Reflection and transmission of elastic waves by the spatially periodic interface between two solids (Numerical results for the sinusoidal interface)

WAVE MOTION 3 (1981) 3348 @ NORTH-HOLLAND PUBLISHING

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REFLECTION AND TRANSMISSION OF ELASTIC WAVES BY THE SPATIALLY PERIODIC INTERFACE BETWEEN TWO SOLIDS (NUMERICAL RESULTS FOR THE SINUSOIDAL INTERFACE) J.T. FOKKEMA* Delft University of Technology, Delft, The Netherlands

Received 21 April 1980

The integral-equation method for calculating the reflection and transmission of elastic waves by the spatially periodic interface between two solids, developed in a previous paper, is applied to a sinusoidal interface, and numerical results are presented. The computations have been carried out for four different heights of the profile (the plane interface included), a single frequency of operation, two combinations of elastic solids, and the four types of excitation. We have considered the interface between granite and slate, the interface between copper and flint glass, and P- as well as SV-wave incidence in either of the media. A peaked behaviour of the reflection and transmission factor occurs at angles of incidence where an elastodynamic spectral mode changes from propagating to evanescent and vice versa. An additional anomaly occurs in cases where the horizontal wave number of one of the spectral orders coincides with the horizontal wave number of a Stoneley wave along the corresponding plane interface. The latter phenomenon is the more pronounced, the shallower the corrugation of the interface is.

1. Introduction

glass. A fundamental difference between the two cases are the surface waves of the Stoneley type, that are absent in the first case, but do occur in the second. As incident fields, from both sides of the interface, P- and SV-waves are considered separately.

In this paper we present numerical results pertaining to the reflection and transmission of elastic waves by the spatially periodic boundary between two elastic solids. The sinusoidal interface is taken as an example. The analysis is carried out with the boundary integral-equation method. For the relevant theory and the notations employed we refer to our previous paper [l]. The computations have been carried out for four different heights of the periodic profile (the plane interface included), a single frequency of operation and two combinations of elastic solids. The first combination arises from a seismological application and considers the interface between two kinds of rocks, viz. granite and slate. The second has its application in materials science and considers the interface between copper and flint

2. The sinusoidal interface and the introduction of the acoustic power scattering matrix of the propagating spectral orders method The boundary integral-equation developed in our previous paper (11 applies to the elastodynamic reflection and transmission by a periodic interface of arbitrary profile. In the present paper we shall report about computations that have been performed for the sinusoidal interface x2 = (h/2) sin(27rxr/9),

* Now at Network Laboratory, Department of Electrical Engineering, Netherlands.

Delft University

of Technology,

(2.1)

in which .h is the distance from its top to its valley (Fig. 1).

Delft, The

33

34

J.T. Fokkema / Reflection by periodic interface of two solids

s”-wave =f_q: ’ P-wave

‘P

)

3

Fig. 1. Sinusoidal

interface

As to the incident field, we choose a particular value of kinC (through which, according to k, = kin’ + 2m/iB, all spectral orders are specified) and distinguish the four following cases: (i) Incident P-wave in medium I with amplitude factor AP’(k’“‘). (ii) Incident SV-wave in medium I with amplitude factor AS’(k’“‘). (iii) Incident P-wave in medium II with amplitude factor AP”(kinC). (iv) Incident SV-wave in medium II with amplitude factor AS”(k’““). Since we are mainly interested in the redistribution of incident acoustic power among the reflected and transmitted waves, we introduce the acoustic power scattering matrix of the propagating spectral orders. Let [F’], denote the column matrix of vertical intensities of the scattered field of the propagating spectral order n (cf. Table 1 of

PI>:

between

two solids.

where it E {propagating spectral orders}. Let further [fin”] denote the column matrix of the vertical intensities of the incident field: r ]AP’(ki”c)]21P’(ki”c)1

then the power scattering duced through

matrix [n],, is intro-

[I”‘]” = [n],[l’““].

