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The projection method for nonlinear surface acoustic waves
Wave Motion 16 (1992) 151 162 Elsevier
151
The projection
for nonlinear surface acoustic waves
D.F, Parker, A.P. Mayer Department of Mathematics and Statistics. University of Edinburgh, Ma):field Road. Edinburgh EH9 3JZ, Scotland. UK
A.A. Maradudin Department of Physics, University oj" Ca!~¢ornia. Irvine. CA 92717. USA Received 16 September 1991
Linear elastic surface waves have complicat~ structure even in homogeneous halfspaces. Many applications involve waves on anisotroptc, layered or piezoelectric materials, in analysing nonlinear effects, an economical procedure for deriving propagation equations is paramount. Since the asymptotic procedures naturally yield linear, inhomogeneous elliptic boundary value problems, compatibility conditions must be imposed to ensure solvability. These are provided by the "projection method", which yields the propagation equations with minimal recourse to determining field perturbations introduced by nonlinearity. The method is applied to surface waves in dispersive and nondispersive systems.
1. In~oducfion Effects of nonlinearity on the propagation of surface acoustic waves (SAWs) are weak. They influence the evolution of wave profiles over length scales much larger than the wavelength of an input wave. Asymptotic methods are therefore appropriate, yielding equations for the gradual evolutioe, of ~hc amplitudes of sinusoidal components of wave profile at the surface. In general, the absence or presence of dispersion crucially affects the nonlinear behaviour of SAWs, especially the evolution of an initially sinusoidol wave. In the first case, higher harmonics couple resonantly to the fundamental wave. Nonlinear SAWs may then be represented as a superposition of SAW solutions O_~e" the linearized equations of motion and boundary conditions, with the amplitudes at the various wave numbers varying slowly along the propagation direction. This variation is then governed by an integro-differential equation over all wave numbers [1-3]. The integra{ic~a kernel incorporates the details of the nonlinear coupling. In the second , ~ c , dispersion inhibits the growth of higher harmonics. Nonlinear wave solutions described by a single sinusoidai carrier wave with slowly yawing amplitude are possible; the evolution of ,u . . . . . t~,,a~ ~,,,;. . . . . . . . . ,,a by ~ on~r~ii~'atiem nf th~ n~nline'ar ,~chrSdin~,er eouation [4 8l. Although the evolution eq~mtions for nonlinear SAWs in the presence or absence of dispersion have forms essentially simik,lr to their ccdnterparts for nonlinear bu!k waves (BAWs), the considerably more complicated structure of the displacement field (and electrostatic potential in the case of piezoelccmc . . . . . . . , causes the coefficients and integration kernels in the evolution equations to depend on the elastic and electroelastic nonlinearity in a more complicated way. This can give rise to qualitatively new effects like the appearance, in :he absence of dispersion, of nonlinear periodic surface waves of non-distorting form [9]. 0|65-2125/92/$05.00 ~' 1992
Elsevier Science Publishers B.V, A21 rights reserved
D.F. Parker et al. / Projection method Jot nonlinear waves
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For any quantitative estimates of effects of nonlinearity on the propagation of SAWs, i.e. in technical devices, numerical values for the coefficients and kernel are required. It is therefore desirable to have expressions for these quantities which can be easily evaluated for the particular system under consideration. The following derivation procedure for the evolution equations, which combines the multiple scales technique with a projection method based on Green's theorem [10], yields such expressions in the form o f overlap integrals of appropriate strains and electric fields.
2. Basic equations The following derivations are based on the equations of electro-elasticity for a semi-infinite medium filling the region x3 < ((.,~, x,.) [11, 12]. Following Taylor and Crampin [13], a four-component field u(x, t) is introduced, the first three components of which are the Cartesian components of the mechanical displacement field, and the fourth component corresponds to the e~ectrostatic potential. The position vector x is referred to the material frame, and t denotes time. Furthermore, a generalised Piola Kirchhoff stress tensor Tsj ( J = 1 . . . . . 3 ", j = 1 . . . . . 4) is defined. The components T j j for j = 1. . . . . 3 are the components of the mechanical Piola-Kirchhoff stress tensor, whereas Tsa are the three components of the electric displacement field in the material frame. The nonlinear constitutive laws relating Ts~ to the gradients of the u~ may then be written in the compact form (cf. [14, 15]) I
T~, = Sis,,.,tu,,,..u + ~.S,j,,,.,t,,xu,,.,uu,.x
I
~
4
+ ~ S~,,,;u, xe.~u,,,..,~ ..... vup.e + O(u,,..u).
