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Progress in Surface Science, Vol. 32, pp. 39-110 Printed in the U.S.A. All dghts reserved.

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EFFECT OF SURFACE PHENOMENA IN SOLIDS ON SURFACE ACOUSTIC WAVES VICTOR V. KRYLOV Department of Physics, Moscow M.V. Lomonosov ~tate University, Moscow 117234, USSR

Abstract

A review of the modern status of the t h e o ~ of surface acoustic waves in solids with accounting the surface effects is presented. The main approaches are discussed to the determination of surface parametere~ surface tension, surface elasticity, and surfaoe mass density. The careful derivation of nonclassical boundary conditions describing the influence of surface phenomena is given. One of the topics for discussion is the specification in boundary conditions, which is due to the modes of description of the prestressed subsurface layer. The effect of surface phenomena on the properties of Rayleigh surface waves is analyzed in comparison with the results of classical approaches to the waves in layered media. ~ e a r surface acoustic waves caused by surface effects are also considered. One of the problems to discuss is the comparison of the phenomenological or semlphenomenological results with the results followed from lattice dynamics theories. The analysis is presented on the influence of surface phenomena on the reflection of bulk acoustic waves. Some speoial cases are described, in which the role of surface phenomena seems to be the most significant.

Contents

lo 2.

Introduction Thermodynamics of Surface Deformations

A. Gibbs definition of surface effects B. Surface energy and surface tension Surface stress tensor D. Surface elasticity and surface mass density On the nonuniqueness in definition of surface parameters F. A membrane model of surface effects Nonclassical Boundary Conditions Describing Surface Phenomena 3. A. E f f e c t o f s u r f a c e t e n s i o n

39

42 45 45 45 48 49 51 53 55 55

40

V.V. Krylov

So General boundary conditions C. Eulerian and Lagrangian representations D. Comparison wlth other theories 4. A. B. C. D. E.

~o

Shear Surface Waves Caused by Surface Effects A. Continuum model B. Dynamic theories and nonlocal elasticity approach. The role of spatial dispersion

.

ao B. .

ao B. O. D. 8.

Influence of Surface Phenomena on Rayleigh Waves General remarks Rayleigh wave dispersion caused by surface tension Connection with capillary waves in liquids Effect of surface elasticity and surface mass density Influence of crystal lattice discreteness on the contribution of surface phenomena

Effect of Surface Phenomena on Bulk Acoustic tion from a Free Surface Vanishing zero effects at Brewster angles Microscopical approach to the calculation of coefficients Some Special Cases Characterized by a Strong of Surface Effects Surface acoustic waves in highly anlsotropic Flexural waves in thin plates Localized vibrations of a sharp-angled wedge Laser excitation of sound in metals Conclusion

ai Cv c cI Cp ct D d E Eo

83 87 87

reflection 91 Influence 91 crystals

91 93 95 I00 106 106 106

Symbols

As

77 79 79

Waves Reflec-

Acknowledgement s References

A Ao

58 59 63 65 65 66 71 72

free energy of a bounded solid free energy per one atom surface free energy Lagrangian coordinates specific heat per constant volume surface acoustic wave velocity longitudinal bulk acoustic wave velocity plate longitudinal acoustic wave velocity shear (transverse) bulk acoustic wave velocity flexural rigidity of a plate the crystal lattice constant Young modulus nonperturbed value of mechanic energy of elastic semi-

Effect of Surface Phenomena on Surface Waves

41

space per unit surface area P(k),P(~)the Rayleigh determinant ~'(z) the function of inhomogeneity h characteristic thickness of a layer or a plate the space distribution of light intensity I(x) Jacobian of the transformation from Lagrangian coordinaJ K

Kem k

tes to Eulerian coordinates modulus of compression coefficient of electro-mechanlcal coupling projection of the acoustic wavevector on the horizontal plane; wavenumber of surface and wedge waves

ka

wavenumber of flexural acoustic waves

k1

wavenumber of longitudinal acoustic waves wavenumber of shear acoustic waves characteristic depth of penetration of surface acoustic

kt I

waves the mass of adsorbed atoms or molecules

Pc

the number of atoms in the element of volume or surface under consideration the number of atoms per unit volume or surface area the time-averaged energy of nonperturbed surface wave carried through the elastic semispace of unit width the Piola - Kirchhoff stress tensor the Piola - Kirchhoff surface stress tensor

S

the principal radii of surface curvature the surface area eiconal of the wave

N

n

S(x,y) T

tij

t~s ui uij ¥

hi Vlt

absolute temperature tensor of thermodynamic stresses tensor of surface thermodynamic stresses particle displacement vector deformation tensor the volume of the hod.7 the reflection coefficient of a bulk longitudinal wave for the incidence of a bulk longitudinal acoustic wave onto the solid surface the reflection coefficient of a bulk shear wave for the incidence of a bulk longitudinal acoustic wave onto the solid surface

42

V.V. Krylov

Vg

the group velocity of the wave

Ws

the mechanic work on increase of surface area

V 7

~ij ~,~ ~s, ~ s

P ps s Pad

@

el, e2

the coefficient of light absorption in the medium the coefficient of light transmission into the medium piezoelectric modulus the coefficient of surface tension surface stress tensor coefficient of t~ermal expansion Kronecker symbol dielectric constant the coefficient of heat conductivity Lame parameters for the volume of the body surface Lame parameters circular frequency mass density surface mass density additional surface mass density caused by adsorbed particles the Po~sson ratio tensor of mechanic stresses scalar Lame potentials the root of the Rayleigh equation angle of observation; angle of the elastic wedge the Brewster angles I. Introduction

The existence of surface acoustic waves in solids was first predicted theoretically by Lord Rayleigh still in the last century [ 1 ] . However, during more t h a n one hundreed-year h i s t o r y the surface waves were never studied with such intensity as at present time. In the first decades after the Rayleigh's invention they were investigated mainly for applications in seismology [2,~, and since the middle of 50-ths of a current century - for needs of ultrasonic testingS,5], then at the end of 60-ths the modern period of surface waves studies and applications began, which was characterized by the rapid growth of number of publications on this problem. One of the reasons of such increased interest t~ surface waves was the appearance of surface acoustic wave devices


43

for signal processing. This event has stimulated a lot of work on surface waves studies both in isotropic solids and in crystals, including their propagation, excitation,

scattering, electron

flow amplification, etc. [6-9]. Another important factor, which has promoted the accelerated development of surface waves investigations, was understanding the role that they play in surface physics [10-13]• From the very beglnn~ng of surface acoustic waves studies the question has arisen how much are these waves sensitive to the deviations of geometry or elastic properties of real solids from the idealized case of homogeneous elastic semispace, which was originally considered by Rayleigh. In other wor~s, in what degree the characteristics of surface waves do change if instead of a plane free boundary of homogeneous elastic media the real surface having a complicated geometry is considered, and the medium per se, moreover, is characterized by structure inhomogere~ies. For applications to physics of surface phenomena the most significant is the structure inhomogene~y in vertical direction, i.e., in the direction of a normal to the surface. Why, it is of common knowledge that even in the case of a free surface, created by the method of splitting a crystal along cleavage planes, i.e., in the case of perfect surface, a subsurface layer always exists being differed by it,s physical properties from them in the volume, because the upper surface atoms are in different conditions in comparison with the atoms in the volume of the medium [9-14] The subsurface layer under consideration is subjected to the action of constant residual stresses, and the subsurface atoms are characterized by changed coupling constants. The vertical dimensions of damaged subsurface layers are usually of the order of one or two interatomic distancies in a crystal lattice, i.e., @ 2 - 5 A. The layers of much more dimensions are created as a result of technological manufacturing of the surface. For example, even after the most careful mechanical polishing the damaged sub@ surface layer is created with a thickness of 40 - lOO A being provlded by different imperfections and residual stresses [15]. We shall call all above-mentioned phenomena surface phenomena, though historically one usually applies this term only to the case of atomically clean subsurface layers (for this situation the term "capillary phenomena" is also used). Such a generaliza-

44

V.V. Krylov

tion is reasonable since for the description of long enough wavelength acoustic waves the subsurface layers in all above-mentioned cases can be described by the same mechanical parameters: surface tension (for the subsurface layers created due to mechanical manufacturing the term"effective surface tension'is also used ~ ) , and surface elasticity and surface mass density. The study of the features of surface acoustic waves propagation in solids with taking into account the surface phenomena is important. First of all, from the point of view of better understanding the subsurface layer dynamics ~0-13], which is important for many modern branches of science and technology. Besides this, it causes a direct practical interest, mainly in acoustoelectronics, where the presence of technologically damaged subsurface layers influences on the behavior precise surface acoustic wave devices, in particular, resonators on surface acoustic waves. The most evident effect of damaged layers is the frequency dependence of surface acoustic waves velocities, i.e., physical and geometrical dispersion occuring due to the presence of a layer. More significant, however, is the long-time influence of the residual stresses initiated by the mechanical manufacturing. Their relaxation in time is responsible for a long-time nonstability of acoustoelectronic devices, i.e., for their aging [15]. Indeed, the effect of surface phenomena becomes stronger when the surface acoustic wavelength approaches the characteristic layer thickness. Since the resonators on surface acoustic waves are used in the frequency range up to 1 - 2 GHz, i.e., up to wavelengths about 1 mkm, the role of surface phenomena becomes noticeable enough even for typical damaged layer thicknesses of o

the order 50 A. The study of the effect of surface phenomena on surface wave propagation is important also from the point of view of possibilities of experimental estimation of subsurface layer parameters for atomically clean surfaces as for ones created by means of mechanical manufacturing. A special very interesting and practically important problem is the surface wave measuring the adsorbtion of the particles of surrounding gases or liquids onto the solid surface. In present work the attempt is made to describe from a single point of view the present status of the theory of surface acoustic


waves propagation,

45

especially Rayleigh surface waves, with taking

into account the effect of surface phenomena. We shall use mainly the continuum approach to the problem. But in some cases the results of discrete models of bounded crystal lattices will be discussed. 2. Thermod~namlcs of Surface Deformations A. Gibbs definition of surface effects We shall give here some necessary information about the parameters describing the mechanical surface phenomena

[18-22].

