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On the multiple scattering of Rayleigh waves by surface roughness
WAVE MOTION 11 (1989) 371-381 NORTH-HOLLAND
ON THE MULTIPLE SCATTERING SURFACE ROUGHNESS
371
OF RAYLEIGH
W A V E S BY
N.S. C L A R K E and C. DAVIES* Department of Engineering Mathematics, University of Newcastle upon Tyne, UK
J.S. BURDESS Department of Mechanical Engineering, University of Newcastle upon Tyne, UK Received 15 June 1988, Revised 18 October 1988
This paper is concerned with the multiple scattering of surface acoustic waves by small but extensive irregularities in the otherwise plane surface of an isotropic elastic half-space. Usually the reflected wave will also be small but if the irregularities are sufficiently extensive and have Fourier components with wavelength half that of the incoming wave, then the reflected wave will have a substantial amplitude. This case, with arbitrary irregularity profile, is examined herein by use of a multiple scales perturbation scheme applied directly to the field equations and boundary conditions. A uniformly valid first-order approximation is obtained in closed form. This work complements that of Simons [1] who obtained the equivalent solutions for a particular example by a different method.
1. Introduction
Consider a Rayleigh wave propagating 'along' the plane surface of an isotropic elastic half space until it encounters a region of surface roughness. At every change of shape of the surface it is to be expected that the incident Rayleigh wave will be partially transmitted, partially reflected and partially scattered into bulk waves propagating into the interior. It is assumed that the form of this roughness is in lines parallel to the incoming wave front so that the problem remains essentially two-dimensional. Of considerable relevance to the problem is the ingenious paper of Simons [1] in which the roughness is considered to be in the form of thin strips of material laid on the surface at regular intervals. These are of rectangular cross-section and consist of material which may be the same as in the substrate, or differing from it. The method of solution in [1] commences with the assumption of an approximating boundary condition, devised by Tiersten [2], which is formalised into an integral equation to be satisfied on the boundary. This is solved by perturbation methods and a closed form expression obtained for the reflection coefficient in the case when there are a large number of strips of 'small' height. In some ways the present work overlaps that of Simons in that the geometries have similarities. We restrict consideration to the case where the materials are the same but allow the actual shape of the surface roughness to be arbitrary, which includes Simons case of a regular rectangular array. This latter may be considered to be deliberately constructed by attaching strips to the surface, or by cutting grooves by processes such as etching, possibly for the purpose of constructing a resonator. Alternatively we may regard the surface to be inadvertently irregular, that is taking the surface as found. This raises the possibility * Current address: Materials and Structures Department, RAE Farnborough, UK. 0165-2125/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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N.S. Clarke et al. / Scattering by surface roughness
of using the results presented in Section 4 to investigate the surface shape. Another feature which distinguishes the present work from that of Simons is that although we also use a perturbation scheme (and multiple scales) to effect a solution, we do so by applying it directly to the field equations and the boundary conditions. It was noted by Simons [1] (and further references cited therein) that if the amplitude of the roughness is small (O(E), where e is the ratio of the height to the wavelength of the surface), then the amplitude of the reflected wave from each reflection will also be O(~). However, if the region of roughness extends over m a n y wavelengths, and if the wavelength of the array is half the wavelength (AR) of the incident Rayleigh wave, then the multiple reflections of the Rayleigh wave are phased so that they reinforce each other to give a reflected amplitude of O(1). In our case this resonant p h e n o m e n a will arise if in the Fourier representation of the surface shape there are components with wavelength approximately AR/2. The bulk waves generated by this roughness do not sum in this manner and therefore remain much smaller than the reflected Rayleigh wave. These waves are neglected. In Section 2 the field equations and boundary conditions are manoeuvred into a dimensionless form suitable for exploitation by a perturbatio n scheme. The solutions in the regions either side of the roughness are the ordinary Rayleigh wave solutions. For a solution in the roughened region, Section 3, a perturbation scheme is required. As this threatens to be singular the method of multiple scales is used to develop a uniformly valid zero-order solution. In order to obtain a solution covering the whole surface, via determining the reflection and transmission coefficients, the solutions in the three regions must be ' m a t c h e d ' to each other. In Section 4 it is argued that this matching may be achieved to the first order by insisting only that the displacements be continuous at both ends of the roughened region on the surface. This matching is effected and the solutions presented there.
2. The governing equations Consider a half-space occupied by an isotropic homogeneous linearly elastic material. The surface is plane except for a region which is 'rough' or 'etched'. We introduce a Cartesian coordinate system such that the surface is plane in the regions X < 0 and X > L, while on 0<~ X ~< L the surface is etched in straight lines parallel to the Z-axis. The mean surface level in 0<~ X ~< L is taken to be Y = 0 and the plane surfaces are then denoted by Y = eHo, where H0 is a constant and e is a small dimensionless number. The entire surface may be described by Y= ell(X),
for all Z,
(2.1)
with H ( X ) = Ho in -oo < X < 0 and in L < X < oo and ~oL H ( X ) d X = O. The material occupies the region -oo < X < + ~ , -oo < y < E H ( X ) . We shall suppose that a Rayleigh wave emanates from X = - ~ and propagates in the positive X direction. This means that the problem will be independent of Z, i.e. it is two-dimensional.
Y=EH
o
~/ Y
I
r
..
..
i °°°
III;'llllltllllll/I, lff ~ - I l U U l f iJ-~flBlld-i].lllllllllllllllllllllll X-O
X=L
Fig. 1. The boundary geometry.
N.S. Clarke et al. / Scattering by surface roughness
373
The displacements U1 and U2 in the X and Y directions of a point in the material must satisfy the following equations (see Achenbach [3]),
a2UI+/A +
02U1+
/z -7
02U2
02U1
/z ) ox a------V P (2.2)
02 U 1 , 0202 . . . . . . 0202 a 202 (A +2/Z) aX OY - r / z - ~ T-r t^ * z l z ) -~y-T = p - ~ .
Here A a n d / z are the Lam6 constants and p the mass density and 7" is time. On the free surface the traction must vanish, and this gives the boundary conditions on Y = • H(X)
°u'+°vfi-,n, /Z (kaY OX/
+2/z)
OU l
oqV2"~ = o, (2.3)
OU 1
002
,
~OU1 .jt OU2~
A-~-~+(A+2/Z)-~-eH/ZI . ~ -
-~-J =0.
Here H ' is the derivative of H. In addition to these surface conditions we shall impose the condition that the solution is to be a Rayleigh type wave. This requires that UI and [/2"->0
as Y-* -oo.