(2.1)
The components of the S-tensors in (2.1) are material coefficients [12]. We adopt the usual convention of implicit summation over repeated indices and .M denotes the derivative with respect to x.u. The field u obeys a generalised equation of motion. P~l,~ ~7~ = -i), j ,
(2.2)
~i~crc ibm.= d,k -- dj4d,~ and p is the mass density in the material frame. This equation of motion is supplemented by bounda~' conditions at the surface and at interfaces, and by conditions at infinity. For example, the quantities uj(x, t) and h s ( x ) T j , , ( x , t), j = 1. . . . . 4; must be continuous at interfaces between dielectric materials (where h(x) is a material unit normal to the interface at position x). At the traction-free surface x~ = ~, rl~tbTs~ must vanish. In addRlon to this mechanical boundary condition, an electrical condition has to be imposed at the surface. Among various possibilities, the physically most relevant ones are an electr. ically isolated surface and a metallised (electrically earthed) surface. For simplicity, we restric t our considf rations here to the latter case. requiring u4 = 0 at the surface. The detailed form of the boundary conditions in the fe,'mer case has been derived earlier (see e.g. [16, ~7]) up to second order in the field components uj and their gradients. The following derivations of evolution equations can easily be modified to apply also , , ~ case, ~.c. a mat~, ~a~-~u~t'u:~_~micr~ ace Since we consider the propagation of surface waves, we require that ~ vanishes as x~ ~ends to -- ~'~. or. ;n special circumstances, that the 3-cemponent of the energy flux is r~e~.~:~ive for v, <<0. [ h e propag?.ti~n dircctien of the ,:ur~kcc ..va..:: i.: i:Je,-tifi,..-d ~ith the x~-direction, and, for simplicity of presentation, we treat fields L¢which are periodic in x~ of period La. By letting L, ~ ~.., we obtain restdls for general wa~eforms. 3"he linearised equation of motion and boundary copditions admit solutions of the form
D.F. Parker et al. / Projection method for nonlinear waves
~53
with w ~ 0 as x3 ~ - ~ , with ~ ( - k ) = -o2(k) and with w ( x 3 l - k ) = w*(x3[k), where * denotes the complex conjugate. These correspond to time harmonic surface waves with wave vector k = (k~, k2. O~ and corresponding frequency og(k). For waves propagating in the x~-direction we take k, = 0 and write k~ = k. If the c o m p o n e n t s of the S-tensors in (2.1), the surface profile function ~ or the density p depend on x~ a n d / o r xz, w depends also on these coordinates. In the following section the material properties and surface profile, and hence w, are independent of x~ and x:. In Section 4, we shall allow the material parameters, like the surface profile function, to depend periodically on x~ with period a. In this case, the solutions of the linearised equation o f m o t i o n and b o u n d a r y conditions can be chosen to be of the Bloch-Floquet type, i.e. w is then a periodic function of x~ with period a.
3. Nonlinear surface waves in the absence of dispersion Electro-acoustic waves Iocalised at the surface of a h o m o g e n e o u s medium filling the half space x3 < 0 are non-dispersive, since this system does not define any length scale. If the medium is isotropic, Rayleigh waves are the only surface wave solutions o f the linearised equation of motion and boundary conditions. T h e y are polarised in the sagittal plane. In anisotropic and possibly piezoelectric media, generalised Rayleigh waves exist which, depending on the propagation direction and the symmetry of the surface (the "'cut"), are not of purely sagittal polarisation and m a y be accompanied by an electric field (see e.g. [18]). T o investigate the propagation of generalised Rayleigh waves under the influence of nonlinearity, we expand the generalised displacement field ~ in terms of a small parameter e << 1 in the following way [ 15] :
u(x, t) = sU(O, x~, X~ I~, X~ I~, T ~ ~) + s2d21( 0, .x'~, X~ I~ X~ n~, T ~t~) +O(~3).
(3.1)
Here 0 = x ~ - vt, with v = vR being the phase veloctty of the linear generalised Rayleigh waves. X~ ~= s.v~, X~ ~= cx2 and T q~= st are stretched coordinates. The dependence on X ~ t a k e s accoant of effects due to the finite lateral width o f the wave profile. By inserting (3.1) into the equation of motion and boundary conditions and collecting terms of first order in s, it becomes obvious that U car~ be written in the form
U( O, x3, X ¢1~, T ~l~) =.3-'.eikOw(x3 Ik)A(k, X ~l~, T ~ ) ,
(3.2)
k
where A ( - k ) = A * ( k ) ensures that U is real and where per,odicity requires that the wave numbers k are integer multtpl~s o~ , n / L ~ . (More generally, the summation m a y be ret~laced by an integration over all real k.) T o second order in s, the equation of motion takes the form fro Oim~m,ll -"~ hlmMllm tM - 20.v;
s,.,,,,.,,,,,u,,,.,,~u,,.,.+ 2ovrt,,, ~,~ u,,,.~ + ~s;~,,,:+ s,,~,,o) :-~;!-, ~;,:j,
f3.31
and the corresponding boundary conditions ;~ the surface are 0 ~, = 2', Sj3,, w,,~'~/,,,,,; - c,,.x -r + S;3,,,r g;-,-& -S,~,,,.~.~u,;,..~ U .... j = 1, 2, 3, at x3 = ~.