(2.4)

We write [ZI], as (2.5) in which the submatrices are written as

mxr’l”= [ “n$ w*l*l” = [Egg:

n,;$, ;g;::],

(2.6) (2.7)

(2.2) (2.8)

J.T. Fokkema / RefIection by periodic interface of two solids

IIPS’,“” flSS’,‘*” I -

(2.9

The submatrices [n’*‘], and [flllvll], denote the acoustic reflectances in the vertical direction in the media I and II, respectively. The submatrix [n”*‘], denotes the acoustic transmittance in the vertical direction from medium I to medium II and the submatrix [nlvll], denotes the acoustic transmittance in the vertical direction from medium II to medium I. From the structure of [n], it is clear, that the first column of [n], yields the acoustic reflectances and transmittances originating from an incident P-wave in medium I, the second column those originating from an incident SVwave in medium I, the third column those originating from an incident P-wave in medium II and the last column those originating from an incident SV-wave in medium II. In our numerical results, we present the values of the elements of the scattering matrix of the zeroth spectral order [L$, only, although the contributions from all spectral orders, as far as they contribute significantly, have been obtained with the aid of our computer program. The reason for this is, that we mainly want to investigate the influence of the ‘roughness’ of an interface on its reflecting and transmitting behaviour and hence the results obtained are to be compared with those of a plane interface, for which only the zeroth spectral order exists. Further, some space limitations on the presentation of results should be observed and therefore we confined ourselves to presenting the results pertaining to the zeroth spectral order only. It follows from eq. (3.8) in [l] in conjunction with the symmetry of the sinusoidal profile, that [n], is a symmetrical matrix. This leads to the relations l7SP2’ = IIPS:‘, zJPP:‘=

~sp;L”

= &rp#$.“,

17PPi”,

l7SPF’ = lJP$“,

nPsg~’ = 17SP2”,

17ss:*” = nssI”.

(2.10)

Further, the power relation (4.10) of [l] can, for

35

the four cases of incidence, be written as c nPPf;‘+C nsP:‘+C LrPP’,“’ n n n + c nsp’.“’ = 1 n

(2.11)

for an incident P-wave in medium I, 1 nPs:‘+C nss:‘+C ZIPS’,‘.’ ” ” ” +CnssY=l n

(2.12)

for an incident SV-wave in medium I, c 17PPk”+C I;Isp’,.“+c 17PP’,“” ” ” R + 1 LCSP’,“”= 1 ”

(2.13)

for an incident P-wave in medium II, z z7Ps’,‘“+~17ss’,.“+~17Ps:*~” n ” n + c Lrss’,“” = 1 n

(2.14)

for an incident SV-wave in medium II, where all summations are to be carried out over the collection of the pertaining propagating orders. Although our computer program is quite general, we have, in our study, considered in more detail two out of all possible combinations of elastic materials. One combination has its origin in a seismological application, the other in a situation arising in materials science. For the seismological application we have taken the one for which the plane interface is discussed by Knott [2], by Gutenberg [3] and by Ewing, Jardetzky and Press [4, pp. 87-891. It considers the. plane interface between the two kinds of rocks: granite (medium I) and slate (medium II). Although the elastic parameters of these media vary with their geological sites (cf. [5]), we have taken those used by Gutenberg [3], who in turn has taken them from Knott [2]. They are reproduced in Table 1. This combination of materials does not satisfy the conditions for propagation of surface waves (of

J. T. Fokkema / Reflection by periodic interface of two solids

36 Table

1

Elastic

parameters

Material

granite slate

pertaining

to granite

and slate

Poisson ratio Y

P-wave velocity

SV-wave velocity

Mass density

cP (m/s)

CS(m/s)