For X ~ +oo the solution should be an outgoing (transmitted) Rayleigh wave, whilst for X ~,-oo the solution is the sum o f an incoming Rayleigh wave and an outgoing (reflected) Rayleigh wave. Dimensionless variables are now introduced by 7"= to-it,
(X, Y)= ~p-~(x,y),
L= ~-~w21, (2.4)
H(X)=~-~w2h(x),
H o = 4 ~ w 2 ho,
( UI, U2) = 4p-~2 (Ul, u2),
where to is the frequency of the incoming wave. It follows that parameter a defined by
=/z/(a +2/z),
H'(X) = h'(x). We also introduce a material (2.5)
and so is related to Poisson's ratio v by ot = ( 1 - 2 v ) / ( 2 - 2 v ) . We note that 0 ~< r, ~<0.5. In terms of these dimensionless variables the governing equations and surface conditions are, in y < • h (x)
021/1 ~_ 02Ul , z4 02U2 021/1 Ox 2 ot-~-y2~- ti -- a ) aX Oy = o r -Ot - ,2 (1__O0 02U! ,
02U2, 02U2
02U2
OX Oy t Ot OX---T-t-Oy----" ~ = a at 2 ,
(2.6)
N.S. Clarke et al. / Scattering by surface roughness
374
whilst on y = ~ h(x)
03U-O I'lU2=hf(x')oLOX -Ioy
+(1--2a)
oy J' (2.7)
(1-23)
OUl+OU2= ~ h'(x)a ~OUl+OU2~. Ox Oy [Oy Ox )
It is the solution of (2.6), subject to (2.7) and the far field conditions, which is sought. The above dimensionless scheme was adopted so that the mathematical problem (2.6) and (2.7) has the smallest possible n u m b e r of independent parameters overtly involved.
3. The Rayleigh wave solutions
The incoming Rayleigh wave emanates from x = -oo and travels undisturbed until it strikes the etched region in 0 <~x ~
(3.1) u2 and for
i(2k 2 - 1) {ei(k~_t) - R e i ( - ~ - ° } { ( 2 k 2 - 1 ) e p(y-~h°) - 2 k 2 eq(y-'h°)}, 2kq
l
eq(y-'h°)}, (3.2)
U2 =
i(2k 2 - 1) T ei(k~-~){(2k 2 - 1) 2kq
e p(y-'h°)- 2k 2 eq(Y-'h°)}.
Here R and T are the (complex) reflection and transmission coefficients, as yet unknown. Also p and q are related to the wave n u m b e r k by p = x / ~ - - ce,
q= ~ ,
(3.3)
and the wavenumber satisfies the equation (2k 2 - 1) 2 =
4kEpq.
(3.4)
It is well known (Achenbach [3]) that for each ~t in (0, 0.5) there is just one real root to the equation (3.4). This roof is illustrated in Fig. 2. It should be noted that we have chosen without loss of generality the amplitude of the incoming wave u~ to be unity on the surface. We need now to discuss the solutions within the etched region, where h ' ~ 0 and hence the above solutions are not available. I f a solution based u p o n the smallness of e is sought, secularity is encountered, and the method of multiple scales is used to circumvent this difficulty. Moreover, if the etched region is
N.S. Clarke et aL / Scattering by surface roughness
375
0.5
0"4 ~"---k
0.3
I vs
t/
0-2
0.1
0"!
0.2
0"3
0"4
0'5
Fig. 2. a,/3 and k as functions o f Poisson's ratio J,.
large in the sense that El = 0(1) and the surface has periodic components with wave-number (approximately) twice that of the incoming Rayleigh wave, then it is found that the zero-order solution is significantly affected by the presence of the etchings. This will become apparent in what follows. To anticipate the difficulty encountered with the secular terms we introduce the 'slow' variables =~x
and
~=ey,
(3.5)
and regard uj(x, y; ~) as uj(x, y, ~, 7; ~). Equations (2.6) are then 02u1+ a 02u1+ (1_ a) 02u2
Ox2
Oy"
:- {
2 02ul "~-"
. c)2ut
(l-a)
02ul Ox Oy - a Ot~
02Ul " / 02192 02U2~ ) aOyOn+(1-a laXO,?÷;yOe) +O('2),
02U2 02U2
O2Uz
ox Oy+a~x2 +-~fy2-a Ot2
=-E{(1-a)\OxO
~
O~y/+2aOxO~+2
y ~J
We seek a solution to these equations in the form of a power series in ~,
uAx, y, 6, ,7; ~)= uJ°)(x, y, ~, 7)+ ~u)'(x, y, ¢, ,7) + o(~). Upon substitution of this form into (3.6) and comparison of coefficients of like powers of r we find that u~°) and u~°) satisfy equations (3.6) with ~---0 and hence admit solutions of the form U(1o) _-- kA (o) epy+i(kx-o + qB (°) eqy+i( k:x-t ) + k C (0) e py+i( -k-x-t ) .~. qD (o) eqy+i(-/cx-0,
u~°) = -iP A ~°) epY+i~-° - ikB~°) eqy+i~-') + ipC ~°~e py+i~-~-° + ikD ~°~e ~+i~-~-'),
(3.7)
376
N.S. Clarke et al. / Scattering by surface roughness
where A (°), B (°), C (°), and D (°) are unknown functions of ~¢and 7/. We have anticipated the results of the application of the surface condition by retaining the symbols k, p and q with their previous usage. The equations for u~~) and u(:~) are found from the O(~) terms in (3.6), and with use of (3.7) these become
: u l ')
:ui"
--+a
OX 2 :
Oy 2
: u l ') +(l-a)
OX Oy
: u l ') -a--
Ot 2
E 1 e p y + i ( k x - t ) + E 2 e qy+i(kx-t) -~- E3 e p y + i ( - k x - t ) + Ea e qy+i(-kx-t),
(1 - a) 82u~1) ..q2..(l) o~2.u 2(1) t, ~2 Ox Oy + ct Ox---T+ Oy----T-
(3.8)
~2U(I ) a
Ot 2
---- F 1 e p y + i ( k x - t ) + F 2 e qy+i(kx-t) ..~ F 3 e p y + i ( - k x - t ) q_ F 4 e qy+i(-kx-t),
where 0A (°) --(l+alkp
E'=-i{(l+a)k2+a(1-a)}--~
,gA (o) 0---~-'
E 2 = - i k q ( l + ° O - - ~0 B ( °-) - {(1 + a ) k 2 - 2 ° ~ } 0B(°) aV ' OC (o)
Ea=il(l+a)k2+a(1-a)}-~ E4= ikq(1 + a ) -OD(°) -~--
F~ = - ( 1 + a ) k p
OA (o)
{( 1 + a ) k 2 - 2 a }
+i{(l+a)k2-2t~}
)}
FE={-(l+a)k2+(1-a F3 = - ( l + a ) k p
c3C (o)
o~
oc(O) OV '
-(l+a)kp
-OD(°) Or/ ' OA (o) -,
(3.9)
c30~°)+1(1 " + a ) k q ~B(°) c3rl '
--
+i{-(l+a)k2+2a}
OC (o) ,
on
~D (°) . 0D (°) F, = {-(1 + a ) k 2 + (1 - a)} - - ~ - - 1 ( 1 + a ) k q - 0~7 Solutions to these equations are sought in the form u~')= A}') e pr+i(~,-,, + B~') e qy+i(~,-,) + CJ ') e ny+i(-~-,) + _D Oj )
eqy+i(-kx-0
for j = 1, 2.