We

shall use the Gibbs definition of surface phenomena, following the spirit of Ref. [21]. Let us suppose the elastic solid to be bounded by the plane surface ~ , and let us define in this solid the volume V cutting on the surface a plot of the area S (see Fig. I). We shall define now the internal energy or the free energy of the semibounded sample with the volume V. In what follows we shall not - ~ e differences between these two quantities,

supposing for simplicity

X3

.-......." V

J

Fig. I. On the definition of surface effects.

that the absolute temperature of the body T is zero. Then, for example, for the free energy A of a real semibounded sample we may write

46

V.V. Kryl ov

A = NA ° + SA s,

(2.1)

where A ° is a free energy per one atom in the depth of the body, N is the number of atoms inside the volume V. The value A s represents the so called specific surface free energy in the Gibbs sense [18]. In the case of liquids this quantity is often known also as surface tension [21,23] (the explanation of this fact will be given below). Similarly, the definitions of surface entropy

and surface e n t h a l p y m a y be introduced to develope the sur-

face thermodynamics [21,22,24]. It must be underlined that surface quantities introduced by such a manner are not the characteristics of any two-dimenslonal hypothetic surface or a film, as, e.g., in Ref. [25-27], but they are realy surface excesses of corresponding physical quantities caused by the material properties deviations from homogeneity. During the reversible isothermic processes in the body the change in a free energy dA is equal to the work done with a body. Let us at first suppose this work for simplicity to be determined only by changes of a volume of the body dV and by changes of it's area dS. Then the change in a free energy can be written as follows:

aA = (aA/aV)sdV

+

(aA/~S)vdS

.

(2.2)

Here the supposition is used that the volume and the surface area of the body can be changed independently. With accounting the relation

(2.1)

the work on surface area increase takes the form

dW s = (SA/SS)vdS =

A s + S(SAS/~S)v dS .

(2.3)

B. Surface energy and surface tension Still Gibbs has indicated two different ways to increase the surface area: by creating a new surface, for instance, due to a splitting, and by the deformation of the initial surface. In the first ease 8AS/Ss = 0, but in the second case ~As/~s ~ O. Using the notation 7 = As + S(~AS/SS)v

,

(2.4)


47

one can write Eq. (2.3) in more compact form: dW s = 7~s

The quantity

(2.5)

.

,( is usually called surface tension. It is evident

from Eq. (2.4) that the surface tension ,( is equal to the free surface energy A s only if ~As/as = 0. It is the same situation, which takes place for liquids, where these two quantities in fact are identical. For solids the surface tension, in general, is not equal to the surface energy, and, in particular, may be negative, though the free energy is always positive. We shall turn now to more detailed discussion of the introduced surface parameters, in particular A s. Let us use a continuum model and consider a free energy per one atom as a function of a normal coordinate

x 3 , i.e . ,

AO

=

AO(x3)

, according to Ref. [ 21 ] .

Denoting the. number ofgatoms per unit volume through that

n(x 3) , so

N = Jn(x3)dV = S]n(x3)dx 3 , we obtain ?

-m

A = __S]A°(x3)n(XB)dX 3 . Since in the depth of the body (2.6) can be rewritten as

(2.6)

A°(x 3) = A ° = const, the equality

(2.7)

A = NA° + S~[A°(x 3) - A°]n(x3)dx 3 . From the comparison of Eqs. (2.7) and (2.1) it follows that As

:j[A°(x

I f the dependences of A°(x])

3) - A°]n(x3)dx 3

.

(2.8)

and AO on S are known, f o r

example as a result of microscopic calculations, t h e equality (2.8) can be directly used for the calculation of T and excess work dW s due to the variation of surface area dS. From another side, this work can be determined if the stresses 6i~(x 3)~ in the volume of the medium are known. One has to note that, according to the boundary conditions on a free surface 6i3 = 0 for x 3 = 0, in all the body only the components 60~(X3) differed from zero, where Greek indices ~ and take the "surface" values 1 and 2. Such a conclusion follows

48

V.V. Krylov

from the solution of the static equilibrium problem for the case under consideration. Let us suppose now that the body is subjected to a depth-independent deformation, which is described by the small deformation tensor

uij = !(aui/ax j + ~uj/ax i) , 2

where Latin indices i and J take the values 1,2,3. And let us be limited for a beginning by the case of symmetric stress state dc~(x 3) = T(x3)6 ~ , where ~ is a Kronecker symbol. Then the change in free energy of the body dA associated with the variation of a displacement vector u i by a small quantity 6u i will take the form [281 °

dA =

~(x3)6uo~V

= ~T(x3)dx3]dS

,

(2.9)

where dS = S6uc~ ~ . Note, that Eq. (2.9) is exact, in spite of the simplification followed from the use of the above written linearized deformation tensor uij [28,29]. One must take into account, however, that tensor 6c~ is defined relative to the deformed surface. With the help of the relation T(-~) = T = const, the equality (2.9) can be rewritten in the form analogous to Eq. (2.2): 0

The work sed from

dW s on the increase of surface area is evidently exprem Eq. ( 2 . 1 0 ) a s 0

(2.11) Comparing this relation with face tension

Eq. (2.5), one can obtain

3)

-

for sur-

.

C. Surface stress tensor If the stress state in a solid has an arbitrary

symmetry, the


49

surface tension only is not enough to describe the excess work done with a solid. For a description of this situation a more general quantity such as the surface stress tensor is used: O

This tensor was at first introduced by Shuttleworth [19] and Herring [20] , and then it was discussed widely by other authors [21,22]. With the use of the tensor written in the form

~c~

the expression for

dA

can be

(2.14) Note, that for surface stress tensor ~c~ , as well as for usual bulk stress tensor ~iJ ' it is valid the equality Tc~ = ~pm , which follows from the static equilibrium condition [19]. Therefore, from four components of ~c~ only three are independent. D. Surface elasticit~ and surface mass densit~ It is evident that definition (2.13) does not exclude a possible dependence of ~c~ on deformations. u c~q , that wa S also pointed out by Shuttleworth and Herring ~19,20J. In the first approximation in uc~ we have °

+

S

,

(2.15)

S

where c o ~ ~ are the so-called surface elastic moduli. In particular, i f t h e s u r f a c e i s i s o t r o p i c or if the corresponding face of a crystal has third- or high-order symmetry axes, then Vo%e

=

"f~o~3 + kS~o%eUvv + 2~Suo%e '

(2.16)

where ~s and ¢u s have a sense of surface Lame parameters, and is the above-introduced surface tension determined now as a half-trace of the surface tension tensor 7~°r~ = ~ . The presence of the tensor T°~B, independent of deformations, in the expansions (2.15) and (2.16) describes the fact that the subsurface layer is, in general, under stress state. The absolute

50

V.V. Kryl ov

values of surface tension of capillary origin are of the or~er 103 erg/cm 2 (or dyn/cm) [21], whereas for techaologically clean e d surfaces they may be by 2 or 3 orders greater [17]. The surface elastic moduli in their turn may in principle be determined by the relations S

O

[c 6(x 3) -co

=

]a= 3 ,

(217)

where c o ~ 8 are the elastic moduli in the depth of the medium. One must take into account, however, that moduli c ~ 5 , which are used in Eq. (2.17) , are, generally speaking, different from the corresponding moduli Cijkl , where indices i,j,k, and i take the values I and 2. The reason for this is caused by the fact that due to the presence of a free surface the deformations in the plane x I , x 2 are accompanied also by the deformations in x 3 direction, which give their contribution to the stresses 6~ through the corresponding elastic moduli. Therefore, the quantities co~ ~ in (2.17) represent the renormalized values of usual bulk elastic moduli with taking into account the solution of the static boundary problem [30]. We shall return %o this question once more in dlscuSslng the relation between the boundary conditions obtained with the help of Eq. (2.16) and the approximate boundary conditions for the case of homogeneous thin films on solid substrata [31]. In what follows we shall need also the surface mass density ps, which can be introduced, for example, as a quantity describing an excess work against the inertial forces, when the body is displaced as a whole by the distance

dl : 0

= P~iVdl It follows from

+

Eq. (2.18)

~is~S[2(x 3) -9]~3

•

(2.18)

that O =

It

_

•

(2.19)

is evident that the above-lntroduced surface parameters y , kS,~uS, ~s may be in general positive or negative, because they describe the "surface excesses" of physical quantities in Gibbs sense; the relations between these parameters may be arbit-


51

rary enough. For perfect atomically clean crystal surfaces, a freedom in signs of ~ s M s , ~ s is limited; for example, the nonequality M s ~ 0 is true. This condition is responsible particularly for the possibility of the existence of shear surface waves of Love-type even in crystals with a center of symmetry, i.e., in nonpiezoelectric crystals. We shall speak about this in detail in the section 5. The values of surface elastic moduli of capillary origin, for instance, ~ s , can be estimated as dud , where ~ is the bulk shear modulus, d is an interatomic distance. Then in the case under consideration ~ s is of the order of 103 dyn/cm. The relative density change in a subsurface atomic layer is usually not large (about some percents [21] ); and one can use the estimation s~ 0.i ~ d ~ 10 -7 g/cm 2. In the case of mechanically cleaned surfaces and, of course, in the case of thin films covering the solids the dispersion in the values of surface elastic moduli and surface mass density may be very wide. E. On the nonuniqueness

in definition of surface parameters

During the consideration above we used the plane x 3 ~ 0 , which coincides with a free solid boundary, as a reference plane for determination of all surface parameters. In principle, for the problems of surface waves propagation along the free surfaces, which will be considered in present work as a rule, one could be restricted by such a definition. But in general case it is indeed not necessary to choose the reference plane along the free solid surface. Moreover, for example, in the case of interfaces between two different solids, where some diffusion processes occur, the natural reference plane for determination of surface effects (in this case it is more correct to call them boundary effects) does not exist. It was Gibbs ~18], who has pointed out at first on this fact of nonuniqueness in definition of surface effects (see also Ref. E32-34]). This nonuniqueness, of course, has no influence on the description of physical phenomena related to a surface. In particular, it has not to be displayed in surface acoustic waves characteristics. The uniqueness in description of physical phenomena in all these cases is provided by quite unique relations between the values of surface parameters determined relative to the different planes of reference.