P (kg/m31

0.28 0.17

4177 4762

2309 3003

2.70 x IO3 2.70x lo3

the Stoneley type) along their plane interface [see e.g. [6, 71, [4, p. 1121 and [8, p. 481). Our second example considers the interface between copper (medium I) and flint glass (medium II). The elastic parameters are taken from the American Institute of Physics Handbook [9] and are reproduced in Table 2. For a plane interface between these materials, the conditions for Stoneley-wave propagation are satisfied and with the aid of the formula given by Ewing, Jardetzky and Press [4, p. 112, eq. (3.139)], we calculated the, non-dispersive, Stoneley-wave velocity to be cST = 2266 m/s. As input parameters for the computer program, we have taken the normalized wave numbers kP%, kS’9, kP”$B and kS”$%, the normalized depth of the sinusoidal profile h/9 and the ratio of mass densities pn/pl. Due to limitations on storage capacity and computing time, we have chosen the frequency of operation such that in each interface problem we assigned to the smallest wave velocity the normalized wave number 9 being the maximum value that could be handled within our restrictions. Then, the other normalized wave numbers follow from the ratios of their wave velocities. It is remarked, that

parameters

Material

copper flint glass

Table 3 Normalized wave pertaining to the computer program

numbers and normalized wave granite/slate interface, as used

lengths in the

Material

kPg

kSL@

API9

AS/9

granite (medium I) slate (medium II)

4.97 4.36

9 6.92

1.26 1.44

0.70 0.91

Table 4

Table 2 Elastic

this restriction is not a theoretical but a practical one. The normalized wave numbers kP9 and kS9 are related to their corresponding normalized wave lengths through APIg = 2?r/kPg and AS/G@= 2?r/kSG@. For the granite/slate and the copper/flint glass interfaces these quantities are listed in Tables 3 and 4, respectively. In the case of the copper/flint glass interface we obtain for the normalized Stoneley-wave number and the corresponding normalized wave length the values kSTg =9.01 and AS=/9 =0.69, respectively. The computations have been performed for the values h/9 =O, 0.1, 0.3 and 0.5. Again it is remarked, that the upper limit of this normalized depth sequence is not dictated by the theory but by practical limitations. The results for the plane interface (h/9 = 0) have been obtained by directly applying the boundary conditions pertaining to the plane interface to the spectral combinations of order zero. The input parameters for the computation of the results pertaining to the plane interface are cP”/cP’, cS”/cS’, Y*or VI*,and pn/p’, and, hence, they do not depend on the frequency of operation.

pertaining

to copper

and flint glass

Normalized wave numbers pertaining to the copper/flint computer program

and normalized glass interface,

wave lengths as used in the

Poisson ratio Y

P-wave velocity

SV-wave velocity

Mass density

CP (m/s)

cS (m/s)

P (kg/m3)

Material

kPLb

kS9

hP/ka

AS/9

0.37 0.22

5010 3980

2270 2380

8.90 x lo3 3.89 x lo3

copper (medium I) flint glass (medium II)

4.08 5.13

9 8.58

1.54 1.22

0.70 0.73

J.T. Fokkema / Ref7ection by periodic interface of two solids

Finally, we let the pictorial presentation of the results be accompanied by a diagram indicating the range of propagation of the different spectral orders, expressed in terms of angles of incidence and emergence. For the copper/flint glass interface ‘horizontal phase matching’ occurs. With this, we mean that either for some positive spectral order the relation k,9 = kST9 (m > 0) or for some negative spectral order the relation k,9 = -kST9 (m O:

k,9

= kST9 if

cST/cP G s/(mhP) m-CO:

k&3 = -kST9

< cST/(cP- cST), (2.15)

if

cST/(cPfcST)~~/(jm]AP)~cST/cP, - for an incident SV-wave m>O:

k,9

= kST9 if

cST/cS~9/(mhS)~ccST/(cS-cST), mC0:

k m9 = -kST9

(2.16) if

From (2.15) and (2.16) it follows that the boundaries of the inequalities are independent of the frequency of operation; however, they do depend on the material properties of the two media and on the type of incident wave. In the case of the copper/flint glass interface, it is easy to verify that only k19= kST9 and k&? = -kST9 can occur for the chosen frequency of operation. The angles of incidence for which horizontal phase matching then appears, are indicated in the diagrams. 3. Numerical results pertaining to the sinusoidal granite/slate and copper/flint glass interfaces The results pertaining to the granite/slate interface are presented in the following figures:

