An addition to the theory of the Rayleigh surface waves attenuation by surface roughness

•"i•

Solld State Communications, Vol. 68, No. I0, pp.903-907, 1988.

Printed in Great Britain.

0038-I098/88 $3.00 + .00 Pergamon Press plc

AN ADDITION TO THE THEORY OF THE RAYLEIGH SURFACE WAVES ATTENUATION BY SURFACE ROUGHNESS V.V. Kosachev, Yu.N. Lckhov, V.N. Chukov Moscow Engineering Physics Institute, 3 1 K a s h i r s k o e shosse, Moscow, USSR, 115409 (Received ~I September 1988; in revised form ~ ~ep~emoer 1988 by G.S.Zhdanovj

The problem of the attenuation of Rayleigh surface waves by surface roughness is solved in the first Born approximation of perturbation theory. A technique of the Green function developed by Maradudin and Mills [3] has been used on account of a principal correction whose mathematical treatment is given. When the correlation function of surface roughness has a Gaussian from, the attenuation coefficient in the short-wavelength limit ~ ~ ( ~ is the wavelength of Rayleigh wave, ~ is the transverse correlation length of the surface roughness) does not depend on the frequency, it tends to the constant value. The results of this paper coincide with the results for the attenuation constant obtained by the Rayleigh method by Eguiluz and Maradudin[6] . It is shown that the scattering of the Rayleigh wave into other Rayleigh waves in the region ~ i s a much more effective attenuationmechsnism than scattering into bulk waves. A strong dependence of the attenuation constant on the properties of an elastic medium is found.

I. Introduction

~(~l,~z)va~lshes and ~z=(~2(~, ~z) > , where the angular brackets denote an average over the ense~mble of realizations of the function ~ ( , ~ , ~ z ) . It is assumed as well that ~(z~,~2)is nonzero only within a rectangular region of the surface with linear dimensions Lz,Az It is necessary to calculate the scattered displacement field, the energy flow in the scattered wave, and attenuation coefficient of Rayleigh waves as a function of ~ and a , the wavelength ~ a n d Poisson constant ~ . In other words, the scattering problem must be solved, and the surface roughness is considered as the scattering centre. Such a formulation of the Rayleigh sUrface wave attenuation problem differs from that considered in the works [I,2,6,7], in which the surface l o c a l l z e d e l a s t i c displacement field satisfying the stress-free boundary oondltloms is obtained from the equations of motion of a semi-infinite elastlo medium with the infinite surface roughness. The formulated scattering problem will be solved in the first Born approximation of perturbation theory by a Green's function method developed b~ Maradudin and Mills [3]- However [5 ] pointed out a possible error. Eguiluz and Maradudin T6] also write about their

The present paper is devoted to the theoretical investigation of the attenuation of Rayleigh surface waves by randomly rough, stress-free surface of an isotropic elastic medium. Such investigations arouse cousiderable interest in connection with the study of the surface properties of solids and different technical applications. This problem has been studied theo r e t i o a l l y b y different methods in the works [I-7J. Contradictory results for the attenuation of Rayleigh surface w~ves by the slightly rough surface ~<~ ($ is the mean-square amplitude of the surface roughness) were obtained in these workse In the present work, an attempt is made to find the source of these contradiotions. 2. A Green's Function Method Let a plane monochromatic Rayleigh surface wave propagate across a stressfree randomly rough surface of an isotropic elastic medium, which occupies the half-space ~ 3 > O. The h e i g h t ~ 3 o f a point on the surface above the ~ 2 plane is given by the relation ~3=~ri.~l~. It is assumed that the average value of 903

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THEORY OF THE RAYLEIGH SURFACE WAVES ATTENUATION

disagreement with [3~. To clarify the reasons of these discrepancies we sugge~ to evaluate the technique in more detail. It should be noted that underlying a Green's function method [3] is the statement [8] that the surface states in the theory of elasticity may be studied by two equivalent methods. The first method consists in seeking the solution of the equations of motion of a semiinfinite isotropic medium, which occupies the half-space ~B • O

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space coordinates ~ ~ o g e t h e r with the elastic moduli ¢ ~ (~) in Eqs.(~) and Eqs.(5). And lot"the homogeneous elastic medium occupy~-~ the half-space :E 5 > O it must have the form

? By means of Eq.(6) Eq.(5) may be written as follows

(7)

satisfying the stress-free boundary O conditions on the plane surface Xo= =

o

where ~ (Z~) is the ~ Cartesian component of the displacement of the medium at point ~ and time t ; ~ is the mass density of the medium; C a ~ / are the elements of the elastic modulus tensor;

F= (x~, ~ , ~ ).

and the equivalence of the two approaches under discussion becomes obvious. The equations of motion (3), ~herefore, should be written in the form

The second method is to obtain the solution of the equations of motion of an inhomogeneous space

.-.