52

V.V. Kryl ov

To establish these relations in most simple way one can start directly from the definitions of surface parameters. Let us show, for example, how the surface stress tensor ~ e determined for a free crystal surface is changed after transformation from the reference plane x 3 = 0 to the plane x 3 = -h, where h does not exceed the thickness of a subsurface inhomogeneous layer. We shall start from Eq. (2.14) for the work dA , rewriting it in the form

da = do~6u ~ Here

~

? + hdo~ua~S

+ T~Su~S

.

(2.2o)

is the volume of the medium bounded by the plane x 3 = -h

instead of the plane x 3 = O. The second and the third terms in the right-hand side of Eq. (2.20) describe evidently the contribution of surface excesses to the quantity dA relative to the plane x 3 = -h. Therefore, the surface stress tensor in this plane is described by the expression

T~(-h) = V~s(O) , ~o~h

,

(2.21)

which characterizes the above-mentioned relation between the values of surface parameters. If the stresses in the bulk of a crystal are absent in the absence of deformations, i.e., the Hooke's law holds

go~ = Cc~klUkl

,

where Ukl = ( i / 2 ) ( S u ~ S x I + 8Ul/SXk) is the tensor of small deformations (or the linearized deformation tensor), and Cijkl are bulk elastic moduli, then, with taking into consideration the boundary conditions on the free crystal surface ~i3 = 0 giving the relations between some of the components of the deformation tensor Ukl , one can express the components g~s only throug~ the surface deformations u~5 , as was mentioned above. For example, in the case of isotropic body, where Cijkl = X g i j S k l + ~ ( S i k S j l + 6ilSjk ) , X and ju are Lame elastic constants, one finds ~c~s = k ' U w S o ~

+ 2~um9

,

(2.22)

Effect of Surface Phenomenaon Surface Waves

53

where )~' = 2 ~ j u / ( ~ + 2Ju) is the renormalized value of the modulus ~ with taking into account the effect of free surface. In particular, for one-dimensional deformation Ull it follows from Eq. (2.22) that 611 = (~' + 2jU)Ull =

4~u()~+J~)/(k+

2J~) Ull.

(2.23)

Note, that in relations (2.22) and (2.23) we have used the boundary conditions in classical form ~i3 = 0 , i.e., without considering any contribution of surface phenomena, since they could result only in the corrections of the order h 2 in the relation (2.21). Using the expansion (2.16) for a surface stress tensor T ~ , one can rewrite Eq. (2.21) for an isotropic body in the form

T~(-h)

= ~

+ ( k s + hk')~o~u w +

(2.24) + 2(ju s + hJa)uc~e • Thus, the surface elastic moduli depend on the location of the reference plane according to the relations ~S(-h) =)iS(o) + h~' and juS(-h) = ~uS(o) + hj~ . The surface tension T , as we can see, does not depend on h being quite unique invariant characterizing the surface tension forces £33]. Underline, that such a situation takes place only in the absence of a stress condition in the bulk of the body. In conclusion we shall give the relation, which connects the values of surface mass density ralative to the different planes of reference. Writing the expression (2.18) in the form similar to Eq. (2.20), we obtain

pS(-h)

= pS(o)

+ hp .

(2.25)

F. A membrane model of surface effects The influence of inhomogeneous subsurface layer may be considered also with the help of the model of elastically-stressed thin plate (or elastic membrane) of another material rigidly connected with a homogeneous elastic semispace (see, e.g., Ref. [25-27]). In such a model the mass density of the medium as a function of a normal coordinate

x3

has the form

~(x3) = ~,

if 0 > x 3 ~ - h ,

54

V.V. Krylov

and ~(x 3) = ~ if x 3 < -h , where h is the thickness of a plate. The surface mass density ~ s is determined by the relation

j)s = l i m ( ~ ' h ) h~,O where it is supposed that The surface tension du s

~

,

(2.26)

jo, -~ i/h . and the surface elastic moduli )~s and

(we shall be confined here to the

isotropic case) have not

such a clear sense in the framework of a membrane model, and they are introduced

only

formally as corresponding e@efficients in

the expansions of some "surface" tensor

To~ , which characterizes

the stresses acting per unit length of the elastic membrane contacting with ~ body ( or the "material surface", according to the terminology of

Ref.[25,26]):

•~

= Tgo~ + kSuw~o~ + 2@ASuo~ .

It is also supposed here that the layer thickness

(2.27) h

tends

to

zero. She physical sense of the moduli )ts and d~s introduced in this way is usually clarified by comparison of equivalent boundary conditions obtained with the use of M s and ju s = 0

in a case

of

with the well known Tiersten,s boundary conditions [31].

Note, that the reference plane in the cases considered in [25-27] is the plane cribed in

Ref.

x 3 = -h . With the use of simple rules des-

previous paragraph all above-mentioned surface para-

meters can be easily transformed into their values referred to the free surface x 3 = 0 . One of the disadvantages of the introduction of surface parameters from a membrane model, in addition to the above-mentioned artificiality of determination of

.f , )s, and @A s , is the need

to consider a subsurface inhomogeneous layer as a continuum. Note, that the Gibbs definition of surface parameters does not need such a limitation. A subsurface layer in the last case may consist of individual nonconnected inclusions of adsorbed atoms or any other defect.

It is only important to suppose that the number of

such inclusions N per macroscopic plot of the surface with an area S is large enough. In particular, for the layer of sparsely placed surface

adsorbed atoms with a mass

M

one can choose the "clean"

x 3 = 0 as a reference plane. Then, relative to this plane

Effect of Surface Phenom~ena on Surface Waves

55

the i ~ l u e n c e of adsorbed atoms will be displayed through a~ditional surface mass density P~d = MN/S 3. Nonclassical Boundar~ Conditions Descrlbin~ Surface Phenomena A. Effect of surface tension The contribution of surface phenomena ~O the acoustic fields formation near the solid surface is naturally to be described By the use of specially constructed boundary conditions. It is useful for a beginning to obtain the boundary conditions with accounting 9 ~ the surface tension [3~. Together with methodological r e a s , ~ soncerned to the simplification of derivations, such a consideration causes also an independent interest, since, e.g., the influence of residual stresses in subsurface layer on wave propagation may be sometimes greater in comparison with other parameters, which will be discussed later. Consider any deformed body and suppose the displacement vector u i to be changed by a small quantity 8u i . The corresponding work done by the elastic stresses can be written in the form[28] ~A =

(adij/~xj)auidV

(3.1)

Integration by parts yields 8A = ~ 6 i j ~ u i d f j - fdij(~Sui/~xj)dV ,

(3.2)

where dfj are the components of the vector d~ of surface element, directed along the outward normal to the surface. It is evident that the second term on the right-hand side of Eq. (3.2) describes the work done by the elastic forces in changing the strain tensor in the volume of the body. By exact analogy, the contour integral in Eq. (3.2) can be regarded as the work done by the elastic forces in changing the particle displacement of the surface by a small quantity ~u i . To take into account the surface effects interpreted in Gibbs sense it is necessary to add here the work of the above-consldered surface stresses done in deforming the body. Assuming this work to be characterized only by the surface-tension tensor ~ o and considering for simplici-

56

V.V. Krylov

ty the isotropic surface, we can wrlte it in the form ~ S , where 5S is the variation of the surface area in deformation and T is the usual surface tension. We shall write the work of the surface-tension forces, introducing normal surface coordinates x n , x t , x¢ , which are measured along the outward normal and two mutually perpendicular tangents to the surface, respectively. The strain-induced variation of a surface element can be caused by both tangential and normal d/splacements of points of the surface. In the first case the area variation is proportional to the variation of the trace of the corresponding components of the strain tensor 5u w , associated with the displacement

8u v :

= dfn Uv where the subscript

v

= dfn(a uv, Xv)

takes the values

t

,

and •

(3.3) . The work of

the surface-tension forces in this case has the form

BW(1) =

7(a

%/a

v)df n

(3.4)

Integrating by parts and assuming that the boundary variations ~u v

are equal to zero, we obtain ~W (I) ---~(~/SXv)~uvdf n .