37

Figs. 2 and 3 for an incident P-wave in granite, Figs. 4 and 5 for an incident SV-wave in granite, Figs. 6 and 7 for an incident P-wave in slate, Figs. 8 and 9 for an incident SV-wave in slate. They show, for h/9 = 0,0.3 and 0.5, the values of the elements of [ZI], as a function of the angle of incidence (varying between 0” and 90’). For h/9 = 0.1 the results nearly coincide with those for h/9 = 0; for this reason we have not included the relevant curves in the figures. The results pertaining to the copper/flint glass interface are presented in the following figures: Figs. 10 and 11 for an incident P-wave in copper, Figs. 12 and 13 for an incident SV-wave in copper, Figs. 14 and 15 for an incident P-wave in flint glass, Figs. 16 and 17 for an incident SV-wave in flint glass. They show, for h/La = 0, 0.1, 0.3 and 0.5, the values of the elements of [J7], as a function of the angle of incidence. For the different cases, a few prominent properties will be referred to below. 3.1. Granite/slate interface From Figs. 2 and 3 that present the results for an incident P-wave in granite, we observe that from normal incidence up to the critical angle 19Pi = 61.3”, the transmitted P-wave of spectral order zero is the most important one and the influence of the roughness of the interface is negligible. For larger values of the angle of incidence, the transmitted P-wave of spectral order zero becomes evanescent and from the critical angle onward up to grazing incidence, the reflected P-wave of spectral order zero takes over the main part of the incident energy. This part increases with increasing depths of the profile. The conversion energies ZISPir and L?3Pi**’ remain small and have their extreme value in the vicinity of the critical angle.

38

J.T. Fokkema / Reflection by periodic interface of two solids

I

.

I

.

I

.

1

1

.

I

.

1

.

I

.

--__ Or.

z

s------3

G---a

---_ - ------_ “0

l-

Or- 0 f O4

r:

4

-2

0

-2

_,

-I

i

8.4’

I I 1 ( I

I

,

03).6°

i,-9001

’,

’

123.3 .,

11 17,.6?

I

I I I

i

I I

57.50

-23.3O

17.6’

9o”

eSf2

4,

es;

reflected

reflected

P-wave

SVrave

SV-wave

P-wave

transmitted

incident

SV-wave

transmitted

Fig. 4. Range of propagation of the different spectral orders for the reflected and transmitted waves and their corresponding angles of emergence as a function of 6: for an SV-wave incident in granite on a granite/slate interface.

!

I I

-4413O

o 00 f

0’

order

spectral

02 -

0.‘ -

0.6 -

= 6.92 =1

gyp’

fsv

1

II

= L.36

kS=D

kS’D kP%

~4.97 =9

kP’D

e$ -

600

.

900

J

9o"

:

0

_

0.0‘ -

0.08 -

0.17 -

TTPSH'

0.16

0.2

1

300

Fig. 5. Acoustic reflectances and transmittances, in the vertical direction, of the zero spectral orders as a function of t$ for a sinusoidal interface between granite (medium I) and slate (medium II) for an W-wave incident in granite.

sv

.

es;-

600

1

-h/D=0

I 60'

.

