equation
(A-1) is solved by factorizing n(a)
=fi(a)S(a)
the vertical slowness yi ( (Y) as (A.21
9
in which #(a) and $( (Y) are analytical functions in D”, and D”,, respectively as 1a 1-+ CCin their regions of regularity. By the additive decomposition
Qiwz(a, $1 =T%W~,,=Q~,<~~
s> +Q%w
in which QE( LY,s) and Qkn( (Y,S) are analytical Eq. (A.2) changes into -
Y&j
@G(CG s)
-Q&SW
(see Eq. (38) ), and behave as o( 1)
~1 , functions
s> = - Y%W&(W
(A.31 in D”, and in D”, respectively
s> +Q:&,
~1 .
(see Eqs. (39)-(
42) ) ,
(A.41
The left hand side of Eq. (A.4) is an analytical function in D”, with asymptotic behaviour o( 1) as 1aI -+ ~0in Dk, while the right hand side is an analytical function in DE with asymptotic behaviour o( &‘*) as 1a I --f 00 in DE. Consequently, they form an entire function which according to Liouville’s theorem (Tit&marsh [ 211) vanishes. As a result we have
(A.3
160
K.F.I. Haak, B.J. Kooij/
Wave Motion 23 (1996) 139-164
Appendix B Conditions for intersection
of the modijied Cagniard contours with the real axis
In this appendix the conditions to obtain a geometrical field contribution are investigated. Starting with the integral form expressions, the points of intersection of the modified Cagniard contours on the one hand and the real q,,,-axis or a-axis, respectively, on the other hand are determined. In the last section a condition for the presence of headwaves is derived. B.1. General geometrical
condition for the geometrical$eld
We consider the integral form expression
u^,tx,, x39 s> =-
A,( a, qo,o)
-qo,~~s’l~ exp{-s[yl(~)D,+~ll~ X expt - s[ 35 ( 40,0) Dqo,o
dq,,,da,
(B.1)
in which Ao=
~WYwlo,o) Pl(40.0 - a> .
U3.2)
The modified Cagniard method now aims at parametrizing
T,=Y,t~m,+~x,
U3.3)
3
in which u E [a, qo,o] and rU is real and positive. Eq. (B.3) represents a branch of a hyperbola, XL,
U f-- ~2
1,
rL,fi $$ ($-Tz)“*, I,
ru a T, .
which is given by (B-4)
Here, R,= (X~+D~)“2,
Re(R,)
>O,
(B.5)
-G=x, >
v3.6)
x,,, = - xy
(B.7)
and D,, is some vertical propagation T,, =
3. Cl
distance. Further,
(B.8)
Due to the occurrence of a singularity in the integrand of Eq. (B.l) it is necessary to investigate the points of these points are given intersection S, and S,,, with the real (Yand qo,o-axis, respectively. From Eqs. (B.4)-(B.8) by
K.F.I. Hank, B.J. Kooij/ Wave Motion 23 (1996) 139-164
XL‘
161
1
&I=(x;+D;)l’*
(B.9)
;’
It can be verified that the condition (B.lO)
I&I 5 I%0,0I is satisfied if
WI
5 1x1I 40.0 D, D
(B.ll)
and vice versa. Now we are in a position to transpose the conditions concerning real axis in conditions concerning the location of the source and the receiver. B.2. The vertical partical velocity$eld In case 2 the vertical propagation D,=
(2n-
l)a+x,
the points of intersection
with the
in D”’ and 0”’ (n < m) distances D, and D,,,, are given by
,
(B.12) (B.13)
DqO,O=a+~~S) . Consequently,
the condition
(cf. Eq. (52) )
ILI > I~,o,oI
(B.14)
transposes in
IXP I < Ix1I
LZ+J$’
B.3. General geometrical
(B.15)
’
(2n-l)a+x,
condition for a headwaue contribution of the geometrical field
Let us consider the integral form expression
fi33(%x3 s) =
W-h
2(2ni)
~2):
of a geometrical
field
exp[-s(Y~D~+LYX,)I da,
(B.16)
Lt in which the integration path L, coincides with the imaginary a-axis. Using the modified Cagniard method the integration contour L, deforms into the complex a-plane. The presence of the double valued functions 3/, and x in the integrand causes branch cuts along Br,=
]Re(a)]
> $,
(B.17)
k=1,2.