(3.10)
These forms are substituted into (3.8) and coefficients of like exponentials compared to yield four pairs of forced, over-determined, simultaneous linear algebraic equations for the AJ 1), BJ l', CJ l' and DJ 1' . For
N.S. Clarke et al. / Scattering by surface roughness
377
these equations to be consistent we must have that . OA(°) OA(o) lk---3~+ p = O, on ot
ik OB(°)+
0e
OB(°)
q--if-= 0, (3.11)
OC(°) OC(°) ik -p =0,
0~
ik OD(°)-
on
0~
q
OD(°)
on
=0.
It also follows that A(21)= - i --pA (1) a OA(°) k ' kp c9~ '
B(2')=
-i k q
1 0 B (°) B~')'i q2 O ~ ' (3.12)
C(21)_i__Pco ) ot OC (°), k ' -~ 0-~
kD~l) 4 1 0D (°) D~l)=i q2
Equations (3.11) represent the conditions on the zero-order coefficients in order that secular terms do not arise in the first-order solutions as they would thereby invalidate the perturbation scheme. To proceed further we must now consider the boundary conditions on the surface y = • h(x), that is, equations (2.7). However in the spirit of the perturbation scheme we shall apply the boundary conditions at the mean surface y = 0 rather than on the actual surface y = • h(x) and accommodate the discrepancy by writing
uj(x, h(x)) = uj(x, O) + ~ h(x) ~ (x, 0) + O(e2),
(3.13)
and similar expressions for the derivatives of uj. When such expansions are used in (2.7), along with the multiple scales, we find that the zero-order boundary conditions are
Ou(o) Ou(o) ( l _ 2 a ) , , . . l +v,,2 =0, Ox Oy
Ou(O) 1 + ' .Ou ' ' 2'°' = 0 , Oy Ox
(3.14)
on y = 0 (and n = 0), for 0 < x < l, and the first-order boundary conditions are . (1) z . O) fOu(O) ou(o)) "1 +~'"2 = 0U~°) Ou~°)~_h'(x)~""l +(1_2ot)"*'2 ~ Oy Ox On o~ a [ Ox Oy J /,,
f ~ 2 . (o) a2.,(o)) x / t , ~a I o ~2 /
0U(21)
(1-2or) Ou~l)+ Ox Oy
= -(1-2a)
Ou~ - - °)
O~
fz, (o) a. (o)'1 ou~2 °) ÷oth,(x)~O.., _ _ +0-2
On ~a2 . (0)
-h(x)
( Oy
(3.15)
Ox J
,~2.. (0)"1
o ul +~, " 2 [ ( 1 - 2 a ) Ox Oy Oy2 J'
on y = 0 (and ~/=0), for 0 < x < L With (3.7) in (3.14) the equation for the wavenumber is found to be the same as in (3.4) and hence justifies the notation used (as remarked after (3.7)). Also it is found that B(°)(~, 0) - -(2k2-1)A(°)(~, 0), 2kq
D(°)(~:,0 ) - - ( 2 k 2 - 1 ) C(°)(~, 0). 2kq
(3.16)
378
N.S. Clarke et al. / Scattering by surface roughness
Consider now the first-order conditions (3.15). It is apparent that secular terms will arise, due to the zero-order terms on the fight-hand side, unless they are suppressed by further conditions on the zero-order coefficients. This suppression will usually lead to the zero-order coefficients being independent of the slow variables, and hence the Rayleigh wave will only be affected to O(~). However, if the etched surface function h(x) has periodic components with wavenumber double that of the incoming wave, then it is found that this is no longer the case. It is found that the zero-order coefficients will depend upon the slow variables and that the Rayleigh wave will experience appreciable modification if ~l = O(1). This will be born out by the following analysis. At the same time we shall investigate the effect of a slight 'mismatch' in the wavenumber. We shall represent h(x) by a complex Fourier series and display explicitly only those terms which give rise to secularity. Let
h(x) = h (+) ei2k'xF h (-) e-i2k'x + ~'~(X),
(3.17)
where h (+) and h (-) are complex conjugates, and where /2(x) is here, and in what follows, a generic symbol representing terms which will not enhance secularity. Also the mismatch is expressed by k ' = k + ~kl + O ( e 2) which means that h(x) = h (+) ei2kx+i2kt~ Jt- h (-) e-i2kx-i2k~-~-~'~-Jl- O(~2). It is convenient at this stage to point out that doubts could be raised about the validity of the expansion scheme if h'(x) were to be large at any point. However in such cases we may always interpret the results by use of deferred limits in the spirit of generalised functions. With the zero-order forms put into (3.15) the latter are .,(1) ,,~.,(1) "1 +0'42 =pei(kx_O+Qei(_kx_O+~-2, ' Oy
Ox
(3.18) ( 1 - 2 a ) 0U~I._"t¢ ~ .ji 0U~I.__~, ~ ¢ = R ei(kx-t)"~" S ei(-kx-t)]- a ,
Ox
Oy
where p=i/2k2-ot
p
O = _ i ~ 2 k2-t~ t P
2k 2 - 1 } 0 A (°) ~1
q
O~ +,~,~{l+4k2(1-a)}h(+)ei2k'¢C(°)"
2k2-1}OC(°)+_~k{l+4k2(l_a)}h(-)e-i2kdA(O) ' q
O~
R=fl-2ot-2k2(1-a)]0A (°) i a ( 2 k 2 - 1 ) h(+) ei2kl~ic(O), J O~ 2kq 2 4k2 q
(3.19)
S = { 1 - 2a - 2k2(1 - a) } OC(°) + ia(2k2-1) h(-) e_i2k16A(O) 2kq 2 O~ 4k2q evaluated on 7 / = 0. The forms (3.10) are inserted into the conditions (3.18), and when (3.12) are used two pairs of coupled overdetermined linear algebraic equations are derived involving the pairs A~1), B~~) and C~ 1), D~ ~). For these equations to be consistent it is necessary that (after a little manipulation)
OA(o)
iflh (+) ei2kl~:C (0) = 0,
OC (o)
--"~-
iflh (-) e-i2k, eA (°) = 0
(3.20)
N.S. Clarke et al / Scattering by surface roughness
379
on ~ = O, where
fl = 8k3pq3/{4k:(2k2- l)p -
(4k 2 - 1 ) q }
(3.21)
depends only upon a (and hence ~,), i.e. it is a material parameter. This dependence is illustrated in Fig. 2. The system (3.20) has the following solutions: A(°)(6, O) = M