. 900

-1

do

I I

;-900

r I i I 26.1'

-22.7' ePI,

SV-wave

P-wave

transmitted

transmitted

P-wave

P-wave

reflected

incident

SV-wave

reflected

Fig. 6. Range of propagation of the different spectral orders for the reflected and transmitted waves and their corresponding angles of emergence as a function of @Pi’ for a P-wave incident in slate on a granite/slate interface.

order

specrra1 -

=

= 6.92

kS=D

O.OL

TrsP,g’

t

0 05

K

II

300

SV

I!__

P

Kg. 7. Acoustic reflectances and transmittances, in the vertical direction, of the zero spectral orders as a function of BP; for a sinusoidal interface between granite (medium I) and slate (medium II) for a P-wave incident in slate.

= 4.36

kP=D

le.97 kS'Dr9

k P’D

60’

n

O0

10.9’

16.1’ t

i.3’

I 23.1’

I

’

, 1

54.6’

39.1’

\ 45.9O

I

1

es;

,.40:zl

-3a.a06sf2

9o”

transmitted

transmitted

P-wave

SV-uave

SV-wave

P-wave

reflected

incident

W-wave

reflected

Fig. 8. Range of propagation of the different spectral orders for the reflected and transmitted waves and their corresponding angles of emergence as a function of OS~ for an W-wave incident in slate on a granite/slate interface.

order

spectra1

60°

t

ITPsi,=

0 16

021

01.,1.,

0.2 -

iiII I

sv

300

P

Fig. 9. Acoustic reflectances and transmittances, in the vertical direction, of the zero spectral orders as a function of OS~ for a sinusoidal interface between granite (medium I) and slate (medium II) for an W-wave incident in slate.

es,” w

0.06

900

0.4 -

-

-

0.03 -

0.12

300

es,”

600

t

0.6 -

0

0.2 -

0.L -

ilP!p

O.OL

0.05

60’

sv

es,”

60’ -

ye n; I

,

900

c

O0

-2

I I I

I

1 .f 190 ( I I I es:

9o"

for the reflected and transmitted waves and their corresponding angles of emergence as a function of 0P’, for a P-wave incident in copper on a copper/flint glass interface.

orders

reflected P-wave

reflected SV-wave

incident P-wave

spectral

-14.2' OS1 -I -70.6' ,& -2

26.9' OS1 0

7i.60 I

of the different

I c-90'

I I

I

1 61'

6d.60

Fig. 10. Range of propagation

-44.3O -1 '

0

I

44.3O

O0

I I, 32.7' :l.B'

34!3O Lo 16140 I _I t

transmitted P-wave

1.0 +

kS’D

l-rPP,n.l \

\

\

/’

300

‘*I

-._.-

,--.

P

I

It

*___-‘“‘-“..__. . ..\

9P,’

-

p \:

I

4

\ ‘: \: \’ C

\. \ \ ‘*.

-..*

60'

--

60'

91p

9

I

0

0.04 I/

0.08

0.12-

llSP,n*’

0.16

0.2

7

P r___*-.\ --:----

II I

300

_

sv

300

i

\ =--._

BP;

60" -

eP:

60' __)

--__ -.".--...---____ _y-

4 $.

L

i i i !

i

i

i

I

Fig. 11. Acoustic reflectances and transmittances, in the vertical direction, of the zero spectral orders as a function of 0Pk for a siksoidal interface between copper (medium I) and flint glass (medium II) for a P-wave incident in copper.

\

\‘=.*

2 0.2

0.1

0.6

;.. ._._._

304

=L.OB = 9

kP’D

-. I 0.8..-_._.\ \

f

0.6

.