(3.5)

The variation of the surface area under the normal displacement 8u n can be expressed in terms of the principal radii of curvature of a given point of the surface - R 1 and R 2 (see Ref.[23] ): 6f~ 2) = dfn(i/R 1 + i/R2)Su n

.

(3.6)

With the help of Eq. (3.6) one can obtain the following expression for the corresponding work: ~W (2) = ~ I I R

1 + llR2)gUnd~n •

Thus, the resulting work of the surface-tension forces = SW (1) + ~W (2) can be written as

(3.7)

SW s =


57

Let the surface under consideration be free and in a thermodynamic equilibrium state. It is known, that the total surface work must be zero in this case. On the basis of the above-obtained expressions, we write the thermodynamic equilibrium condition ~

[6in~U i - ( ~ / S X v ) S U v + y(I/R I + i/R2)SUn]dfn = 0,

where the subscript

i

takes the values

(3.8)

n , t , and • . Inas-

much as equality (3.8) must hold for any ~u i , we obtain the following boundary conditions with regard to surface tension:

~vn"

b7/bXv

~ o , (3.9)

6nn÷V(1/R

1

+ 1/R 2) = 0 .

The physical sense of the conditions

(3.9)

is clear enough.

The first equality implies that the surface-tension gradient, which can occur, for example, on a curved surface of a crystal or in the presence of temperature gradient, m~st be equalized by the internal shear stresses. But if the surface tension ~ is constant everywhere, then the internal shear stresses do not appear. The second equality (3.9) is the analog of the familiar Laplace formula for liquids and does not require further comments. It is clear, that for ~ = 0 the expressions (3.9) coincide with classical boundary conditions imposed on the components of the elastic stress tensor in the case of a free surface of a solid. Note, that the second boundary condition in Eq. (3.9) can be used directly for the determination of static stresses and strains in a bounded solid with a constant value of ~ . In particular, for small spherical particles of radius R the above-mentioned boundary condition must be added to a static equilibrium equation for a spherically symmetric case ~ 6 n ~ n = 0 and the HookS's law in the form 6nn = KUmm , where K = A + (2/3)#/ is ~he modulus of compression. Taking into account that the trace of a deformation tensor represents the relative change of the volume

58

V.V. Krylov

of the body ~ = dV/V : 3dR/R , one o&~ easily obtain the simple relation connecting the relative variation of a particle radius, i.e., deformation, with surface t@nSiOn: dR/R = -2T/3KR . This relation coincides in the notation with the correspohding expression for a capillary deformation of a liquid drop, Wl~ich was used by Nicolson [36] for experimental evaluation of ~ in solids by means of a diffraction of Roentgen rays by small particles. Returning to the first boundary condition (3.9) , one must note that it describes also the static action of surface-tension forces on the bodies with plane boundaries in those cases where the surface regions with a non-zero ~ are limited in size. Such a situation occurs, for example, ib thin plates, which are polished on one face and ground on another. Shear stresses ~vn = 8~/~Xv ' resulting on the region boundaries, give rise to an interesting effect as one-face spherical flexure of plates in the direction of polished surfaces - the so-called Twyman effect [17,37]. B. General boundar~ conditions Let us now take into account the influence of surface elasticity and mass density, using the expression (2.14) for a surface work in terms of a surface-stress tensor ~c~ ' which depends on surface deformations u75 . Repeating the previous speculations and adding the inertial forces _gsGi caused by a surface mass density pS we obtain the expressions[38]

d v n - ~7v~,p(uy~)

,9SGv = o ,

(3.lO) 6nn + 7(u~6)(i/R 1 + I/R2) + 9SUn = 0 , which generalize the boundary conditions (3.9) . Here comma denotes the differentiation with respect to X~ ' points are the differentiation in time, and the qu?ntity ~(uv~) denotes the deformation depending surface tension. ~(u~5) = (1/2)T~(uv~) . If we shall suppose that in nondefo~med state the surface under consideration is plane and the curvature arises only due to the normal displacements u n , then the sum 1/R 1 + 1/R 2 can be approximately written in the form [23] 1/R 1 + 1/R 2 = -Un, c~c • With this relation the boundary conditions (3.10) take the form


-

+ PscW

= o

59

,

(3.11) 6nn - ~an, crc + PS{in = 0 . If we use now the expansion of surface-stress tensor ~ on degrees of surface deformations (for certainty consider an leotropic case and write only terms, linear versus u ~ (see equality (2.16))), then we can transfer from Eq. (3.11) to the description

dvn -

).Su~, v - 2 ~ S u v p , p

+ ]geev = 0 , (3.12)

dnn - ~fUn,~¢

+ ]gs~n

= 0 .

For T = 0 the conditions (3.12) coincide in form with the well-known approximate Tiersten,s boundary conditions [31] , which are valid for uniform thin films of thickness h with the mass density ]9' and the elastic constants ~' and da' . The quantities ]gs, ~ s , and )~s in Eq. (3.12) have in this case the next i n t e ~ pretation: )~s.

]gs = (]9' -]9)h , ~ s _ (ju, -jm)h , and 2/a') - 2 ~ / ( > . + 2 ~ ) ] h . Note, that

[2k'd~'/(k'+

in

Ref. [31] a reference plane x 3 = - h was used rather than x 3 = O, as in present work. Therefore, the original Tiersten's boundary conditions have some other appearance, according to paragraph 2E . Without accounting the surface effects the obtained boundary conditions (3.12) go over to traditional boundary conditions of the classical theory of elasticity: dvn = 0 , dnn = O . The same result is obtained evidently after the transition to a lowfrequency limit in Eq. (3.12) as well, if the displacements ui describe the wave fields. The additional terms caused by surface effects become negligibly small in this case. C. Eulerian and La~ran6ian representations In this paragraph we shall discuss one delicate question that one must keep in mind by using the above-considered boundary conditions. The fact is that we by no means used the specific laws and terms of nonlinear elasticity, whereas the presence of a prestressed subsurface layer per se indicates that nonlinearity might play there a definite role. The different approaches to a description of such nonlinearity cause the different forms of boundary

60

V.V. Krylov

conditions. In particular, the first boundary condition in Eq. (3.12) differs from the corresponding condition of Ref. [33] by the absence of the additional term -~Uv,~ s , which was obtained by the author of Ref. [33] as a result of the accounting a nonlinear addition in the expression for a full deformation tensor in a term containing the surface tension ~ in the expansion of a surface free energy on degrees of deformations. As will be shown below, the above-mentioned disagreement is caused by different determinations of surface effects in both works. By the transformation to a uniform system of description the boundary conditions become identical. In the approach that we use all the nonlinear speciality of the problem is described by the surface stress tensor ~o~s ' which is determined versus a length of a deformed curve. This tensor characterizes the truthful stress condition of the surface; in other words, the quantity

~o~s is the symmetric Cauchy stress

tensor

[29] , or, according to Truesdell,s terminology, the tensor of mechanic surface stresses (see, for example, Ref. [9,39,40] ). As was mentioned earlier, the expression for a surface work dW s (see formula

(2.14)), which uses this tensor, is redly exact if

under uc~s a tensor of small deformations, namely uc~s = (1/2)" "(u~,~ + u~,~), is understood [28,29]. This tensor is derived from the full deformation tensor G ~ = (1/2)(u~,~+ u~,o~+ Uv,~Uv, ~) as a result of neglecting the nonlinear term. It is supposed in this case that in boundary conditions

(3.9)-(3.12)

the space deriva-

tives are taken with respect to Eulerian coordinates x v , and the value of surface mass density j~s is related to a deformed state. Returning to the first boundary condition (3.11) , it is easy to see that for the variables describing surface effects the full analogy occurs with a mechanical equation of motion in the volume of the medium if the mechanic Cauchy stresses are used: P~i = gij,j "

(3.13)

As it is well known, in nonlinear theory of elasticity this equation remains valid. However, one must clearly understand that corresponding space derivatives are taken with respect to Euleriis determined an coordinates [39,40] , and the mass density ~


61

versus a deformed volume. With the use of Truesdell introduced Piola - Kirchhoff stress tensor Pij connected with a mechanic stress tensor 6kq by the relation 6kJ = (i/J)Xk,iPiJ ' where J = d etlaXk/aail is the Jacobian of the transformation from Lagranglan coordinates a i to Eulerian x k (note, that Eulerian and Lagranglan coordinates are related through the displacements u i = x i - a i , and the Jacobian binds the mass densities in deformed 2 and in nondeformed ~o states: ~o/~ = J), one can transfer by the well-known way [9,39] from the equation (3.13) to the equivalent equation ~oUi = PiJ,j

(3.14)

'

where a comma denotes now the differentiation with respect to Lagrangian coordinates aj . Introducing an analogous Piola - Kirchhoff surface stress tensor it is easy to be convinced that the first boundary condition (3.11) in a Lagrangian description takes the form J~vn - Pv~,~ S S- = 0 . ÷ pour

(3.15)

To express the surface Piola - Kirchhoff stress tensor p s through deformations with the help of surface moduli of elasticity c ss ~ one must transfer from a mechanical stress tensor dvn and s to symmetfrom the surface Piola - Kirchhoff stress tensor Po~ ric tensors of volume and surface thermodynamic stresses tij and %s 6 , which are connected with d and pS by the relations s

dij = (1/J)Xj,kXi,mtkm and P ~ relations into equality (3.15) + u~, v , we obtain

Since (3.16)

s

ow

t" t

s + + ua,.s)tom- ~(~vm + Uv,~)to~

c,Sv. + + po %

vn

= Xobvt~v . Subs i u t i n g these and accounting that X#,v = ~ v +

=

(3.16)

o .

t ~ n = O(uF?) , b u t t ; c ~ = 0 ( 1 ) , t h e n n e g l e c t i n g i n Eq. so t h e t e r m s q u a d r a t i c i n u ~ , and d e n o t i n g t h r o u g h t c ~

the zero-number terms in the expansion of deformations, we obtain the equality

se tc~

in degrees of

62

V.V. Kryl ov

tv ~ - tvj,,j s

-

~

tso~

+

.P

po %

=

o

(3.17)

which coincides with the corresponding boundary condition of Ref.