e(~+ikl)~-t -
N e (-~'+ikQf, (3.22)
C(°)(~, 0) = fiT+) {(~ +
ikl)M e (~-ikp~ -
(~-
ikl)N e(-~-ikpe},
for arbitrary constants M and N and where
=~/fl2h(+)h(-)- k 2
(3.23)
may be real or imaginary depending upon the extent of the mismatch. The surface values B(°)(¢, 0) and D(°)(~, 0) may also be determined from (3.22) and (3.16). These boundary values now enable us to find the solutions to (3.11). If we seek such solutions by the method of separation of variables, and use (3.22) to yield the separation constant, then we readily determine the zero-order coefficients as A(°)(~,
71)=M exp{(¢+ikl)~+k (-i~'+kl),l}+ N exp{(-~'+ikl)~+k (i~'+k~)~l},
B(°)(~' ~1)-
-(2k2-1)[Mexp{ } 2 k (~'+q ikl)~+k-(-ir+k~)~l q +Nexp{(-~'+ikl)~+k(i1"+kl)l?}],
(3.24)
,7)=
-(7"-ik,)Nexp{(-~'-ik,),+k(-iv+k,)l?}], P i (2k2-1)[(~.+ik,)Mexp{(~._iki),+k(iT+k,)l?} D(°)(~' ~/) - / 3 h (+) 2----~ -(~'-ik,)Nexp{(-~'-ik,)~+k(-i~'+k,)~l}]. When these coefficients are used in (3.7) we have the zero-order solutions within the etched region. This has now to be 'matched' to the solutions (3.1) and (3.2) on either side of this region in order to determine the complex constants M, N, R and T. 4. Matching the solutions
It is not possible to amalgamate the solutions in the three regions to form an exact solution, if only because these solutions are only approximate. However it is possible to perform an approximate matching valid to the same order as the individual solutions themselves.
N.S. Clarke et al. / Scattering by surface roughness
380
In what follows we consider terms only of O(1) and ignore all terms proportional to e unless accompanied by l, i.e. we retain terms depending up el as these contribute to the O(1) terms. Bearing this in mind it will be seen from the forms of (3.1), (3.2) together with (3.7) and (3.24) that matching to this order may be effected by insisting only that the displacements be continuous at x = 0 and at x = l on the surface, i.e, at y = O(e). This ensures that, to the first order, the displacements are continuous at x = 0, l for all y. We may set x = 0, y = O(e) in (3.1) and equate the resulting values of ul and u2 to those obtained from (3.7), together with (3.24), at the same location. This yields a pair of equations involving R, M and N. Another pair of equations involving T, M and N are obtained by a similar process by setting x = l, y = O(e). These four equations are readily solved to give M = {2k(r - ikl) e-:~'t}/{(r + ikl) + (~" - ikl) e-2~'t}, N = 2k(~'+ ik,)/{(z + ikl) + (~- - ik,) e-2~'t},
(4.1)
T = • eikl~l/{'r c o s h ( T d ) + ikl sinh(~'el)}, R = iflh (-) sinh(~-el)/{~" cosh(zel)+ ikl sinh(~-el)}.
We may note that exactly the same values of M, N, T and R would have been obtained had we matched the separate solutions using the more 'rigorous' (and more laborious) matching procedure of Tiersten [4] based u p o n a variational principle. The determination of M, N, T and R completes the derivation of the leading order representations of the displacements. From (4.1) it is seen that for [R[ to be a m a x i m u m it is required that k~=0, that is for m a x i m u m reflection the wave length of the incoming wave should be exactly twice that of the etchings. This m a x i m u m value is given by
IRI -- tanh D,
(4.2)
where D = / 3 ~ el is proportional to the (dimensionless) area removed by etching. In order to assess the sensitivity of IRI to a small mismatch we introduce the mismatch parameter K = kl/fl~. We then have that
2) -K2} -1/2,
for 0 < K < 1,
IRI = sinh(Dx/1
for K = I ,
[ R l = O ( l + O 2 ) -1/2,
for K > 1,
IRI = I s i n ( D x / K 2 - 1 ) I { K 2 - cos2( D x / K 2 - 1 ) } -1/2.
-K2){cosh2(Ox/-~-K
(4.3)
It is evident from the above that IRI is a slowly decreasing function of K for 0 < K < 1 but thereafter it falls away rapidly. Hence we may conclude that provided
kl <[3x/h(+)h (-)
(
(4.4)
then IRI will not depart significantly from its m a x i m u m value. It should also be noted from (4.1) that as the length of the etched region is increased then the amplitude of the transmitted wave decays exponentially fast, provided (4.4) holds, as 2z
ITI 4~ 2+k2
e -'l
as el~oo.
N.S. Clarke et aL / Scattering by surface roughness
381
These results are in complete agreement with those o f Simons f r o m his case o f rectangular strips or grooves, i.e. where h (+)=i ,ff
and
h (-)=-i. 'IT
Further consideration is given to these results in a subsequent p a p e r where they are used to discuss the p e r f o r m a n c e o f a resonator, comprising two such etched regions either side o f an 'excited cavity'.
Acknowledgment One o f us (C.D.) w o u l d like to thank the S.E.R.C, for an a d v a n c e d studentship during the time w h e n this work was c a r d e d out.
References [1] D.A. Simons, "Reflection of Rayleigh waves by strips, grooves and periodic arrays of strips or grooves", Y. Acoust. Soc. Am. 63, 1292-1301 (1978). [2] H.F. Tiersten, "Elastic surface waves guided by thin films", J. AppL Phys. 40, 770-789 (1969).