900

90'

reflected

transmitted

P-wave

!N-vave

Fig. 12. Range of propagation of the different spectral orders for the reflected and transmitted waves and their corresponding angles of emergence as a function of 19s; for an W-wave incident in copper on a copper/flint glass interface.

spectral

.I

0

0.2

0.4

0.6

0.9

900

0

0.0‘

0.06

0.12

Fig. 13. Acoustic reflectances and transmittances, in the vertical direction, of the zero spectral orders as a function of OS: for a sinusoidal interface between copper (medium I) and flint glass (medium II) for an SV-wave incident in copper.

es;-

60’

llP$’

0.16

I

f

sv

I

II

es;-

60'

P

I 908

J.T. Fokkema 1 Reflection by periodic interface of two solids

44

L 0

-900

-27.6' ,,I;

00

-2

-1

2

* o'l? o"0 m*.D* .Gcd.Gd -_ NN

I

6:' 1

I

I transmitted SVvavc

8SX1 incident SV-wavc 0 .31.5°900

Fig. 16. Range of propagation of the different spectral orders for the reflected and transmitted waves and their corresponding angles of emergence as a function of @St for an SV-wave incident in flint glass on a copper/flint glass interface.

1

1 I *J, . 0; '1 .‘.7O r. 36.7O

f

.

4 ‘.O I

ofjijj$J+~ ,jiref1ected Pvave

-2

spectra1 order kS'D.9

kP'DzL08

O.OL -

0.00-

0.12-

P&L=

10.16 -

I

i

i i i

! I

i

I

I ! ! !

i! ',I

j !

f i

30"

..-

e

,

’

\

II I

I

II

’

sv

600 es:-

’

sv

P

P

’

LL-

Fig. 17. Acousticreflectances and transmittances, in the vertical direction, of the zero spectral orders as a function of OS~’ for a sinusoidal interface between copper (medium I) and flint glass (medium II) for an SV-wave incident in flint glass.

Tl

t

02

0.12

1 0.16 TrPsy

900

46

J.T. Fokkema / Reflection by periodic interface of two solids

From Figs. 4 and 5 that present the results for an incident SV-wave in granite, we learn that this situation is similar to the one of Fig. 3. Up to the critical angle &= 50.3”, almost all incident energy is transmitted into the zeroth spectral order of the SV-wave in medium II (slate). From the critical angle onward up to grazing incidence, all energy is reflected in the SV-wave of zeroth spectral order and this phenomenon becomes less pronounced with increasing depths of the profile. From Figs. 6 and 7 that present the results for an incident P-wave in slate, we observe that in the range O”-70” of the angle of incidence, the transmitted P-wave of spectral order zero has the most significant contribution and then suddenly falls off. In the range 70”-90” of the angle of incidence, the reflected P-wave of spectral order zero sharply rises from the value 0 to 1. Further it is concluded, that the influence of the roughness of the interface is negligible. From the Figs. 8 and 9 that present the results for an incident SV-wave in slate, we conclude that the influence of the profile depth is very small. 3.2. Copper/flint glass interface From Figs. 10 and 11 that present the results for an incident P-wave in copper, we observe rapid variations in the curves for h/9 = 0.1 in the vicinity of f3Pb= 42” being the angle of incidence for which kl9 = kST9. A condition of this kind implies horizontal phase matching between the relevant evanescent spectral order and a Stoneley wave along the corresponding plane interface. These variations are very pronounced in the curve representing the transmitted SV-wave of spectral order zero. This is the more remarkable since in the case of the plane interface the conversion energy l7SP~~’ almost vanishes. Further, for h/9 = 0.1 a small variation is observed in ZISP:’ at @Pi = 60.6”, being the value for which k--29 = -kST9. The variations in the curves for h/9 = 0.3 and 0.5 are a consequence of the changing of spectral orders from propagating to evanescent or vice versa.