[33]. In a symmetric case, i.e., for Eq. (3.17) that tvn -

S

tv~,p - ~fUv,~ +

tsn~ °~ = ~ o ~

S**

#oUv =

0 .

, it follows

from

(3.18)

The second boundary condition in Eq. (3.11) in the approximation under consideration conserves it,s form during the transformation to a Lagrangian description and thermodynamic stresses: tnn-

~Un,~+

s,.

poUn

= 0.

(3.19)

Thus, the equalities (3.17) dud (3.11) are related by standard transformations, and this indicates their physical equivalence. The choice of one or another mode of description is determined therefore only by the matter of a taste and comfort. Note however, that the tensor of thermodynamic stresses t ~s s does not reflect the truthful deformed state of a surface and has in significant degree a formal sense, relating the change of thermodynamic potential with the change of a full (nonlinearized) deformation tensor. The surface moduli of elasticity characterizing the linear terms in the expansion of a tensor t L in degrees of deformation u ~ are the moduli of elasticity for elastically stressed medium. In principle, they can be expressed in terms of inhomogeneous (versus x 3) linear and nonlinear bulk moduli of elasticity for an inhomogeneous layer with a known value of T . However, this problem is very difficult and tedious, and we sh~ll not discuss it in detail (the main ideas and techniques of calculations can be found in Ref. [3, 15, 39-43] ) • The same conclusions may be related also to the tensor of surface stresses in Eulerian representation ~ s ' which coefficients of expansion on degrees of deformations, i.e., surface elastic moduli (see Eqs. (2.15) and (2.16)), are also determined in a stressed condition. No one of both methods of description has advantages over the other; the results obtained with their help will differ from each other by a presence or absence of an


63

addition T to usually unknown surface elastic moduli in a stressed state (of course, the differences between dij and tij , as well as between 2s and ~ , in the linear approximation versus u i may not be taken into account). In what follows we shall use for clarity the boundary conditions in the form of Eq. (3.11) and Eq. (3.12) , not speaking more about their nonlinear nature. D. C o m ~ a r i s o n w i t h

other theories

One of the first derivations of nonclassical boundary conditio n e has been made by the authors of Ref. [25,26], which was based on the membrane model of surface effects. In this derivation the above-mentioned nonlinear speciality due to the presence of a prestressed subsurface layer has been taken into account in terms of Piola - Kirchhoff stress tensor. As a result, the authors have obtained the boundary conditions in Lagrangian representation, which were used then for solving the wave problems E44-46]. These boundary conditions, however, seem to be not quite correct, since they do not transfer after the standard transformations into corresponding boundary conditions for surface stresses in Cauchy sense. The authors of Ref. ~27], who had also used the membrane model, have obtained the boundary conditions, which coincide in fact with the above-written boundary conditions in Eulerian representation (see Eq. (3.11)). But in this work there was not taken an attention of the speciality of an elastically stressed subsurface layer. The generalization of nonclassical boundary conditions to the case of boundaries between two rigidly connected solid semispaces has been made in Ref. E27, 32-34, 46] for the description of boundary waves of Stonely-type. Thus, in Ref. E32-34] it has been pointed out a need to take into account the jump of displacements on the boundary caused by surface effects, that has not been noticed in Ref. [27, 462 . Meanwhile, the account of a jump in the displacements appears to be a principal need to describe the elastic waves localized near the plane inhomogenelies of the stacking fault type ~47]. By such a way the disagreements, indicated in Ref. E48] by comparison of the results obtained by different methods [27, 49, 50] , are eliminated. Let us turn again towards the Tiersten,s boundary conditions for structures of thin films on solid semlspaces [31J, in which

64

V.V. Krylov

all the above-mentioned nonclassical boundary conditions containthe case T = 0 . If the film thickness h is a function of coordinates along the surface (Fig. 2), then the Tiersten's conditions, as it is known [51], can be generalized to this case by adding the terms, which are proportional to the first derivatives of h . If the film is of the same material as the substrate,i.e., )~, = ~ , ja, = ju , and ~ , =jo , then from the boundary conditions of Ref. [51] follow the well-known boundary conditions describing the influence of small topographic surface inhomogers~ies.

X3 X£

t

Fig. 2. Inhomogeneous layer like a film of variable thickness. These conditions, which are usually obtained by using the Rayleigh hypothesis [52, 53], play an important role in the theory of surface acoustic wave scattering at the i n h o m o g e r ~ i e s like grooves,

[52-56].

pits, etc. It is not difficult to show that if one will suppose the surface parameters )~s , ~A ,s ~s, and T in Eq. (3.11) to be variable, then in spite of can be obtained: 6vn-

Eq. (3.12)

~r/Sxv

-

the more general boundary conditions

)~Suctc~,v - 2juSuvs,js +

v

=

o,

(3.2o) ~ n n - 7Un,~o~ +

pS~n

= o .


If

T = 0

and

x2

and the dependences of )~s , ju s , and

j~s

65 versus

xI

are caused only by variations in thickness of the layer

h = h(x I , x 2) , then the boundary conditions (3.20) coincide in form with the conditions of Ref. [51] (note, however, that in Ref. [51] only two-dimensional case h = h(x l) was considered). In particular, if the elastic properties and the mass density in a layer do not differ from their values in the depth of the body, the expressions (3.20) represent the equivalent boundary conditions describing the effect of surface roughness in the first approximation relative to h(x I , x 2) and ~h/~x I , ~h/~x 2 . Note, that in contrast to the conditions of Ref.[51], the expressions (3.20) include simultaneously possible changes of elastic constants and mass density with x I and x 2 , according to the relations (2.17) and (2.19) . The reference plane for the determlnation of the parameters )~s, Ju s, and jos in writing Eq. (3.20) may be chosen from the point of view of comfortability, dance with the relations

(2.24) and (2.25)

in accor-

.

4. Influence of Surface Phenomena on Raylei~h Waves A. General remarks Let us now consider the problem of Rayleigh surface wave propagation with accounting the surface effects. It must be noted that there are some works concerned to Rayleigh waves under the influence of surface effects. In a short note of Craig [57] the frequency shift of R a y l e i g h w a v e s (or the relative change in their velocity ac/c) due to the effect of surface tension was evaluated by using the well-knownquantum-mechanical analogy concerning the relation between the frequency shift and the energy variation. In the case under consideration this variation was associated with the effect of surface tension. In Ref. [44] the attempt was made to analyze the Rayleigh and Love surface waves propagation on the basis of mathematical formalismus using the model of elastic membrane for the description of surface effects [25, 26]. However, the main attention there was given to the discussion of the relations between the parameters of the problem, which provide the obtained dispersion equation for Rayleigh waves to have one, two, or zero roots, without physical foundation of a validity of such an analysis. This dispersion equation itself was not solved

66

V.V. Krylov

either analytically or numerically. Similar analysis was then applied to Stonely waves [46]. Note in connection with the earlier discussion, that the expressions used by the authors of Ref. [44] for equivalent boundary conditions in I~grangian variables do not transfer by standard transformations to the above-written boundary conditions in Eulerian description. This indicates possible incorrections in the results obtained in Ref. [44], in spite of presence there of some right qualitative conclusions. In the above-mentioned Ref. [27], which was also based on the membrane model of subsurface layer, the Stonely waves propagation was discussed and the Green's function for two contacting bodies with taking into account the surface effects was derived. In Ref. [35], which appeared practically at the same time with Ref. ~7], the Rayleigh waves propagation with accounting only surface tension was analyzed. In Ref. [38], based on the boundary conditions (3.11) , the contribution of surface elasticity and surface mass density was taken into account as well as the contribution of surface tension. In this work the influence of a crystal lattice discreteness on Rayleigh wave propagation was also discussed together with the effect of subsurface layer. In later investigations the attention was attracted to the some questions of Rayleigh wave propagation in strongly anisotropic crystals [58, 59], to the contribution of surface effects into the low-temperature specific heat of a crystal [58], to the study of the influence of surface phenomena on Rayleigh waves in the framework of lattice dynamics [60], etc. Except these, a lot of works dealing with nondestructive testing applications exist in the literature. In these works the effect of subsurface inhomogene~ies of different types and scales are investigated by introducing the inhomogene~ies directly into the equations of elasticity for the volume of the medium. These works will be mentioned below from the point of view of the comparison of the results obtained by different methods. B~ Raylei~h wave dispersion caused by surface tension Consider for simplicity the elastic half-space under consideration to be isotropic and occupy the domain z ~ 0 , where the notations x = x I , y = x 2 , and z = x 3 are introduced. Then, for the description of Rayleigh wave propagation in such a half-

Effect of Surface

Phenomena on

Surface Waves

67

space with accounting the surface effects one must use the mechanical equation of motion

~i

=

6iJ,J

,

(4.1)

the linearized constitution equation (the Hooke's law) 6ij = CijklUkl

(4.2)

and take into account the above-written boundary conditions for a free surface z = 0 with taking into account the surface effects (see Eq. (3.12)). Note, that for the isotropic solid Cijkl = =X~ijSkl

+ Ju(~ikSjl

+

8ilSjk ) , where

~

and ~

are the

Lame elastic constants. For simplicity, let us consider at first only the effect of surface tension T , supposing that k s = ju s = 2s = 0 (see Ref.[333). Then, the boundary conditions (3.12) will have the form dxz

= dy z

= 0 ,

(4.3) dzz - T(82Uz/~x2 + 82Uz/~Y2) Supposing into account plane xz , scalar Lame

= 0 .

that the wave propagates along x direction and taking that the particles in Rayleigh wave move in sagittal express the displacements u x and u z through the potentials T and T [3-5, 8, 9]: u x = 8T/Sx-

~/~z

,

~'/~x

.