[3] J.D. Achenbach, Wave Propagation in Elastic Solids, NorthHolland, Amsterdam (1973). [4] H.F. Tiersten, Linear PiezoelectricPlate Vibrations, Plenum Press, New York (1969).
ON THE MULTIPLE SCATTERING SURFACE ROUGHNESS
371
OF RAYLEIGH
W A V E S BY
N.S. C L A R K E and C. DAVIES* Department of Engineering Mathematics, University of Newcastle upon Tyne, UK
J.S. BURDESS Department of Mechanical Engineering, University of Newcastle upon Tyne, UK Received 15 June 1988, Revised 18 October 1988
This paper is concerned with the multiple scattering of surface acoustic waves by small but extensive irregularities in the otherwise plane surface of an isotropic elastic half-space. Usually the reflected wave will also be small but if the irregularities are sufficiently extensive and have Fourier components with wavelength half that of the incoming wave, then the reflected wave will have a substantial amplitude. This case, with arbitrary irregularity profile, is examined herein by use of a multiple scales perturbation scheme applied directly to the field equations and boundary conditions. A uniformly valid first-order approximation is obtained in closed form. This work complements that of Simons [1] who obtained the equivalent solutions for a particular example by a different method.
1. Introduction
Consider a Rayleigh wave propagating 'along' the plane surface of an isotropic elastic half space until it encounters a region of surface roughness. At every change of shape of the surface it is to be expected that the incident Rayleigh wave will be partially transmitted, partially reflected and partially scattered into bulk waves propagating into the interior. It is assumed that the form of this roughness is in lines parallel to the incoming wave front so that the problem remains essentially two-dimensional. Of considerable relevance to the problem is the ingenious paper of Simons [1] in which the roughness is considered to be in the form of thin strips of material laid on the surface at regular intervals. These are of rectangular cross-section and consist of material which may be the same as in the substrate, or differing from it. The method of solution in [1] commences with the assumption of an approximating boundary condition, devised by Tiersten [2], which is formalised into an integral equation to be satisfied on the boundary. This is solved by perturbation methods and a closed form expression obtained for the reflection coefficient in the case when there are a large number of strips of 'small' height. In some ways the present work overlaps that of Simons in that the geometries have similarities. We restrict consideration to the case where the materials are the same but allow the actual shape of the surface roughness to be arbitrary, which includes Simons case of a regular rectangular array. This latter may be considered to be deliberately constructed by attaching strips to the surface, or by cutting grooves by processes such as etching, possibly for the purpose of constructing a resonator. Alternatively we may regard the surface to be inadvertently irregular, that is taking the surface as found. This raises the possibility * Current address: Materials and Structures Department, RAE Farnborough, UK. 0165-2125/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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of using the results presented in Section 4 to investigate the surface shape. Another feature which distinguishes the present work from that of Simons is that although we also use a perturbation scheme (and multiple scales) to effect a solution, we do so by applying it directly to the field equations and the boundary conditions. It was noted by Simons [1] (and further references cited therein) that if the amplitude of the roughness is small (O(E), where e is the ratio of the height to the wavelength of the surface), then the amplitude of the reflected wave from each reflection will also be O(~). However, if the region of roughness extends over m a n y wavelengths, and if the wavelength of the array is half the wavelength (AR) of the incident Rayleigh wave, then the multiple reflections of the Rayleigh wave are phased so that they reinforce each other to give a reflected amplitude of O(1). In our case this resonant p h e n o m e n a will arise if in the Fourier representation of the surface shape there are components with wavelength approximately AR/2. The bulk waves generated by this roughness do not sum in this manner and therefore remain much smaller than the reflected Rayleigh wave. These waves are neglected. In Section 2 the field equations and boundary conditions are manoeuvred into a dimensionless form suitable for exploitation by a perturbatio n scheme. The solutions in the regions either side of the roughness are the ordinary Rayleigh wave solutions. For a solution in the roughened region, Section 3, a perturbation scheme is required. As this threatens to be singular the method of multiple scales is used to develop a uniformly valid zero-order solution. In order to obtain a solution covering the whole surface, via determining the reflection and transmission coefficients, the solutions in the three regions must be ' m a t c h e d ' to each other. In Section 4 it is argued that this matching may be achieved to the first order by insisting only that the displacements be continuous at both ends of the roughened region on the surface. This matching is effected and the solutions presented there.
2. The governing equations Consider a half-space occupied by an isotropic homogeneous linearly elastic material. The surface is plane except for a region which is 'rough' or 'etched'. We introduce a Cartesian coordinate system such that the surface is plane in the regions X < 0 and X > L, while on 0<~ X ~< L the surface is etched in straight lines parallel to the Z-axis. The mean surface level in 0<~ X ~< L is taken to be Y = 0 and the plane surfaces are then denoted by Y = eHo, where H0 is a constant and e is a small dimensionless number. The entire surface may be described by Y= ell(X),
for all Z,
(2.1)
with H ( X ) = Ho in -oo < X < 0 and in L < X < oo and ~oL H ( X ) d X = O. The material occupies the region -oo < X < + ~ , -oo < y < E H ( X ) . We shall suppose that a Rayleigh wave emanates from X = - ~ and propagates in the positive X direction. This means that the problem will be independent of Z, i.e. it is two-dimensional.
Y=EH
o
~/ Y
I
r
..
..
i °°°
III;'llllltllllll/I, lff ~ - I l U U l f iJ-~flBlld-i].lllllllllllllllllllllll X-O
X=L
Fig. 1. The boundary geometry.
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373
The displacements U1 and U2 in the X and Y directions of a point in the material must satisfy the following equations (see Achenbach [3]),
a2UI+/A +
02U1+
/z -7
02U2
02U1
/z ) ox a------V P (2.2)
02 U 1 , 0202 . . . . . . 0202 a 202 (A +2/Z) aX OY - r / z - ~ T-r t^ * z l z ) -~y-T = p - ~ .
Here A a n d / z are the Lam6 constants and p the mass density and 7" is time. On the free surface the traction must vanish, and this gives the boundary conditions on Y = • H(X)
°u'+°vfi-,n, /Z (kaY OX/
+2/z)
OU l
oqV2"~ = o, (2.3)
OU 1
002
,
~OU1 .jt OU2~
A-~-~+(A+2/Z)-~-eH/ZI . ~ -
-~-J =0.
Here H ' is the derivative of H. In addition to these surface conditions we shall impose the condition that the solution is to be a Rayleigh type wave. This requires that UI and [/2"->0
as Y-* -oo.