(3.4a)
u~:"~=O
(~.4bl
atx3=0.
Here a~d in the foii,~wing, Gi-eek subscripts run only over I and 2.
154
D.F. Parker et aL / Projection method for nonlinear waves
For the derivation of an evolution equation for slow variations o f the amplitudes, it is not necessary to determine the field u (') explicitly. The system (3.3) and (3.4) is a linear, inhomogeneous boundary value problem for # ('). To ensure its solvability, compatibility conditions have to be imposed. They may be obtained by projecting (3.3) onto solutions fi(x, t) of the linearised system (i.e. multiplying by fi*, s u m m i n g over j and integrating over the region - c ¢ < x3 < 0, 0 < 0 < L~.) M a k i n g use of Green's theorem then gives
f' f ] "x. lPO"l~lJ'l [ .,_,
f
,,,
l~Jml¢;;;;I -- Ojdmj, fg,lj,jl.~m,M,__2SjJmMnNllj.j~m.MUn, =, ° (2) I -* r ,
+ 2Pv'~*rb., ~T(~) u,,,.,
-
~
~ U., +
"[
=0,
(3.5)
since the boundary terms which arise during application of Green's theorem vanish, in the present case, because of the boundary conditions imposed on the fields u and ft. By applying Green's theorem again, it is readily shown that the first two terms in (3.5) cancel, so yielding an equation which no longer contains the field u °). This form o f the solvability condition then provides the desired evolution equation since, by using the explicit expressions (2.3) for ~ and (3.2~ for U, it m a y be recast as
i{VrO~rt) + ~ }
~(k)= E KR(k,k')A(k')A(k-k').
(3.6)
The kernel KR for genera!ised Rayleigh wa~es may be written explicitly as an overlap integral over products of the functions wi(xs ]L) which describe ,he depth-dependence of linear surface waves. It has the ferm
= N S,s,,,,v
I
O
^
!iOA--,~)'.,i(x,l--'"~ ~ " ..... ' x
x [O,(k--t,')~,,,(x,l&-&')]} dx3
k-RR(k'/k)
(3.7)
~vhcn t,h,c fm:.ctions w arc normahscd so that O
N = 4pvk
f
~*)(x3 [ --k) rljmw~(x 3 ]k) dx3
(3.8)
--.c
is independent of k and where the operator {)j(k) is
Q.s(k)= 8suik + 8j~c32-~' O
(3.9)
The vector ( Va. V2, 0) is the group vele,city vector for generalised Rayleigh waves having wave vector in the positive.st-direction. ~,~ . . . . . . . ;,.~i ..... h,.,,;,,,, ,,¢,h~ ~. . . . . . 1 ~- ~ I . . . L ~ .a:._l . . . . . . . ~ ,~ _, . . . . . . . . -_ ~,. poCentiM and phase velocity of ,meat generalised Rayleigh waves need be known. In the. limit of infinite period L~ when the sum in (3.2) is replaced by an imegral, the sum over k' in (3.6) is likewise replaced by an integral. Under certain circumstances, the kernel KR vanishes identically tbr reasons of symmetry. This is the case for Bleustein Gulyaev (BG) waves. Linear BG waves are of shear-horizol.~tal polarisation, i.e. only w., and w~ are non-zero in (2.3) [19, 20]. These surface waves occur if, for example, the x:-axis is a two-fold rotation axis. We shall consider this geometry here. The fi~ st term on the right-hand side of (3.3) and of (3.4a) then
D.F. Parker et al. .1 Projection method for nonlinear waves
155
vanishes for j = 1 and 3, and the projection of (3.3) onto the generalised displacement field of a linear BGwave yields the result
8X~
~T
(3.1o)
J
To derive an evolution equation for slow variations of A(k) incorporating nonlinearity, we have to carry the expansion (3.1) to third order in e (i.e. we consider the term e3d 3)) and we introduce further stretched . (',) ., T(2) = ~:t. Equation (3.10) implies that the amplitudes A(k) depend on coordinates X~') = ~'xt, A~." = ~'x:, the stretched coordinates X] ~), T (t) only via the combination ~ :=X] t) - V~T"k Since |"2 = 0, they may also depend on X ~o. In order to proceed further, the second order field d ') has to be determined explicitly. It may be decomposed into a shear-horizontal/electric part d "x) and a sagittal part d ''s) as d :)= d"x)+ u("">, where