(4.4) u z = ~TlSz

+

Then the Hooke,s law takes the form =

(2a

/axaz

+

2 -

2)

,

(4.5)

6zz = 2 j u ( & 2 ~ / ~ z 2 + c~2~/~x(3z) + ~ . ( ~ 2 ~ / ~ x 2 + ~2~/~z2)

Substitution of Eqs. (4.4) and

(4.5) into

,

Eq. (4.1) with suppo-

68

V.V. Krylov

sition on the harmonic time dependence of potentials ~ and yields, as it is well--known, the usual Helmholtz equations for and ~ :

~2~/~x2 +

2

+

=

o,

=

o

(4.6)

52~/8x2 + 82Y/~z2

+

k~

,

where k I = (M/c I and k t = ~O/c t are the wavenumbers of longitudinal and shear bulk acoustic waves, respectively, cI = = ~ ) ~ + 2ju)/30]1/2 and c t = (juJ~) 1/2 are their phase velocities, ~) is the circular frequency omitted).

(the factor

exp(-i~t)

is

We seek a solution of the problem (4.3) - (4.6) in the form of plane waves propagating along the x axis and decaying exponentially into the depth of the medium: = Aexp(ikx + VlZ)

,

= Bexp(ikx + vtz)

,

(4.7) where

v I = (k 2 - k~) 1/2 ,

constants.

v t = (k 2 - k~) 1/2

Since the expressions

and

A, B

are

(4.7) satisfy the wave equations

(4.6) , then, substituting Eqs. (4.7) into the boundary conditions (4.3) with account of Eqs. (4.4) and (4.5) , we arrive at a system of two homogeneous algebraic equations versus A and B : -2ikVlA + (k 2 + v2)B

= 0 , (4.8)

[(v~ - k2)k+ 2jUVl2 + ~Vlk2]A + (i~k 3 + 2iJukvt)B

+

= 0 .

Setting the determinant of the system (4.8) equal to zero and performing simple transformations, we obtain the next dispersion law for Raylelgh waves with accounting the surface tension:

(2k 2 - k~) 2 - 4VlVtk2

- avlk~k 2 = 0 ,

(4.9)

where a = ~/ju . Clearly, for a = 0 , i.e., in the absence of surface tension, the equation

(4.9) goes

over to the w e l l - k n o w n R a y l e i g h

equation

Effect of SurfacePhenomenaon SurfaceWaves

6g

describing the velocity of a surface wave in an isotropic halfspace. For further analysis~ it is useful to transform the equation (4.9) , dividing it by k 4 and introducing the notations ~ = o is

yet un-

knownvelocity

of Rayleigh wave. After a few simple manipulations we obtain

(2 - ,~)2 _ 4 ( l - ~9)112 [(1 - 7 )112 + "b(UJ)qZl2/4]

= o. ( 4 . l o )

Making use of the smallness of b ( ~ ) , we solve Eq. (4.10) by the perturbation method, writing the solution in the form 7 = = 7o + 71 ' where 7o is the root of the Rayleigh equation, corresponding to the solution of Eq. (4.10) for b ( ~ ) = 0 , and 71 is a small addition caused by the surface tension. Substituting this expression into Eq. (4.10) and neglecting the second order terms in b(0U) and sion determlning 71 z

7l =

71 ' we obtain

the following expres-

o)I/27(w)

.

(4.ll)

• '(7o )

Here

Y'(7o)

is

the derivative

of Rayleigh determinant

dF/d~ ,

taken in the point 7 = 7o • W r i t i n g t h e e x p r e s s i o n f o r F'(7o) a n d b ( ~ ) , we c a n r e w r i t e t h e f o r m u l a f o r 71 i n t h e f o r m

~ l = 2PT'W'~uct

(4.12)

,

where

p ~-(I =

-

(2-

. ,i/2 1/2[ - UTo s :qo

(i - Q~o )I12

(}(l(1-- %)112 (}:~o)I/2

+

( i - ~o ) z / ~ (4.Z3)

qo)l "Ij

is

a c o n s t a n t d e p e n d i n g on t h e P o i s s o n r a t i o o f t h e medium 6 . e x p r e s s i o n f o r t h e R a y l e i g h wave v e l o c i t y c with allowance f o r s u r f a c e t e n s i o n h a s t h e f o r m The f i l l

o / o r " qo 1 / 2 (1 + p~t,V'qoj"C t )

.

(4.14)

70

V.V. Krylov

The calculation shows that as the Poisson ratio is varied

from

0 to 0.5 the value of p decreases monotonically from 0.24 to 0.09 . Thus, if the surface tension is positive, it produces anomalous dispersion of the Rayleigh wave velocity. Let us calculate, for example, the dispersion caused by capillary surface tension T • Since reliable experimental data on the surface tension of solids are not available in the literature, we shall estimate the values of the dispersion in Eq. (4.14) on the basis of theoretical calculations. For example, in the case of (I00) face of an

NaCl crystal, the value of

~

at

0 K

is 438

dyn/cm [21~. The corresponding relative increase in the velocity of a Raylelgh wave of frequency I00 ~Hz propagating in the [00~ direction (calculated in quasiisotropic approximation) is equal to ~ 1 . 6 x l O -6 . Relative measurements of the sound velocity within

lO -7 error

limits do not pose a problem at present time C61~. Therefore, by performing measurements at a fixed frequency one can use expression (4.14) to determine with sufficient accuracy the variations of the surface tension under the influence of various factors: temperature, pressure, etc. In this case, of course, it is necessary to include in the calculations the influence of the indicated factors on the elastic moduli and linear dimensions of the crystal sample. ~easurements at several frequencies should in principle provide a source of information for estimating the absolute value of the surface tension. The main difficulty in this connection is the need to take into account a dispersion caused by other mechanisms such as variations of elasticity and mass density in a subsurface layer, surface roughness, scattering by defects in the volume of a crystal, thermoelastic effects, etc. The influence of these mechanisms on the dispersion of a Rayleigh wave is quite comparable in order of magnitude with the influence of surface tension and can often become predominant, particularly at high frequencies. The account of all these disturbing factors and reduction them to minimum must be performed specially in every particular case. For example, to exlude the influence of surface roughness by performing measurements of capillary surface tension, it is advisable to use atomically clean crystal surfaces prepared, for example, by the method of splitting along cleavage planes. In this case the measurements based on the analysis of Nandelstam -


71

Brillouin scattering of light on the long-wave surface phonons [62, 63] may be available. C. Connection with capillar~ waves in liquids Let us return again to the dispersion equation (4.9) and let us analyze it's behavior in the limiting case of transition from solids to liquids, i.e., by ju -~ 0 (or k t - P ~ ) (see Ref. ~5] and [6~ ). It is easily seen that from Eq. (4.9) follows the dispersion equation for pure capillary waves in ideal compressible liquids:

00 2 = ( ~ / p ) k 3 ( 1 - k~/k2) 1/2 .

(4.15)

Taking into account that k I = (A)/cI , where c I denotes in this case the velocity of sound in liquids, and solving Eq. (4.15) relative to 00 2 , we obtain

032=-

(~/p)2k4/2c 2

For small

002

+

[(~/~)4k3/4c4

(T/j~)2k6] 1/2

(4.16)

k, it follows from Eq. (4.16) approximately

{T/p)k3

and for large

+

+ (T/p)3 5/8c ,

(4.17)

k: 0 0 2 = k2c~

.

(4.18)

The last equality describes the asymptotic approach of phase velocity of capillary wave to the velocity of sound in liquid with increase of k . If one takes kl-- 0 (or c i - - o o ) in Eqs. (4.15), (4.16), and (4.17) , then one obtains the well-known dispersion equation for pure capillary waves in incompressible liquid 00 2 = ( 7 / ~ ) k 3 .

(4.19)

Note, that in the above-considered case of limiting transition to liquids one must put B = 0 in the solution given in terms of potentials (see Eq. (4.7)), since the wave m u s t v a n i s h in the negative direction of z axis for k t ~ o o as well as in general case of arbitrary k . This fact describes the well-k~mwnfeature

72

V.V. Krylov

of capillary waves in ideal liquid: only one scalar potential is sufficient for their theoretical characterization in such a medium. Comparing the surface Rayleigh waves and pure capillary waves in liquids, one must remember about different directions of particles rotation in these waves. Indeed, if in Rayleigh waves propagating in positive direction of x axis the particles move along eliptical trajectories against the hand [91, then in capillary waves the particles rotate along the circles in opposite direction ~231. Thus, in the limiting process of transition from a solid to liquid in Eq. (4.9) , such value of ju must exist, which corresponds to a linear polarization of particle displacements of Rayleigh waves on the surface. D. Effect of surface elasticit~ and surface mass densit E In general case the analysis of the effect of surface phenomena on Rayleigh waves must include also the surface moduli of elastic i t y ~ s, ju s, and surface mass density ~ s , i.e., instead of the boundary conditions (4.3) one must use the general boundary conditions (3.12) , which describe all mechanic surface phenomena ~8]. In this case the analytical manipulations become more tedious, and the dispersion equation for surface wave of Rayleigh type takes the more complicated form

k2 +

-2ikv I - i ( k s + 2jus)k3/~u +

+ ik~2~S/ju

- c si ) 2vt÷

; =0° 2i~v

t + iTk 3 -

- i0~2j)Sk .