For X ~ +oo the solution should be an outgoing (transmitted) Rayleigh wave, whilst for X ~,-oo the solution is the sum o f an incoming Rayleigh wave and an outgoing (reflected) Rayleigh wave. Dimensionless variables are now introduced by 7"= to-it,
(X, Y)= ~p-~(x,y),
L= ~-~w21, (2.4)
H(X)=~-~w2h(x),
H o = 4 ~ w 2 ho,
( UI, U2) = 4p-~2 (Ul, u2),
where to is the frequency of the incoming wave. It follows that parameter a defined by
=/z/(a +2/z),
H'(X) = h'(x). We also introduce a material (2.5)
and so is related to Poisson's ratio v by ot = ( 1 - 2 v ) / ( 2 - 2 v ) . We note that 0 ~< r, ~<0.5. In terms of these dimensionless variables the governing equations and surface conditions are, in y < • h (x)
021/1 ~_ 02Ul , z4 02U2 021/1 Ox 2 ot-~-y2~- ti -- a ) aX Oy = o r -Ot - ,2 (1__O0 02U! ,
02U2, 02U2
02U2
OX Oy t Ot OX---T-t-Oy----" ~ = a at 2 ,
(2.6)
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374
whilst on y = ~ h(x)
03U-O I'lU2=hf(x')oLOX -Ioy
+(1--2a)
oy J' (2.7)
(1-23)
OUl+OU2= ~ h'(x)a ~OUl+OU2~. Ox Oy [Oy Ox )
It is the solution of (2.6), subject to (2.7) and the far field conditions, which is sought. The above dimensionless scheme was adopted so that the mathematical problem (2.6) and (2.7) has the smallest possible n u m b e r of independent parameters overtly involved.
3. The Rayleigh wave solutions
The incoming Rayleigh wave emanates from x = -oo and travels undisturbed until it strikes the etched region in 0 <~x ~
(3.1) u2 and for
i(2k 2 - 1) {ei(k~_t) - R e i ( - ~ - ° } { ( 2 k 2 - 1 ) e p(y-~h°) - 2 k 2 eq(y-'h°)}, 2kq
l
eq(y-'h°)}, (3.2)
U2 =
i(2k 2 - 1) T ei(k~-~){(2k 2 - 1) 2kq
e p(y-'h°)- 2k 2 eq(Y-'h°)}.
Here R and T are the (complex) reflection and transmission coefficients, as yet unknown. Also p and q are related to the wave n u m b e r k by p = x / ~ - - ce,
q= ~ ,
(3.3)
and the wavenumber satisfies the equation (2k 2 - 1) 2 =
4kEpq.
(3.4)
It is well known (Achenbach [3]) that for each ~t in (0, 0.5) there is just one real root to the equation (3.4). This roof is illustrated in Fig. 2. It should be noted that we have chosen without loss of generality the amplitude of the incoming wave u~ to be unity on the surface. We need now to discuss the solutions within the etched region, where h ' ~ 0 and hence the above solutions are not available. I f a solution based u p o n the smallness of e is sought, secularity is encountered, and the method of multiple scales is used to circumvent this difficulty. Moreover, if the etched region is
N.S. Clarke et aL / Scattering by surface roughness
375
0.5
0"4 ~"---k
0.3
I vs
t/
0-2
0.1
0"!
0.2
0"3
0"4
0'5
Fig. 2. a,/3 and k as functions o f Poisson's ratio J,.
large in the sense that El = 0(1) and the surface has periodic components with wave-number (approximately) twice that of the incoming Rayleigh wave, then it is found that the zero-order solution is significantly affected by the presence of the etchings. This will become apparent in what follows. To anticipate the difficulty encountered with the secular terms we introduce the 'slow' variables =~x
and
~=ey,
(3.5)
and regard uj(x, y; ~) as uj(x, y, ~, 7; ~). Equations (2.6) are then 02u1+ a 02u1+ (1_ a) 02u2
Ox2
Oy"
:- {
2 02ul "~-"
. c)2ut
(l-a)
02ul Ox Oy - a Ot~
02Ul " / 02192 02U2~ ) aOyOn+(1-a laXO,?÷;yOe) +O('2),
02U2 02U2
O2Uz
ox Oy+a~x2 +-~fy2-a Ot2
=-E{(1-a)\OxO
~
O~y/+2aOxO~+2
y ~J
We seek a solution to these equations in the form of a power series in ~,
uAx, y, 6, ,7; ~)= uJ°)(x, y, ~, 7)+ ~u)'(x, y, ¢, ,7) + o(~). Upon substitution of this form into (3.6) and comparison of coefficients of like powers of r we find that u~°) and u~°) satisfy equations (3.6) with ~---0 and hence admit solutions of the form U(1o) _-- kA (o) epy+i(kx-o + qB (°) eqy+i( k:x-t ) + k C (0) e py+i( -k-x-t ) .~. qD (o) eqy+i(-/cx-0,
u~°) = -iP A ~°) epY+i~-° - ikB~°) eqy+i~-') + ipC ~°~e py+i~-~-° + ikD ~°~e ~+i~-~-'),
(3.7)
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N.S. Clarke et al. / Scattering by surface roughness
where A (°), B (°), C (°), and D (°) are unknown functions of ~¢and 7/. We have anticipated the results of the application of the surface condition by retaining the symbols k, p and q with their previous usage. The equations for u~~) and u(:~) are found from the O(~) terms in (3.6), and with use of (3.7) these become
: u l ')
:ui"
--+a
OX 2 :
Oy 2
: u l ') +(l-a)
OX Oy
: u l ') -a--
Ot 2
E 1 e p y + i ( k x - t ) + E 2 e qy+i(kx-t) -~- E3 e p y + i ( - k x - t ) + Ea e qy+i(-kx-t),
(1 - a) 82u~1) ..q2..(l) o~2.u 2(1) t, ~2 Ox Oy + ct Ox---T+ Oy----T-
(3.8)
~2U(I ) a
Ot 2
---- F 1 e p y + i ( k x - t ) + F 2 e qy+i(kx-t) ..~ F 3 e p y + i ( - k x - t ) q_ F 4 e qy+i(-kx-t),
where 0A (°) --(l+alkp
E'=-i{(l+a)k2+a(1-a)}--~
,gA (o) 0---~-'
E 2 = - i k q ( l + ° O - - ~0 B ( °-) - {(1 + a ) k 2 - 2 ° ~ } 0B(°) aV ' OC (o)
Ea=il(l+a)k2+a(1-a)}-~ E4= ikq(1 + a ) -OD(°) -~--
F~ = - ( 1 + a ) k p
OA (o)
{( 1 + a ) k 2 - 2 a }
+i{(l+a)k2-2t~}
)}
FE={-(l+a)k2+(1-a F3 = - ( l + a ) k p
c3C (o)
o~
oc(O) OV '
-(l+a)kp
-OD(°) Or/ ' OA (o) -,
(3.9)
c30~°)+1(1 " + a ) k q ~B(°) c3rl '
--
+i{-(l+a)k2+2a}
OC (o) ,
on
~D (°) . 0D (°) F, = {-(1 + a ) k 2 + (1 - a)} - - ~ - - 1 ( 1 + a ) k q - 0~7 Solutions to these equations are sought in the form u~')= A}') e pr+i(~,-,, + B~') e qy+i(~,-,) + CJ ') e ny+i(-~-,) + _D Oj )
eqy+i(-kx-0
for j = 1, 2.