From Figs. 12 and 13 that present the results for an incident SV-wave in copper, again we observe rapid variations in the curves for h/9? = 0.1 at 13s; = 17.7”, again being the value for which kl9 = kST9. With increasing depths of the interface profile, this phenomenon becomes less pronounced. For larger depths, the variations can be ascribed to the fact that the reflected SV-wave of spectral order 1 becomes evanescent. In the range 70”90” of the angle of incidence, almost all incident energy is converted to propagating waves of higher spectral order. From Figs. 14 and 15 that present the results for an incident P-wave in flint glass, we observe variations in the curves for h/9 = 0.1 at 0Pt = 32.1”. For this angle of incidence we have again horizontal phase matching to the Stoneley wave, i.e. klLB = kST9. This phenomenon is decreasing with increasing depths of the profile. Further, we remark that from the figures it follows that with increasing depths of the profile, the propagating waves of higher spectral order collect an increasing part of the incident energy. From Figs. 16 and 17 that present the results for an incident SV-wave in flint glass, again we observe variations in the curves for h/9 = 0.1, when kIGB= kST9 at fX#= 18.5”. Again with increasing depths of the profile, the propagating higher spectral orders take over the main part of the incident energy. 4. Conclusions From the results presented in the preceding section we may conclude that the two configurations show a marked difference in their elastodynamic reflection and transmission properties, especially for non-vanishing, but small depths of the corrugated interface. Now, as far as their elastodynamic properties are concerned, the two configurations differ only in one respect: along the granite/slate interface no Stoneley wave can propagate, while along the copper/flint glass interface a Stoneley wave does exist. Apparently, at horizontal phase matching of a particular

47

J.T. Fokkema / Reflection by periodic interface of two solids

evanescent spectral order to the Stoneley wave along the corresponding plane interface (i.e. when k,,g = ltkSTg) for some spectral order m), the combined P- and SV-wave motion of that particular spectral order almost shows, in the two media, the spatial structure of a Stoneley wave. When this occurs, the redistribution of incident energy is significantly influenced by that particular spectral order, which itself is evanescent and hence carries no energy in the vertical direction. Since Stoneley waves propagate along a plane interface, we expect the relevant phenomenon in our configuration to be only pronounced for small depths of the profile; this is confirmed by our results. Further rapid variations occur when one of the spectral orders changes from propagating to evanescent or vice versa. This phenomenon causes a significant change in the redistribution of energy over the propagating spectral orders and is the more pronounced for larger depths of the profile. The two configurations have in common that the conversion energies (P to SV and SV to P) are small. Both configurations satisfy the reciprocity conditions discussed in Section 2. As a consequence of the reciprocity properties (2.10), the following correspondences exist: - granite/slate interface ZISP:I in Fig. 3 corresponds to 17PS$’ in Fig. 5, I7sPg*” in Fig. 7 corresponds to ITPSA1*”in Fig. 9, 17PPgg1in Fig. 3 corresponds to 17PPi” in Fig. 7, ITSPt*’ in Fig. 3 corresponds to I7PSp in Fig. 9, 17PSA1*’ in Fig. 5 corresponds to I&P$” in Fig. 7, 17S!#” in Fig. 5 corresponds to 17SS~” in Fig. 9, - copper/flint glass interface 17SP$’ in Fig. 11 corresponds to I7P!# in Fig. 13, IISP:,” in Fig. 15 corresponds to Z7PS~1’11 in Fig. 17, Z7PPL1.’in Fig. 11 corresponds to I7PP:” in Fig. 15, zTsP;*.’ in Fig. 11 corresponds to IMPS:” in Fig. 17,

ITPSF” in Fig. 13 corresponds to Z7SPi1’ in Fig. 15, zIss~*’ in Fig. 13 corresponds to LQ!$‘” in Fig. 17. It is noted, that this correspondence is to be invoked on different scales along the abscissas. This is due to the fact that the figures have not been plotted as a function of kin’9 but as a function of the angle of incidence. The reciprocity relations have served as a first check on the accuracy of the computations. As a second check, we have verified the power relations (2.1 l), (2.12), (2.13) and (2.14) for each type of incident field. For the type of interface under consideration, the numerical solution of the integral equations (6.14) of [l] has been based upon the method of cubic spline approximation, combined with a pointmatching technique. As a result, the integral equations are replaced by a system of linear algebraic equations. The integrals in the relevant matrix elements have been computed with the aid of the trapezoidal rule, with two integration points between two successive mesh points of the chosen cubic spline approximation. The series representations of the Green tensors UT, and ryaP have been truncated, in combination with a technique for accelerating the convergence. For a detailed discussion on the relevant numerical techniques we refer to [lo, Chapter 31.