+ ~Vlk2 - ~soo2v I ;

(4.20) Calculating the determinant in Eq. (4.20) and taking into account only the terms of first order i n ~ , )~s , j~s, ~ s , we obtain

2.4VlVtk2 _ + (,os@)k4(Vl + v t )

)Vlk = o.

2 _ [(×s+ (4.21)


Dividing this equation by

k4

73

and solving it by perturbation

method, as earlier, one can easily obtain the following approximate expression for Rayleigh wave velocity c :

clct = _l/2r. 'Io L~ + (P~lqoct) -

(T/. ~

+

g(ks

+

2,~s)/~

,

(4.22) where 7o is the root of the unperturbed Rayleigh equation, constant p is defined by Eq. (4.13), and the constants g s are given by the equalities g = (I - 9o)1/2/(1 - ~ ) i / 2 211 + (i ~o)I/2/(i ~ o )I/2] , where ~ = C ~ F c ~ and s = ~ o

the and .

perfectly clean surfaces, i.e., when the values of surface parameters T ' k s ' jus ' p s are of pure capillary nature, the contribution of )t s , ju s to the velocity change of Rayleigh Yor

wave, being evaluated according to the data of Ref. [65, 66] , may be of the same order as the contribution of surface tension . The contribution of surface mass density ~s is usually of one order lower. In the cases where surface effects describe the mechanical manufacturing of the surface or characterize artificial solid structures like films on the substrate, the relations between surface parameters may be arbitrary enough. For example, in the case of homogeneous films with a thickness of some micrometers and more, deposited on a crystal substrate the contribution of T due to residual stresses in a plane of contact can be neglected relative to the contributions of ~s , ju s , and ~ s , which increase proportionally to the film thickness h . In this situation Eq. (4.22) describes the dispersion of a lowest quasi-Rayleigh mode propagating in a structure film/substrate,i.e., a lowest mode of generalized Lamb waves [9, 31, 67S, for wavelenghts much larger than the film thickness. Recall that it is naturally this condition, which provides the validity of our consideration of the effect of surface phenomena. In comparison with the results of other works dealing with the theory of wave propagation in structures with thin films, the above described results, concerning to the influence of inhomogeneities of elasticity and mass density, have more general character and do not depend on the particular profi~eof subsurface layer.

74

V.V. Kryl ov

Really,according to Eqs. (2.13) and (2.19) , all surface parameters describe a layer in integral manner. Note, that in Ref. [68, 69], which were performed for the purposes of nondestructive ultrasonic testing, the attention was drawn to the fact that in long-wave limit the contribution of variations of elasticity and mass density in a layer to surface wave velocity are expressed through integrals over the layer. This conclusion, however, was obtained by the techniques, different from used above. The authors have performed the direct solution of wave equations for inhomogeneous layered solid medium by Auld's energy perturbation method [70]. Effect of stress state in a layer has not been c o n s i d e r e d there. Note, that the basic relation of Auld [70] for relative change in a velocity of Rayleigh wave in nonpiezoelectric layered medium has the form AC/C o = (Co/4Po)~(-A~0~2ffi~i + ACiJklUi,jUk,l)F(z) dz ,

(4.23)

where A c = c - c o is the velocity variation, c o and u i are the velocity and displacemen~ amplitudes of nonperturbed wave, respectively, PO = l(1/2)~e~u~dijdzl is the absolute value of a time averaged energy of n o n ~ r t u r b e d wave carried throug~ an elastic semispace of unit width, Aft and ACijkl are the maximal changes of mass density and elastic moduli, according to the expressions #(z) = ~o + AgF(z) ' Cijkl(Z) = C~jkl + ACijkl F(z) It can be easily shown %hat Eq. (4.23) can be rewritten in more compact form Ac/c o

= AE/E o

,

•

(4.24)

where A E is the variation of full mechanical energy of elastic semispace of finite width, and E o is the corresponding nonperturbed value of energy. Really, taking into account that Pc = EoVg' where Vg = ~ S k is the absolute value of group velocity, and noting that for unperturbed Rayleigh wave Vg = c o , one can easily derive (4.24) from (4.23) , since the variation of energy, averaged over the period of time, has the form aE

=

(I143S(

di

+

ACijklUi,jUk,l)~Cz)dz

.

(4.25)


75

In cases, where the surface wavelengths are much more than the layer thickness, the expression in brackets of Eq. (4.25) can be extracted out the integral and the values of ~c/c o obtained from Eqs. (4.23) and (4.24) must coincide, of course, with their values, calculated according to Eq. (4.22) for ~ = 0 and with accounting the determination of surface parameters (2.13) ,(2.17) and (2.19) as well as the relations (4.11) and (4.13) . Particularly, in denomenators of Eqs. (4.23) and (4.24) the derivative of Rayleigh determinant F,(~o) , which appears in Eq. (4.11) due to the expansion of F(~) in Taylor series, must present too. It is really so, because the energy leigh wave, is proportional

Pc ' carried by unperturbed Ray-

to F'(~o), namely

Po: (li2)lu:loc(1

-

[71, 72]:

.

(4.26)

Now we shall do also the next evident transformation of Eqs. (4.23), (4.24), taking into account the equality c = 00(k)/k . Then, Eq. (4.24) can be rewritten in the form of a well-known universal relation oo/oo°

= AE/Eo

(4.27)

for relative frequency variation of conservative oscillating system, in particular, quantum-mechanical system. As was mentioned above, the relation (4.27) has been used earlier by the author of Ref. [57] as a basic relation for the calculation of Rayleigh wave velocity change caused by surface tension ~ . The variation in energy A E was connected there with the influence of surface tension. Let us now discuss the Rayleigh wave propagation in a medium, in which the surface parameters T ' Ms' ~ s , and ~ s are the functions of surface coordinates Xl, x 2 . In this case one m~st use the boundary conditions (3.20) with accounting the space derivatives of surface parameters. Inhomogeneities of ~ , k s , ju s, and ~ s complify the problem strongly, causing the need in the account of multiple scattering of R a y l e i g h w a v e s into surface and bulk acoustic waves, which are responsible for the attenuation and additive dispersion of initial Rayleigh waves. If, however, the characteristic dimensions of inhomogeneities are much less than the lengths

76

V.V. Krylov

of propagating waves ~ = 2 ~ k , the effect of scattering can be negl@cted, since in a long-wave limit the attenuation and dispersion associated with it is small enough (see, for example,Ref~73~ In this case it is useful to transfer from the boundary conditions (3.20) to approximate boundary conditions for the averaged field ~ui~ ,• ~ gij ~ , changing the field variables behind surface parameters to their unperturbed values and performing the statistical averaging with respect to space coordinates. Since the terms containing the derivatives of ~ , k s, ju s give zero after avera-

@

ging, we obtain \On~1v --

vo - k-Suo~,

-

2

-s Uv~,~ o

~s~o + ~ _ _v_

=

0

(4.28) -

-

o

T~n,=~

-s..o

+

? un

= o .

Here (in supposition of uniform statistical

distribution)

} = ps(xl,x2) } dXldX 2

,

(4.29)

where S is the area of a plot of averaging. It is evident that the dimensions of this plot must be much less than the wavelength, in order the field could be considered as homogeneous. But at the same time such a plot must be sufficiently large to include a lot of local inhomogeneities. Since the boundary conditions (4.28) do not differ in form from the conditions (3.12), where the values of u i and their derivatives must be changed to the corresponding unperturbed values, the expression for the perturbed velocity in such a smallscale inhomogeneous medium will be described by Eq. (4.22), where the values of ~ , k s, ~ s , audio s must be changed to ~, ~s, In conclusion of this paragraph, note, that in recent time a number of experimental works were performed, where linear dependence of Rayleigh velocity on parameters characterizing surface effects was used for practical purposes. Thus, in Ref. [74] the possibility of measuring the adsozptlon isotherms in systems gas/solid was demonstrated. This opens the possibility of zepla-


c~Ing the usually applied very complicated mass-spectrometers

77

by

compact and simple sensors on surface acoustiC waves, in Ref.[75] and [74 the sensors of chemical composition and humidity of surrounding gases were investigated. The principle of action of all these devices are based on the measuring the relative change of surface wave velocity (or, that is the same, the frequency shift of an oscillator with a delay line on surface acoustic waves) under the influence of selectively adsorbed molecules. For quantitative evaluation of the effect of adsorbed par~$cles on Rayleigh wave velocity by means of formula (4.22), one may be restricted, as a rule, to the account of only the mass loading onto the surface, which is described by an additional surface mass density ps = P~d = MN/S . Here M is the mass of the adsorbed particles, N is their number per macroscopic plot of the surface with the area S . For small concentrations of adsorbed particles n = N/S , when the monolayer is yet not formed, the possibility of restriction by mass loading is quite evident. However, after the formation of continuous adsorbed film the more careful approach is needed to the estimation of the roles of different surface parameters. For example, the possibility of neglecting the contributions of surface elastic moduli relative to reference plane coinciding with a clean surface can be caused there by the smallness of elastic constants of the adsorbed film in comparison with them in the substrate. E. Influence of a cr~stal lattice discreteness on the contribution of surface phenomena As the spatial period of the acoustic field approaches the interatomic spacing, i.e., for the frequencies of the order of I00 GHz and more, it is necessary in general to consider the influence of the crystal lattice. In simplest way this can be done by means Of the nonlocal theory of elasticity [77, 78 S , the semiphenomenological theory, in which the classical elastic moduli are replaced by corresponding integral or differential operators. Let us use an elementary model proposed in Ref. [78] for calculation of the dispersion of Rayleigh waves in a homogeneous discrete semispace, where the nonlocal character of the medium is considered only in the direction tangent to the surface. Here, of course, information is lost about the detailed structure of the