u('-,c)=-~ e~k°i[ ~--w(x~[k)] ~k A(k ) ,
LSk
(3.11a)
"
d'~'") = Z dk°iw( x-~Ik) ~
A (k) + d"'~ L
(3.11b)
k
The functions g,(x3[k) have to be determined from a pair of linear inhomogeneous orcfinary differential equations and bound?+ry conditions at x~ = 0. The part u t''~) of the second order solution may be constructed as
u"-,.+)(x, t ) = ~ e ~° E ~,(x, Ik, k')A(.~-')A(,~ -k'), k
(3.12)
k'
where the functions g do not depend on the stretched coordinates, and are unique, They are convenienfl~ determined by first seeking a particular solution of the equation of motion (3.3) and then adding a solution gU,) of the corresponding homogeneous equation to satisfy the boundary conditions. The latter will be of the form -)
g'"'" " Ik, k')= ~ ; tt,~3
HJ~)(k, k') exp(kfl'~)x3),
(3.13)
(k > 0), where the quantifies fl(') are the roots of the secular equation det(Sj.:.v.:_g_jg,::+ 6j,~,po')= 0
~3.14)
with positive real part. Here, ;~I= i6jn + :36j~ and v is the phase velocity of the linear BG waves. Equatioa~ (3.14) need not necessarily have two solutions with positive real part. One of the two ,B~ may, for example, be purely imaginary. In this case, nonlinear BG waves are not fully localised at the surface. For the same reason, the functions ff(x3!k) are then not fully iocalised at the surface. This has the consequence that if a BG wave is excited with high intensity at the surface of a piezoelectric medium, it will radiate ener~' into the bulk medium via the nonlinear coupling.
D.F. Parker et al. / Projection method.lbr nonlinear waves
156
Having determined the second order field u(2k we now consider the shear-horizontal/electric part of the equation of motion of third order in e,
O
+ 2pvoj,,, ~
u,,,., + (sj.,,, +
sj,.,~) ~
~2
Urn,.
~2
+ S~...,: .~.~z
0 . (2.~)
U . , - {O V~ rt,,,- Sj,.,, } a---5 U . , - 2pv Vt rb.,-~g .,,,.,
+ (&.,.,,+ &~,,,~)~
(2,o)u,,,.., -,- (S,,,,2 + s~,.,,,.,) ~
a
(,,) u,,;::;
(3.15)
and corresponding boundary conditions o
131
__¢",
rr
t2.s)..l_l
-~,,,.,,.,,,,,,,.,,- ,_,,~,,,.,,,,.,.~,,,.,,,,,..,. C3
.(2,e).
+ Sj3,,,, a-~ u,, u(4~)=0
.-
~s,,,,.,,,,,,,,.,u,,,..,,u,.Nu,.~+ 0
s,,,,, aXi2 ) u,,,
. (2,s)
-,- S~3,,,2 aXe,-------S),,,, , j = 1,2. 3, at x3=0.
atx~=0.
(3.16a) (3. I ob)
At this stage, we again apply the projection method, using the solution ~* for BG wa'~cs. After appiying Grcca's lacorem twice, the following equation may be derived: f~" i" -'t)
-'
f,,.,
~'
0
0
,
{A
|
+S,~,,,,O,* 02 a (,,) _ _ , a u(_,,) " ax ~' a C;,, + S~...,,.z-~? ~aX._ u,c,'~f. S,M,,,.uj..~-~X~ .,, - ~,Sjj,,,,~,,,~peuj.jU,,,.,~U,,.,~,Up.e dx.~ dO = O.
(3.17)
,.,earn, boundary term~ vanish because of the boundary conditions applying to fi and to the fields occurring in (3.15). By using the explicit expressions (3.2), (2.3) and (3.1 lb, 3.12) for U, ti and g °''', equation (3.17) may bc cast into the desired evolution equation for the amplitudes A l k ) r
C-
..,,,.,.
f
C,
+ Y. K~c~.(k,k',k")A(k')A(k")AIk-k'-k').
(3.18)
Again. fl~r genera] non-periodic waveforms, the single and double summations in (3.18) may be replaced by. sin-le.., and double integrals., respectively. To evaluate the kernels K~mc!~and ,~"(:~n¢~numerically, we require
D.F. Parker et al. ,/Projection method for nonhaear waves
157
functions g(x~l k, k') in addition to the w(x~ Ik). Explicit expressions in terms of these quantities will be given elsewhere. (For K ~ see [21].) Here we state only the scaling properties _ ~(1) , / . l~c;~^, k , ) _-kKn~;(k /k),
K[l)tl.