When the integration contour intersects a branch cut a head wave contribution Hijden [ 221) _In Section B. 1 we found the intersection point to be
arises (see De Hoop and Van der
(B.18) from which we conclude that L, never intersects the branch cut Br,. The branch cut Br, is intersected
when
162
K.F.I. Haak, B.J. Kooij/
s,>
Wave Motion 23 (1996) 139-164
1.
(B.19)
c2
Hence, a head-wave
contribution
of the geometrical
field only occurs when c2 > ci and (B.20)
Appendix C The conversion
into principle
value integrals
In this appendix Eq. ( 56) is evaluated in terms of principal value integrals. This can be accomplished by application of Cauchy’s theorem to the original integration contour, resulting in a principle value integral and half a residue contribution. However, a lucid investigation in this way is no sinecure. Therefore we rewrite the integral-form expression in the equivalent operator form. Then, the relevant decomposition operators are evaluated in terms of identity and Hilbert operators. We start with the operator-form equivalent of one term on the right hand side of Eq. (56)) viz. Int = qa[B,( a) 1, in which ?Pa= -g
I ev( -SKI>
exp( -wa)
[exp( -wxd
L,
+exp(sy,x,)
Y?(a)
I -B,(a) Pl
da
cc.11
and
cc.21 Here, the argument z is given by z=exp(
-2sy,a)~~,~[~(c~)Uo,o(cy,
s)l .
CC.31
If k E [ 0, 11, no integration singularities arise. The following investigation holds for k > 2. In order to keep the analysis lucid, we omit the integration variables qo,k- i, . . ., qo,l and the argument z in the notation. Next, we apply Eq. (42) to the multiple operator, resulting in k-l [exp( -2~y,a)diL,~]~--l= in which we have introduced
0 $
[J+Plk-‘,
(C.4)
the operators J and P as
J= exp( - 2sy,a)l,
cc.3
P=exp(
(C.6)
-2sy,a)iH.
Here, I and H denote the identity and the Hilbert operator, respectively. The norm of the multiple operator in Eq. (C.4) is less than one (contraction operator), due to the presence of the exponential function and the fact that the Laplace parameter s is chosen real and positive. Hence the repeated diffraction operator is a contraction operator and the scheme is convergent. It is noted that Eq. (C.4) does not represent a binomial series, since the operators J and P do not commute. Equation (C.4) is now evaluated as
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
163
k-l
0
[exp( - 2syra) @vH] k-1 = i
Y-l+
-1-
cc.71 Here, the summation m
E [2, . . . . k] E [l, . . . . k+l-m] E [l, . . . . k-m+p-CfZjkJ =k-C;:;‘k,.
4 kP k i nr Consequently, B,(a)
variables range according to
Eq. (C.2) is now evaluated as
= &
nz,k,?km-,
$>I I
~~.,{exp(-2ksyla)~~,,[~(a)U,,,(cy,
@&,,,,{exp( c2ik-’
-2Lw~WL
+
(C-8)
PE [2, . . . . m-l]
,
l
X~ex~(-2k,,-~~~~~)...~~Iex~(-2k~~~~~)~~,~~~(~)~~,~(~,~)llll. Finally, substitution of the integral-form to the integral form
*sf Int=(2)k(27ri)3 .,,k,,?k,..-,
+
X fi
&IY~,
equivalent
(C.9)
of the operators and application
%,o) exp( - 2ksyl (%,r)a)
of Eqs. (C. 1) and (C.2) leads
dq,.,dq,,,da
L Lqa.1ko.0
1j f -“iri)z+“’
(2)k+1
exp[ -2k pw (40,&l
.a.