Table 5 Number of spline subintervals (K), number of linear equations (M), number of terms in the numerical evaluation of the Green tensors (ZN+l), and computing time per angle of incidence (T), pertaining to a single interface problem for various values of h/9 h/9

K

M

2Nil

T

0.1 0.3 0.5

10 13 15

40 52 60

21 21 21

33 s 51 s 70s

For the values of kP9 and kS9 see Table 3 (granite/slate interface) and Table 4 (copper/flint glass interface).

48

J.T. Fokkema / Reflection by periodic interface of two solids

An indication of the computing time needed by our numerical program to obtain the results for a single problem and for a single angle of incidence is given in Table 5, for various depths of the profile. To produce the figures, we have computed the results at every 2.5” of the angle of incidence starting from normal incidence. In the neighbourhood of (expected) rapid variations we have performed the computations at every 0.1”. In the computations, we have used a combination of single (six significant figures) and double (16 significant figures) precision. A fully double-precision run of the program has been made to investigate whether or not round-off errors were significant. It turned out that the remaining error is due to the numerical discretization of the integral equations, the numerical evaluation of the integrals in the matrix elements, and the truncation of the series representations of the Green tensors. An estimate of these errors has been made by increasing the number of linear equations, the number of integration points in the evaluation of the integrals in the matrix elements and the number of terms to represent the Green tensors. As a result, the accuracy of our results is estimated to be one percent or better. The computations have been performed on the I.B.M. 370/158 computer of the Computing Centre of the Delft University of Technology. The program has been written in the PL/I language.

Laboratory of Electromagnetic Research, Department of Electrical Engineering, Delft University of Technology, Delft, The Netherlands for numerous helpful discussions, suggestions and remarks. The financial support of the Netherlands organization for the advancement of pure research (Z.W.O.) is gratefully acknowledged. References [l]

[2]

[3]

[4] [S] [6]

[7]

[B]

[9]

Acknowledgment

The author wishes to thank Professor Dr. A.T. de Hoop and Dr. P.M. van den Berg of the

[lo]

J.T. Fokkema, “Reflection and transmission of elastic waves by the spatially periodic interface between two solids (theory of the integral-equation method)“, Waue Motion 2, 375-393 (1980). C.G. Knott, “Reflection and refraction of elastic waves with seismological applications”, Philos. Mag. [S], 48, 64-97 (1899). B. Gutenberg, “Energy ratio of reflected and refracted seismic waves”, Bull. Seism. Sot. Amer. 34, 85-102 (1944). W.M. Ewing, W.S. Jardetzky and F. Press, Elastic Waoes in Layered Media, McGraw-Hill, New York (1957). F. Birch, Editor, Handbook of Physical Constants, Geol. Sot. Amer. Spec. Paper 36 (1942). R. Stoneley, “Elastic waves at the surface of separation of two solids”, Proc. Roy. Sot. London Ser. A 106,416-428 (1924). J.G. Scholte, “The range of existence of Rayleigh and Stoneley waves”, Monthly Notices Roy. SOL: Geophys. Suppl. 5, 120-126 (1947). L. Cagniard, Reflection and Refraction of Progressive Seismic Waues, McGraw-Hill, New York (1962). (Translation and revision of L. Cagniard, Reflexion and Refraction des Ondes Seismiques Progressives, Gauthier-Villars, Paris (1939), by E.A. Flinn and C.H. Dix.) D.E. Gray, Coordination Editor, American Institute of Physics Handbook, McGraw-Hill, New York, 3rd ed. (1972). J.T. Fokkema, Reflection and transmission of timeharmonic elastic waves by the periodic interface between two elastic media, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands, report number: 1979-6 (1979).