KO.~,~. nov,, k.,, k , , ) = k 3 ~ ( k , / k , k
{3.19!
"/k), '
(3.20)
if a normalisation of the functions w(x3 Ik) is used that leaves (3.8) independent of k. We also emphasise that ~nGv~"~consists of two parts, one depending linearly on the coupling coefficients of the third order nonlinearity, and one depending quadratically on the coupling coefficients of the second order nonlinearity. This latter arises because nonlinear BG waves necessarily excite a sagittal displacement field. It is interesting to note that the evolution equation (3. ! 8), but without the X [~Lderivatives, has recently been derived for nonlinear wedge waves of uneven symmetry [22], which form another example of nondispersive guid,~d acoustic waves. Like BG waves of the form (2.3), such wedge waves do not couple resonantly to their second harmonic. Wedge waves of even symmetry do excite the second harmonic resonantly. Their evolution due to the influence of nonlinearity may therefore be described by an equation of the form (3.6).
4. Nonlinear surface waves in the presence of dispersion
If the system considered in Section 3 is modified in any way which introduces a length scale, the corresponding surface acoustic waves become dispersive. An important effect of dispersion is to inhibit the growth of higher harmonics of an initially sinusoidal wave. One may therefore seek a solution of the equation of motion and boundary conditions for the generalised displacement field u in the form of the asymptotic expansion (3.1), but with the first order term U containing only one harmonic wave with a slowly varying amplitude:
U=e'Vw(x3 Ik)A(X~ '~, X~ ~b, T"', X~ 2', X~ 2', T ~2~. . . . ) +c.¢. = U'A +c.c.
(4.1)
We have here defined ~ = k . x-coCk)t, and we choose as before k = (k, O, O) tor the wave vector of the carrier wave. The following derivation of an evolution equation for the complex amplitude A is kept sufficiently general to account for various mechanisms that lead to dispersion. These mechanisms include layering stratified parallel to the x~-x_,-plane, media with material properties varying continuously along the x3-direction a n d / o r varying periodically in the xrdirection with periodicity a, as well as periodic corrugation of the surface with profile function ~(x~)=((x~ + a ) . Without loss of gener:dity, we may assame L~ to be an integer multiple of a. For simplicity, we require the system to be independent of the x2-coordinate. In the case in which the material properties or the surface profile function are periodic, w depends not only on x, but also is periodic in x~, with period a. Likewise, the higher order functions u ~''~, n ~>2. in the expansion (3.7) depend periodically on xl, in addition to depending on ~, x2 and the stretched coordhlatcs: u'"~( q/, x~, x,, , . ¥ ~ ' . . . .
) = ~ " ~ I ~'.:. x~. ~'~ + a, X~ j~ . . . .
).
~4.2',
Another effect that gives rise to dispersion is the discreteness of the underlying atomic structure of the medium (the crystal lattice). As has been demonstrated by one of us (AAM, [23]) for a semi-infinite simple cubic cDstal, such effects can be incorporated into continuum theory to leading order of kao, where a0 is the lattice constant, by including a higher derivative term "n the bouada15, conditions and, in certain cases.
158
D.F. Parker et aL ,' Projection method for nonlinear waves
also in the equation of motion. The following derivatives can easily be extended to account for such terms, and the projection method is equally applicable. Since nonlinear terms in u q2~ need be retained only ~o terms quadratic in expressior~ (4.1), the field u ~2~ may be decomvosed into three parts as U~2~(l/t. . . . ) = U~2.0~(...) + { ¢2~2.1)(...) eiV, + ~2.2~(...) ¢2i~, + c.c. }.