78

V.V. Krylov

field in the normal direction to the surface. We recall, however, that this is precisely the approximation in which the integral surface parameters used above, i.e., the surface tension and surface elasticity and mass density, are meaningful. Proceeding as in Ref. ~ 8 ] , one can easily show that the expression describing the phase velocity of Rayleigh wave within the framework of the nonlocal theory takes the form (4.22) , in which, however, the quantities ~ , ju s, and k s , which enter into Eq. (4.22) both explicitly and in terms of c t , must be replaced by the Fourier transforms of the kernels of the corresponding integral operators

~(kd), ~S(kd), kS(kd)} = = ~$U,

#u s, k s } (4/k2d2)sin2(kd/2)

,

(4.30)

obtained by comparison of the phenomenological kernels of the nonlocal theory with the theory of lattice dynamics. Here ~ , ju s , and ~ s are the ordinary local moduli, and d is the lattice constant, k = ~/c is the wavenumber [38]. Yt is seen at once that with an increase in kd the discreteness of the medium magnifies the effect of the subsurface layer on the Rayleigh wave phase velocity. It is interesting to note, that in the case of anomalous dispersion, i.e., if the expression in inner brackets of Eq. (4.22) is positive, it can turn out that Rayleigh waves are nonexistent in general at frequencies close to the limit, i.e., ~ l O 13 Hz, where, according to Eq. (4.22~ the inequality

c

> ct

is satisfied.

In this event the

Rayleigh wave is changed into leaky wave and it's energy is radiated into the volume of the medium as in the case of antiwaveguide [79]. The above mentioned effect of the creation of leaky surface wave at the frequencies close to limit is interesting as one of the possible mechanisms of energy distribution of surface phonons and establishing the heat equilibrium in a crystal. One must say that the attention to the possibility of nonexistence of Rayleigh waves at very high frequencies was drawn still . However, since the influence of the discreteness in Ref. ~ was not taken into account in this work, the conclusions made there have to be considered as qualitative and logically nonrigorous. We now turn briefly to the description of Rayleigh waves with accounting surface effects within the framework of the theory of


79

lattice dynamics [60, 80, 81]. In the case of cubic crystals with interactions between nearest and next-nearest neighbours the transition to long-wave limit in the equations of motion and in the boundary conditions allows to obtain the additive corrections to the equations and boundary conditions of classical theory of elasticity, which are proportional to (kd)2 and kd, respectively. The solution of the boundary-value problem for surface waves polarized in sagittal plane gives the expressions for the frequency or for the velocity of Rayleigh wave, which contain the additional terms with principal proportionality to kd (see Ref. ~0]). This implies that the appearance of such an addition is caused by corrections due to the discreteness in the boundary conditions rather than in the equations of motion. The character of the dependence (Nkd) for the correction to the Rayleigh wave velocity in Ref. [60] coincides evidently with the dependence given by Eq. (4.22) even without accounting the latticer discreteness~ in this equality, s i n c e ( A s, ~ s 2s~ N ~ , ~, ~d . The same order characterizes also the quantity . In fact, the expression for the Rayleigh wave velocity in Ref. [60] contains only the contributions of parameters A s, /a s, and partly T , which are connected with the destruction of atomic bonds by the creation of a free surface. For full account of the contributions analogous to the contributions of ~ s and T , it would take into account the change in distances between atomic planes in a subsurface layer. The calculations of such kind are tedious enough even in the main approximation in kd . Therefore, for practical purposes the above-considered phenomenological and semiphenomenological approaches seem to be more preferable. 5. Shear Surface Waves Caused by Surface Effects A. Continuum model Till now we dealt with Rayleigh surface waves, which have as shear longitudinal components of displacements. Recall that Rayleigh waves belong to a most important and most widely disseminmted in nature type of surface acoustic waves. In homogeneous isotropic semispace as well as in the certain symmetric directions of crystal surfaces, the Rayleigh waves are the only waves, which can be described by a classical theory of elasticity. If the crys-

80

V.V. Krylov

t a l i s p i e z o e l e c t r i c , i . e . , without central symmetry, the p o s s i b i l i t y of the existence of pure shear Gulyaev - Bleustein surface acoustic waves appears [82, 83] . These waves are polarized parallel to the free surface and their phase velocity as well as Rayleigh waves velocity does not depend on frequency, being a little lower than the velocity of shear volume waves of the same polarization propagating in the same direction (with the account of the hardening the elastic moduli due to piezoeffect). Their penetration depth 1 into a solid is proportional to Kemk -2 -1 t , where Kem is the so-called coefficient of electro-mechanical coupling characterizing the "force" of piezoeffect. In contrast to Rayleigh waves, which penetrate into the medium till the depth 1 ~ k t , where ~ t = 2~/kt ' the Gulyaev - Bleustein waves are weakly inhomogeneous, since for typical piezoelectric crystals the values of 1 are of the order 50 - l O 0 ~ . The influence of surface effects, which are present in all types of crystals, including crystals with central symmetry, causes even in the case of perfect surfaces the existence once more of weakly inhomogeneous pure shear surface acoustic waves ~7, 38,4~. In principle, these waves do not differ from the w e l l - k n o w n L o v e waves in layered structures. They have a velocity dispersion and decay in depth at the distances I N d-lk~ 2 , where d is the lattice constant. To derive these waves from the solution of the equations (4.1), (4.2) with the boundary conditions (3.12) , we shall take the displacement vector u i to be parallel to the surface and perpendicular to the direction of wave propagation. In particular, if the displacement vector is oriented along y-axis, the solution is found in the form Uy = Aexp(ikx + vtz) =

-

t-2~

,

(5.1)

t~

where v t (k2 ~t" , k and k t are the wavenumbers of surface and shear bulk waves, respectively. In this case the Eqs. (4.1) and (4.2) are satisfied, and the boundary conditions (3.12) take the form

6yz

,y

_

_~

~,Y

o

.

(5.2)


Substitution of Eq. (5.1) in~@ boundary conditions yields %he dispersion equation juv%

+ juSk 2

- ~A~2j~s

= 0 .

81

(5.2)

(5.3)

It is evident from (5.3) that the existence of a pure shear surface wave, which is connected with the positivity of v t , is possible in the case under consideration if ~s/# _ ~uS/ju > 0 . In this event the approximate expression for surface wave velocity c takes the form [38] clo t = 1 -

(~o212c~)(pSlp

- jus/.~) 2 .

(5.4)

The penetration depth of surface wave into the sub~trateis determined by the relation 1 ~ (~s/# _ ~ s / ~ ) - l k ~ 2 • If the quantities ~ s and ~ s are of pure capillary origin, then ~ ~d , ~s ~d , and the penetration depth of the wave into the medium is of the order d-lk~ 2 . The waves under consideration do not differ from the waves of [8, 8 4 . All that differs Love-type in thin inhomogeneous layers is the particular method of analysis, within the scope of which we were not concerned with the structure of the field in the subsurface layer per se. These pure shear nonclassical surface waves seem to be of definite interest for high frequency surface investigations. Note, that for the above-considered surface shear waves of pure capillary origin the derivation of the exact expressions for the velocity and penetration depth requires to take into account the discreteness of a crystal lattice already in the first approximation. To do this, one can represent the quantity c~ 1 on the left-hand side of Eq. (5.4) according to previous paragraph as

c'~1 = (jol~)ll2kd/2sin2(k~2) ~ (pI~)112(I + k2d216) = = (pI3~)112(1

+

~2d2/6o~)

.

I% is clear that since the small addition to co~ on the righthand side of Eq. (5.4) is proportional to O~2d 2 P the above written additional term G~2d2/6c~^ must be also taken into account. Firstly this fact was pointed out by I.M. Gel'fgat, who

82

V.V. Krylov

obtained the nonclassical boundary conditions of the type like Eq. (3.12) paragraph

directly from the microscopic lattice model (see also 5B ). This is in contrast with Rayleigh waves, which

have a linear dispersion in low-frequency limit (see Eq.(4.22)) and do not require an account of discreteness in the first approximation. Returning to the formula (5.4), note, that the velocity of the described pure shear surface acoustic waves does not depend on surface tension T and parameter )~s, being determined only by the quantities ~s and ju s, as usually takes place~in the case of Love waves. This was pointed out still in Ref.444 j , where the correct results for Love-type waves have been obtained in terms of surface parameters, in spite of the fact that the general boundary conditions in this work were not quite correct, as was mentioned earlier. It is not difficult to show that Eqs. (5.3) and (5.4) can be derived from the exact dispersion equation for Love waves in a system consisting of the elastic semispace and the film (see, for example, Ref.[8, 9]) in the limiting case kth