2

(8) Where the e l a s t i c moduli C ~ are l/ position-dependent. Following Ref. / 8 ] the authors of[9] represented the position-dependent elastic moduli in the form

C~p~O(~1

= C.~/O " O(X,)

(4)

where ~(X~)is the Heaviside step function with value unity when its argument is positive, and zero when it is negative. It follows from Eq.(~)that Eq.(~) can be rewritten as follows

)'>

Oa: )

where ~ and C ~ j x ) are the ordinary, position-lndependent mass density and elastic mcduli of the medium, respectively. Then the authors of work[9]state that the solution of Eqs.(5)or Eqs.(~) is equivalent to the solution ofEqs.(I). satisfying the boundary conditions (2). This statement, however, is not quite oorre ct.

For the equivalence of the two al>proache_~ trader discussion the mass density~(~d) must be a function of the

One s h o u l d e m p h a s i z e t h a t t h e m a s s d e n s i t y ~(~)in Eq. 8 is a function of the space coordinates ~ together with the C ~ v ( ~ + ) Ignoring this fact in the ~ ~ ~he plane surface of the medium does not lead to mistakes in the calculations. The situation, however, changes radically in the case of the rough surface. If the surface is rough and described by the r e l a t i o n X S = ~(xx X~)the expressions (4) and (6) become

When the amplitude of the surface roughness is small, C ~ # v (~) and jo(~) may be expanded in powers of the surface profile function ((~ijXz)

D

Only t h e f i r s t

o r d e r t e r m s i n ~(×~,~z)

are kept in Eqs.(ll). It is seen from Eqs.(ll) that, besides the term which is proportional to e(x~), a novel term appears in the expansion of ~(~), and it is proportional to ~(%3)" This term gives rise to considerable changes in the calculations and results.

THEORY OF THE RAYLEIGH SURFACE WAVES ATTENUATION

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It is this fact that was not taken into account in the work of Maradudin and Mills~3]. They supposed that only Ca~)are the functions of the space coordinates, while ~ was considered as the position-indepehdent mass density of the medium. We shall take this correction into consideration in our further colculations (see Eq.(IO) and Eq.(11)). 3. The Scattered Waves Following the technique described in detail in Ref.~3], the scattered wave has been evaluted at a large distance from roughness. The scattered wave is the sum of three contributions: (a) a bulk longitudinal wave; (b) a bulk transverse wave; and (c) a Rayleigh surface wave. The energy flow in the scattered wave has been calculated by means of the asymptotic expressions for the scattered displacement field. It follows from the expression for the energy flow in the scattered wave that the scattered Rayleigh wave vanishes, when the angle of scattering i~ equal to the values given by the relatio~u~

;

/zc

) ,

905

/.

~2

O,:i

o,q

Z.O

q,O

Fig. I. The results of a numerical calculation of the functions ~Bz~n~+~Bt and ~R when C±/.CL= ~ / ~ . '~ ' The ftmctions ~ and ~ are the results of Ref.['~]. -

where C t .C R - are the velocities of transverse and Rayleigh waves i~ the medium, respectively, azimyth T~ r~,n~ from 0 to 2~F. This result coincides with the conclusion of Ref.[5~, b u t is in contrast with that of Ref. 43~. 4. The Attenuation Length Rayleigh Waves

of

It is convenient to introduce the following designations (see Ref.[3] ) ~Z

2

.g

where ,~(0 ~ ( ~ L d ~.(R)are the contributions to the attenuation length from roughness induced radiation into bulk longitudinal waves, bulk transverse waves, and Rayleigh surface w~ves~ respectively; the functions ~B{, ~B~. ~R are dimensionless. We assume a Gaussian form for the correlation function of the surface roughness. The results of a numerical calculation of the functions are depicted in Fig. I and Fig. 2., t R) The attenuation constants I / ~ ~ ' in the long-wavelength limit @ ~ I C R < ~ are proportional to ~z~z ~ / C ~ . This result coincides with the results obtained in Refs. [1-7] • The contTibutions to the attenuation constant from roughness induced