(4.3)
Collecting the terms of second order in e in the equation of motion and making use of the linear independence of the factors e ~"~, n = 0 , 1, 2, yields the following three :ahomogeneous equations:
Oxa
aej.,MU.,.M
-pco rb.,u,,,
- - SjS.,M,,xO*,.M (L, NIAI2, - exj " - -Oxs - o~S,,,MU.,,M = 2~poJ rlj,,, Um ~
(4.4)
A
+ ( S'r'''' ~.,',~ + ~--X~ ~ S,,,,., r) C~,,, ~Xg---~ 5 A,
~..~ - '-X~ SjJ.,MU'.7.';~ ~..~ - - 21 ~X~ j &,.,M.~,O.,,MO.,,vA2" -4pro". rb.,U.T"
(4.5) (4.6)
where u~2'"~=~2'"~e'"~. Boundary conditions have to be appended. In particular, we require that the quantities
,bl S,,,,.,,<[I:~ + S,j,,~.,,,,~,.0,*,,. ,, C",,..,.IA I:
(4.7)
~,, ) arj,,,Mu,,,, w ~
(4.8)
&j,,,r 6.,,
iii A J &V r
fi;', S,,.,,,,,,ul,:.~] + ~:S,j., ,,,,, ~7,.,.,, &,. ,.A:
(4.9)
( j = 1, 2, 3) are continuous across interfaces and vanish at the surface, ck is a vector normal to the corresponding interface or surface.) For piezoelectric media, we require again that u~2)= 0 at the surface. We now project equation (4.5) onto the solution (2.3) of the corresponding linearised equation of motion and boundary conditions. Applying Green's theorem twice and making use of the boundary, conditions, we then obtain: r a e,l iN~ V r - ~ . ~ + ~-~-TF, ( d = 0,
(4.10)
where ~', .... L .~m LnN
r., , o
m
dv, dx 'I
- •
(4.11)
are ~he components of the group velocity vector and where
N=2O, [f' i'""~; ~7,. pr~,,,t~.,,dv, d,., "
(4.12)
Q.F. Parker
eI
ai. / Projecrion merhodfir rlorl~j~earuxmes
159
Equation (4.13) suggests the introduction of variables f l_“” = XF’ -. I/,P’“. can be verified of (2.3) into the linearised equation of motion and boundary conditions an ~~e~~~~~~t~~n wi the wave vector components, the general solution of (4.5) satisfying the correct bo~~d~r~ conditions is given by u(2.1)=
-i ,iw [
-w
a I ad+ _
akr
(4.13)
a@)
The amplitude a depends on the stretched coordinates. To determine the fields #*O)= F’O’IAl”
(4. Ma)
UW) = F(Zj.42,
(4.14b)
the system has to be specified in more detail. The presence of dispersion ensures the existence of unique bounded solutions to the corresponding equations for F”’ and Collecting now, in the equation of motion, all third-order terms having the factor exp(iyr), we obtain
where +j-p = s.,Jn&fnN
f
(2’2” &,,.Md:i? + ~hhwV , + ~sjJn,,,~~,~~'pper;f,.,,u,~,~v,.,.
We now project equation conditions satisfied by t
which no longer contains 8.” us, the n~ust vanish. to t transforming
(4.16)
and again use Green’s tbe0rem together with the boundary (4.15) on s occurring in 14.15). This
esired evolution equation,
D.F. Parker et aL / Projection method for nonlinear waves
160
Equation (4.18) is a generalisation of the nonlinear Schrrdinger equation which includes effects of gradual lateral variation of a wave profile. By some manipulations of the corresponding terms in (4.17), it can be shown that the matrix (.Ora) is given by ~2 1"2ra(k) = ~ k r ~ka co(&)
(4.19)
apart from an irrelevant antisymmetric part. For the nonlinearity coefficient coc2~ the projection method yields the following expression: L~ N
U.,.,~U.,NUp, e fo'-,fj ~,,~.Sjj,,,M.Npp Uj.j -**-+
~,
~
1o)
/'r*
~(2)1
SSs.,X~.N Us,s[ U,,,.MF..x + ,-, .,,M. ..N~ } dx3 dxl ,
(4.20)
N being defined in (4.12). This expression can be evaluated easily once the generalised displacement field /~ of the linear surface wave with wave vector k and the functions F cm and F ~'~ are known. It consists of three parts: a contribution proportional to the coupling coefficients of the third order nonlinearity, a contribution from the static part of the second order solution and a contribution from the second harmonic. The latter two are linear combinations of products of two coupling coefficients of the second order nonlinearity. In the case of weak dispersion, the part resulting from the second harmonic is dominant if, in the limit of zero dispersion, the surface wave couples resonantly to its second harmonic. An example for this situation is the propagation of Rayleigh waves on a surface of a homogeneous elastic medium covered by a thin film of different material, the film thickness being much smaller than the wavelength. The quantity ~c=[2co(k)-co(2~)]/co(&).
I~'l<<1
(4.21)
may then be used as an additional expansion parameter. Projecting (4.6) on the solution of the linearised equation of motion and boundary conditions u ° ' ( x , t) = w(x312k) e'~2~'' -'°~"~m
(4.22)
we obtain, with the help of Green's theorem. (4co2(k) - co2(2k))
- :~
,fLf °
I Ll I 0 1¢(#)*p~jml4('"¿;,, l| "~"~ a,.u.~3dxt } --&
, jj,,,.~i,,xu?,.j ~,,,.,~U,,.xA" d.v~ dxt.