(3.10)
These forms are substituted into (3.8) and coefficients of like exponentials compared to yield four pairs of forced, over-determined, simultaneous linear algebraic equations for the AJ 1), BJ l', CJ l' and DJ 1' . For
N.S. Clarke et al. / Scattering by surface roughness
377
these equations to be consistent we must have that . OA(°) OA(o) lk---3~+ p = O, on ot
ik OB(°)+
0e
OB(°)
q--if-= 0, (3.11)
OC(°) OC(°) ik -p =0,
0~
ik OD(°)-
on
0~
q
OD(°)
on
=0.
It also follows that A(21)= - i --pA (1) a OA(°) k ' kp c9~ '
B(2')=
-i k q
1 0 B (°) B~')'i q2 O ~ ' (3.12)
C(21)_i__Pco ) ot OC (°), k ' -~ 0-~
kD~l) 4 1 0D (°) D~l)=i q2
Equations (3.11) represent the conditions on the zero-order coefficients in order that secular terms do not arise in the first-order solutions as they would thereby invalidate the perturbation scheme. To proceed further we must now consider the boundary conditions on the surface y = • h(x), that is, equations (2.7). However in the spirit of the perturbation scheme we shall apply the boundary conditions at the mean surface y = 0 rather than on the actual surface y = • h(x) and accommodate the discrepancy by writing
uj(x, h(x)) = uj(x, O) + ~ h(x) ~ (x, 0) + O(e2),
(3.13)
and similar expressions for the derivatives of uj. When such expansions are used in (2.7), along with the multiple scales, we find that the zero-order boundary conditions are
Ou(o) Ou(o) ( l _ 2 a ) , , . . l +v,,2 =0, Ox Oy
Ou(O) 1 + ' .Ou ' ' 2'°' = 0 , Oy Ox
(3.14)
on y = 0 (and n = 0), for 0 < x < l, and the first-order boundary conditions are . (1) z . O) fOu(O) ou(o)) "1 +~'"2 = 0U~°) Ou~°)~_h'(x)~""l +(1_2ot)"*'2 ~ Oy Ox On o~ a [ Ox Oy J /,,
f ~ 2 . (o) a2.,(o)) x / t , ~a I o ~2 /
0U(21)
(1-2or) Ou~l)+ Ox Oy
= -(1-2a)
Ou~ - - °)
O~
fz, (o) a. (o)'1 ou~2 °) ÷oth,(x)~O.., _ _ +0-2
On ~a2 . (0)
-h(x)
( Oy
(3.15)
Ox J
,~2.. (0)"1
o ul +~, " 2 [ ( 1 - 2 a ) Ox Oy Oy2 J'
on y = 0 (and ~/=0), for 0 < x < L With (3.7) in (3.14) the equation for the wavenumber is found to be the same as in (3.4) and hence justifies the notation used (as remarked after (3.7)). Also it is found that B(°)(~, 0) - -(2k2-1)A(°)(~, 0), 2kq
D(°)(~:,0 ) - - ( 2 k 2 - 1 ) C(°)(~, 0). 2kq
(3.16)
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N.S. Clarke et al. / Scattering by surface roughness
Consider now the first-order conditions (3.15). It is apparent that secular terms will arise, due to the zero-order terms on the fight-hand side, unless they are suppressed by further conditions on the zero-order coefficients. This suppression will usually lead to the zero-order coefficients being independent of the slow variables, and hence the Rayleigh wave will only be affected to O(~). However, if the etched surface function h(x) has periodic components with wavenumber double that of the incoming wave, then it is found that this is no longer the case. It is found that the zero-order coefficients will depend upon the slow variables and that the Rayleigh wave will experience appreciable modification if ~l = O(1). This will be born out by the following analysis. At the same time we shall investigate the effect of a slight 'mismatch' in the wavenumber. We shall represent h(x) by a complex Fourier series and display explicitly only those terms which give rise to secularity. Let
h(x) = h (+) ei2k'xF h (-) e-i2k'x + ~'~(X),
(3.17)
where h (+) and h (-) are complex conjugates, and where /2(x) is here, and in what follows, a generic symbol representing terms which will not enhance secularity. Also the mismatch is expressed by k ' = k + ~kl + O ( e 2) which means that h(x) = h (+) ei2kx+i2kt~ Jt- h (-) e-i2kx-i2k~-~-~'~-Jl- O(~2). It is convenient at this stage to point out that doubts could be raised about the validity of the expansion scheme if h'(x) were to be large at any point. However in such cases we may always interpret the results by use of deferred limits in the spirit of generalised functions. With the zero-order forms put into (3.15) the latter are .,(1) ,,~.,(1) "1 +0'42 =pei(kx_O+Qei(_kx_O+~-2, ' Oy
Ox
(3.18) ( 1 - 2 a ) 0U~I._"t¢ ~ .ji 0U~I.__~, ~ ¢ = R ei(kx-t)"~" S ei(-kx-t)]- a ,
Ox
Oy
where p=i/2k2-ot
p
O = _ i ~ 2 k2-t~ t P
2k 2 - 1 } 0 A (°) ~1
q
O~ +,~,~{l+4k2(1-a)}h(+)ei2k'¢C(°)"
2k2-1}OC(°)+_~k{l+4k2(l_a)}h(-)e-i2kdA(O) ' q
O~
R=fl-2ot-2k2(1-a)]0A (°) i a ( 2 k 2 - 1 ) h(+) ei2kl~ic(O), J O~ 2kq 2 4k2 q
(3.19)
S = { 1 - 2a - 2k2(1 - a) } OC(°) + ia(2k2-1) h(-) e_i2k16A(O) 2kq 2 O~ 4k2q evaluated on 7 / = 0. The forms (3.10) are inserted into the conditions (3.18), and when (3.12) are used two pairs of coupled overdetermined linear algebraic equations are derived involving the pairs A~1), B~~) and C~ 1), D~ ~). For these equations to be consistent it is necessary that (after a little manipulation)
OA(o)
iflh (+) ei2kl~:C (0) = 0,
OC (o)
--"~-
iflh (-) e-i2k, eA (°) = 0
(3.20)
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379
on ~ = O, where
fl = 8k3pq3/{4k:(2k2- l)p -
(4k 2 - 1 ) q }
(3.21)
depends only upon a (and hence ~,), i.e. it is a material parameter. This dependence is illustrated in Fig. 2. The system (3.20) has the following solutions: A(°)(6, O) = M
e(~+ikl)~-t -
N e (-~'+ikQf, (3.22)
C(°)(~, 0) = fiT+) {(~ +