~SL,~8~R

~8

~R I

~o,o

I

1

,~o,o

I

I

|

~

i

I

zoo,o ~ / X

Fig. 2. The same as Fig° I , but for the region 40 ~ / ~ < 400.

rad~.'a..t~on into bulk longitudinal waves ~/((C}and bulk transverse wave.s ~/{(Oin the short-wavelength limit o-~JIC R > > ~ deca~ exponentially with increasing frequency, while the contribution from

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THEORY OF THE RAYLEIGH SURFACE WAVES ATTENUATION

roughness induced radiation into R~ 7leigh surface waves tends to the constant C~z/~ 3

where

.

)

.

It follows from Eq.(I3) ~hat.~he._. attenuation coefficienC~/~=~/~%~/~$~/( ~' • enas to the constant value C ~ / ~ 3 with increasing the frequency of the incident Rayleigh wave. The scattering of the Rayleigh wave into other Ra~leigh waves in the short-wavelength limit is, therefore, a much more effective mechanism of the attenuation than the scattering into bulk waves° This is in accordance with the conclusion of Refs.[I,7], but in contrast with the conclusion of Ref.[3]. 5. Discussion It is interesting to compare the frequency ~ependence of the attenuation constant I/~ in the short-wavelength limit ~ < < 6 [ given by Eq.(I3) with the corresponding results of the other works o The following evaluation was obtained in Refs.[I,3]

(T~)

In Ref.[2] the multiple scattering was taken into account, and the following evaluation was obtained in the region ~ < T C R , where T is the attenuation time ~

z

(TS)

In Ref, [7]it was fo~uld that ~/~(~) tends t o the constant value C 6 /a 3 ,but the constant C' differs from the consrant C obtained in the present work (see Eqo(I3)). In Polsson's case, when 5 =0,25, one has C'/C ~ 6,5.

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As for the comparison with results of Ref.[6]it is necessary to note that the results of the numerical calculation of the present work in the region O,i- 4 0 (see Figs. 1,2) was not considered in Ref.[6]. Since the asymptotic expression for ~/~(R) in the short-wavelength limit ~ << was not calculated in Ref. [6], we have carried out these calculations independently using Eqs.(~.2~-4.29) of Ref. [6] and have obtained the expression which coincides with that given by Eq.(I3). In addition, the expressions for the scattered displacement field obtained in Ref. [3], Ref. [6J, and the present work are coincident provide~ the dependence of the mass densit-y~(~)(Eqs.I0,II) on the space coordinates is taken into account. It follows from these conclusions that the results of the present work coincide completely with those for the attenuation constant obtained by Eguiluz and Maradudin[6] o That is wh~ the statement of the authors of Ref.[6] that the scattering of the Rsyleigh wave into bulk waves is a much more effective attenuation mechanism than is scattering into other Rayleigh waves is not quite correct. Thus, one may state that when the problem of the scattering of elastic waves by a weakly rough surface is considered, a Green's function method gives correct results provided the mass density becomes a function of the space coordinates together with the elastic modull in transition from the equations of motion of a semi-infinite homogeneous medium to those of an infinite inhomogeneous space. The dependence of the attenuation coefficient on the medium properties • was investigated in the present work. A numerical calculation has shown that it stTong~7: depends on the Poisson constant b . The contmibutions to the attenuation constant from roughness induced radiation into bulk waves, and R ~ 7 1 e i g h .su~.f.age waves in the lofngwaveleng~n l~ml~ ~ > ~ are the same order of magnitude, when 0<~0,2S o When 0 , 2 ~ 0 , ~ " i the Ra~leigh wave/bulk wave scattering contribution is an o~der of magnitude larger than the Rayleigh wave/Rayleigh wave contribution to the attenuation constant. In the short-wavelength limit ~ < ~ the Rayleigh wave/Rayleigh wave scattering contribution to the attenuation constant is the predominant one for a~7 elastic medi&o

Acknowledgment - We are grateful to Yu.V.Gulyaev, Yu.M.Kagan, and L.A.Fal'kovskii for discussions on various aspects o f the problem and their judgements.

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907

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