(4.23)
This relation implies that u12'~'* contains a contribution of order O(~c-~) (unless the right-hand side is incidentally of order O(~')). From the equation cf motion (4.6) and associated boundary conditions, it tbNows that this contribution is of the form bA2g£(2x, 2t). T h e coefficient b can be determined directly from ~4.23L where u ~'-'! may then be reptaced by bA'-~'{2x, 2t). Inserting this result into (4.20) with (4.14b), the following expression is obtained for the contribution to the nonlinearity coefficient e~?'~ to leading order in &-:
~cN2
D.F. Parker et al. / Projection method for nonlinear waues
161
where
n=!
20 o, (x,
'
Lt
_~
"
"
0 dx3 dx,
(4.25)
'
and
N2 = 8N to2(k)
O*(2x. 2t)Ptb.,U,,,(2x, 2t) cL,c3dx,.
(4.26)
gl
It is emphasised that for the evaluation of this leading order contribution, only the frequencies and generalised displacement fields of linear surface waves have to be known. An important implication of the result (4.24) is that the sign of the nonlinearity coefficient co¢2~ in the systems to which (4.24) applies, is determined by the sign of ~c alone. In such systems of weak dispersion, it should be expected that nonlinear terms of higher order in (4.18) will also be important.
5. Conclusions In conclusion, we have demonstrated how the projection method based on Green's theorem in elasticity theory, in conjunction with multiple scales theory, can be used to derive efficiently nonlinear evolution equations for surface acoustic waves in the presence and absence of dispersion. The procedure yields expressions for the coupling coefficient or kernels in these equations ~hat can easily be evaluated. These expressions have the form of overlap integrals and, in ~ystems where nonlinearity of second order dominates, they involve only the displacement fields and electrostatic potentials for linear surface acoustic waves.
References [iI [2] [3] [4]
R.W. Lardner, "Nonlinear surface acoustic waves on an elastic solid", Int. d. Eng~g. Sci. 2!, 1331-1342 (1983). R.W. Lardner, "Nonlinear surface acoustic waves on an elastic solid of general anisotropy", J. Elast. 16, 63 73 (1986). D.F. Parker, "Waveform evolution fo,' nonhnear surface acoustic waves", Int. J. Engng. Sci. 26, 59 75 (1988). T. Sakuma and Y. Kawanami, "Theory. of surface acoustic soliton. I. Insulating solid", ~%ys. Rev. B20, 869 879 (1984), "Theory of surface acoustic soliton. II. Semiconducting solid", Phys. Rev. B29, 880-888 (1984). [5] K. Bataille and F. Lund, "Nonlinear waves in elastic media", Physica 16D, 95- 104 (1985). [6] A.A. Mamdudin, "Nonlinear surface acoustic waves and their associated surface acoustic solitons", in Recent Deuelopmems in Surfiice Acoustic Wanes, eds. D.F. Parker and G,A. Maugin, Springer, Heidelberg (1988) pp. 62 71, [7] H. Hadouaj and G.A. Maugin, "Une onde solitaire se propageant sur un substrat ~lastique reconvert d'un film mince", C R. Acad. Sci. Paris 309, 1877 1881 (1989). [8] A.A. Maradudin and A.P. Mayer, "Surface acoustic waves on nonlinear substrates", in Nonlinear Waves in Solid State Physics, eds. A.D. Beardman, M. Berto!otti and T: Twardowski, Plenum. New York (1991~ pp. 113 161. [9] D.F. Parker and F.M. Talbot, "Analysis and computation for nonlinear elastic surface waves of permanent lbrm", 5. Elast. t5. 289 426 (1985). [101 A.G. Eguiluz and A.A. Maradudin. "'Effective boundary conditions for a semi-infinite elastic medium bounded by a rough planar stress-free surface", Phys. Rev. B28, 711 327 (1983). [lll D.F. Nelson, Etectric, Optic, and Acoustic Interactions #, Dielectrics, Wiley, New York (1979}. [12] G.A. Maugin, Nonlinear Etectromechanical Effects and Applications, World Scientific, Singapore (1985). [13] D.B. Taylor and S,G. Crampin, "Surface waves in anisotropic media: Propagation in a homogeneous piezoelectric halfspace", Proc. Roy. Soc. A 364, 161 179 (1978). [ 14] G.E. Tupholme, "Multiple scale techniques applied to nonlinear elastic piezoelectric surface waves", in Recent Deveh~pmems in Surface Acoustic Waves, eds. D.F. Parker and G.A. Maugin, Springer, Heidelberg (19S8) p~,. [4 20.