ikl)M e (~-ikp~ -
(~-
ikl)N e(-~-ikpe},
for arbitrary constants M and N and where
=~/fl2h(+)h(-)- k 2
(3.23)
may be real or imaginary depending upon the extent of the mismatch. The surface values B(°)(¢, 0) and D(°)(~, 0) may also be determined from (3.22) and (3.16). These boundary values now enable us to find the solutions to (3.11). If we seek such solutions by the method of separation of variables, and use (3.22) to yield the separation constant, then we readily determine the zero-order coefficients as A(°)(~,
71)=M exp{(¢+ikl)~+k (-i~'+kl),l}+ N exp{(-~'+ikl)~+k (i~'+k~)~l},
B(°)(~' ~1)-
-(2k2-1)[Mexp{ } 2 k (~'+q ikl)~+k-(-ir+k~)~l q +Nexp{(-~'+ikl)~+k(i1"+kl)l?}],
(3.24)
,7)=
-(7"-ik,)Nexp{(-~'-ik,),+k(-iv+k,)l?}], P i (2k2-1)[(~.+ik,)Mexp{(~._iki),+k(iT+k,)l?} D(°)(~' ~/) - / 3 h (+) 2----~ -(~'-ik,)Nexp{(-~'-ik,)~+k(-i~'+k,)~l}]. When these coefficients are used in (3.7) we have the zero-order solutions within the etched region. This has now to be 'matched' to the solutions (3.1) and (3.2) on either side of this region in order to determine the complex constants M, N, R and T. 4. Matching the solutions
It is not possible to amalgamate the solutions in the three regions to form an exact solution, if only because these solutions are only approximate. However it is possible to perform an approximate matching valid to the same order as the individual solutions themselves.
N.S. Clarke et al. / Scattering by surface roughness
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In what follows we consider terms only of O(1) and ignore all terms proportional to e unless accompanied by l, i.e. we retain terms depending up el as these contribute to the O(1) terms. Bearing this in mind it will be seen from the forms of (3.1), (3.2) together with (3.7) and (3.24) that matching to this order may be effected by insisting only that the displacements be continuous at x = 0 and at x = l on the surface, i.e, at y = O(e). This ensures that, to the first order, the displacements are continuous at x = 0, l for all y. We may set x = 0, y = O(e) in (3.1) and equate the resulting values of ul and u2 to those obtained from (3.7), together with (3.24), at the same location. This yields a pair of equations involving R, M and N. Another pair of equations involving T, M and N are obtained by a similar process by setting x = l, y = O(e). These four equations are readily solved to give M = {2k(r - ikl) e-:~'t}/{(r + ikl) + (~" - ikl) e-2~'t}, N = 2k(~'+ ik,)/{(z + ikl) + (~- - ik,) e-2~'t},
(4.1)
T = • eikl~l/{'r c o s h ( T d ) + ikl sinh(~'el)}, R = iflh (-) sinh(~-el)/{~" cosh(zel)+ ikl sinh(~-el)}.
We may note that exactly the same values of M, N, T and R would have been obtained had we matched the separate solutions using the more 'rigorous' (and more laborious) matching procedure of Tiersten [4] based u p o n a variational principle. The determination of M, N, T and R completes the derivation of the leading order representations of the displacements. From (4.1) it is seen that for [R[ to be a m a x i m u m it is required that k~=0, that is for m a x i m u m reflection the wave length of the incoming wave should be exactly twice that of the etchings. This m a x i m u m value is given by
IRI -- tanh D,
(4.2)
where D = / 3 ~ el is proportional to the (dimensionless) area removed by etching. In order to assess the sensitivity of IRI to a small mismatch we introduce the mismatch parameter K = kl/fl~. We then have that
2) -K2} -1/2,
for 0 < K < 1,
IRI = sinh(Dx/1
for K = I ,
[ R l = O ( l + O 2 ) -1/2,
for K > 1,
IRI = I s i n ( D x / K 2 - 1 ) I { K 2 - cos2( D x / K 2 - 1 ) } -1/2.
-K2){cosh2(Ox/-~-K
(4.3)
It is evident from the above that IRI is a slowly decreasing function of K for 0 < K < 1 but thereafter it falls away rapidly. Hence we may conclude that provided
kl <[3x/h(+)h (-)
(
(4.4)
then IRI will not depart significantly from its m a x i m u m value. It should also be noted from (4.1) that as the length of the etched region is increased then the amplitude of the transmitted wave decays exponentially fast, provided (4.4) holds, as 2z
ITI 4~ 2+k2
e -'l
as el~oo.
N.S. Clarke et aL / Scattering by surface roughness
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These results are in complete agreement with those o f Simons f r o m his case o f rectangular strips or grooves, i.e. where h (+)=i ,ff
and
h (-)=-i. 'IT
Further consideration is given to these results in a subsequent p a p e r where they are used to discuss the p e r f o r m a n c e o f a resonator, comprising two such etched regions either side o f an 'excited cavity'.
Acknowledgment One o f us (C.D.) w o u l d like to thank the S.E.R.C, for an a d v a n c e d studentship during the time w h e n this work was c a r d e d out.
References [1] D.A. Simons, "Reflection of Rayleigh waves by strips, grooves and periodic arrays of strips or grooves", Y. Acoust. Soc. Am. 63, 1292-1301 (1978). [2] H.F. Tiersten, "Elastic surface waves guided by thin films", J. AppL Phys. 40, 770-789 (1969).
[3] J.D. Achenbach, Wave Propagation in Elastic Solids, NorthHolland, Amsterdam (1973). [4] H.F. Tiersten, Linear PiezoelectricPlate Vibrations, Plenum Press, New York (1969).