L,
LqO,m
L~~,~_,
$o,o.. .dqo,,da
f
j
Am.f(a~40,ol
L‘l0.I Lga,o
(C.10)
9
p=l in which the summation f(a,
%,o) =exp]
variables range according to Eq. (C.8) andA f~, qo,o) is given by -~[n(~o,o)(~+x~))
-~o,dc~“‘l
-s[n(a!)(a*~~)
+a~~]}
.
(C.11)
References [ 11 B.J. Kooij and D. Quak, “Three-dimensional scattering of impulsive acoustic waves by a semi-infinite crack in the plane interface of a half-space and a layer”, J. Math. Phys. 29, 1712-1721 (1988). [2] K. Aki and P.G. Richards, Quantifatiue Seismology, Theory and Methods, W.H. Freeman, San Fransisco ( 1980). [3] A.T. De Hoop, Handbook ofradiation and scattering ofwaues, Section 10.13, Academic Press, London, New York (1995) to appear. [4] A. Sommerfeld, “Mathematische Theorie der Diffraction”, Marh. Ann. 47,317-374 (1896). [5] E.T. Copson, “An integral equation method of solving plane diffraction problems”, Proc. R. Sot. (London) A 186,100-l 18 (1946). 161 B. Noble, Mefhods based on the Wiener-Hopf technique, Pergamon, New York (1958). [ 71 P.C. Clemmow, “Radio propagation over a flat earth across a boundary separating two different media”, Phil. Trans. R. Sot. (London) A 246, l-55 ( 1953). Also in: M. Born and E. Wolf, Principles of Optics, Pergamon, New York (1959).
164
K.F.I. Haak, B.J. Kooij/ Wave Motion 23 (1996) 139-164
[ 81 R. Du Cloux, “Pulsed electromagnetic radiation from a line source in the presence of a semi-infinite screen in the plane interface of two different media”, Waoe Motion 6,459-476 (1984). [9] I.D. Abrahams, “Scattering of sound by two parallel semi-infinite screens”, Waue Motion 9,289-300 (1987). [IO] W.G. Heitman and P.M. van den Berg, “Diffraction of electromagnetic waves by a semi-infinite screen in a layered medium”, Can. J. Ph.?% 53,1305-1317 (1975). [ 111 H. Shirai and L.B. Felsen, “Spectral method for multiple edge diffraction by a flat strip”, Wave Motion 8,499-524 (1986). [ 121 B.J. Kooij, “Scattering of point-source generated transient elastodynamic waves by a semi-infinite void crack in a half-space”, in: Elastic waue propagation, Eds. M.F. McCarthy and M.A. Hayes, Elsevier, Amsterdam (1989), pp. 217-222. [ 131 L. Cagniard, R@ections et R&fractions des Ondes Seismiques Progressives, Gauthiers-Villars, Paris (1939, in French). Translated and revised version: Reflection and Refraction of Progressive Seismic Waves, Eds. E.A. Flinn and C.H. Dix, McGraw-Hill, New York ( 1962). [ 141 A.T. De Hoop, “A modification of Cagniard’s method for solving seismic pulse problems”, Appl. Sci. Res. B 8,349-356 ( 1960). [ 151 A.T. De Hoop, “Theoretical determination of the surface motion of a uniform elastic half-space produced by a dilatational, impulsive, point source”, Proc. Colloque International du CNRS (Marseille) Ill, 21-32 (1961). [ 161 J.D. Achenbach, Wave Propagation in Elastic Solids, North Holland, Amsterdam (1973). [ 171 J. Miklowitz, The Theory of Elastic Waves and Waveguides, North Holland, Amsterdam (1978). [ 181 D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ (1946). [ 191 N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff, Groningen (1953). [20] H. Hochstadt, Integral Equations, Pure &Applied Mathematics, Wiley, New York ( 1973). [ 211 E.C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford (1932). “Generation of Acoustic Waves by an Impulsive Line Source in a Fluid/Solid Configuration [22] A.T.DeHoopandJ.H.M.T.vanderHijden, with a Plane Boundary”, J. Acoustic Sot. of America 74,333-342 ( 1983).