162
D.F. Parker et al. / Projection method for nonlinear waves
[15] E.A. David and D.F. Parker, "Nonlinear evolution of piezoelectric SAWs", in Recent Development in Surface Acoustic Waves, eds. D.F. Parker and G.A. Maugin, Springer, Heidelberg (1988) pp. 21-29. [16] H.F. Tiersten and J.C. Baumhauer, "'An analysis of second harmonic generation of surface waves in piezoelectric solids", J. Appl. Phys. 58, 1867- 1875 (1985). [t7] D.F. Parker and E.A. Dahid, "Nonlinear piezoelectric surface waves", Int. d. Engng. Sci. 27, 565-581 (1989). [18] E. Dieulesaint and D. Royer, Elastic Waves in Solids, Wiley, Chiehester (1980). [19] Yu. V. Gulyaev, "Electroacoustic surface waves in solids", Zh. Eksp. Teor. Fiz. Pis. 9, 63-65 (1969) (So~. Phys.-JETP Lett. 9, 37-38). [20] J.L. Bleustein, "A new surface wave in piezoelectric materials", ,4ppl. Phys. Lett. 13, 412-413 (1968). [21] A.P. Mayer. "Evolution equation for nonlinear Bleustein-Gulyaev waves", Int. J. Engram. Sci. 29. 999-1004 (1991). [22] V.V. Krylov and D.F. Parker, "'Harmonic generation and parametric mixing in wedge acoustic waves", Wave Motion 15, 185200 (1992). [23] A.A. Maradudin, "Surface acoustic waves", in Nonequilibrium Phonon Dynamics, ed. W.E. Bron, Plenum, New York (1985) pp. 395-599.
151
The projection
for nonlinear surface acoustic waves
D.F, Parker, A.P. Mayer Department of Mathematics and Statistics. University of Edinburgh, Ma):field Road. Edinburgh EH9 3JZ, Scotland. UK
A.A. Maradudin Department of Physics, University oj" Ca!~¢ornia. Irvine. CA 92717. USA Received 16 September 1991
Linear elastic surface waves have complicat~ structure even in homogeneous halfspaces. Many applications involve waves on anisotroptc, layered or piezoelectric materials, in analysing nonlinear effects, an economical procedure for deriving propagation equations is paramount. Since the asymptotic procedures naturally yield linear, inhomogeneous elliptic boundary value problems, compatibility conditions must be imposed to ensure solvability. These are provided by the "projection method", which yields the propagation equations with minimal recourse to determining field perturbations introduced by nonlinearity. The method is applied to surface waves in dispersive and nondispersive systems.
1. In~oducfion Effects of nonlinearity on the propagation of surface acoustic waves (SAWs) are weak. They influence the evolution of wave profiles over length scales much larger than the wavelength of an input wave. Asymptotic methods are therefore appropriate, yielding equations for the gradual evolutioe, of ~hc amplitudes of sinusoidal components of wave profile at the surface. In general, the absence or presence of dispersion crucially affects the nonlinear behaviour of SAWs, especially the evolution of an initially sinusoidol wave. In the first case, higher harmonics couple resonantly to the fundamental wave. Nonlinear SAWs may then be represented as a superposition of SAW solutions O_~e" the linearized equations of motion and boundary conditions, with the amplitudes at the various wave numbers varying slowly along the propagation direction. This variation is then governed by an integro-differential equation over all wave numbers [1-3]. The integra{ic~a kernel incorporates the details of the nonlinear coupling. In the second , ~ c , dispersion inhibits the growth of higher harmonics. Nonlinear wave solutions described by a single sinusoidai carrier wave with slowly yawing amplitude are possible; the evolution of ,u . . . . . t~,,a~ ~,,,;. . . . . . . . . ,,a by ~ on~r~ii~'atiem nf th~ n~nline'ar ,~chrSdin~,er eouation [4 8l. Although the evolution eq~mtions for nonlinear SAWs in the presence or absence of dispersion have forms essentially simik,lr to their ccdnterparts for nonlinear bu!k waves (BAWs), the considerably more complicated structure of the displacement field (and electrostatic potential in the case of piezoelccmc . . . . . . . , causes the coefficients and integration kernels in the evolution equations to depend on the elastic and electroelastic nonlinearity in a more complicated way. This can give rise to qualitatively new effects like the appearance, in :he absence of dispersion, of nonlinear periodic surface waves of non-distorting form [9]. 0|65-2125/92/$05.00 ~' 1992
Elsevier Science Publishers B.V, A21 rights reserved
D.F. Parker et al. / Projection method Jot nonlinear waves
152
For any quantitative estimates of effects of nonlinearity on the propagation of SAWs, i.e. in technical devices, numerical values for the coefficients and kernel are required. It is therefore desirable to have expressions for these quantities which can be easily evaluated for the particular system under consideration. The following derivation procedure for the evolution equations, which combines the multiple scales technique with a projection method based on Green's theorem [10], yields such expressions in the form o f overlap integrals of appropriate strains and electric fields.
2. Basic equations The following derivations are based on the equations of electro-elasticity for a semi-infinite medium filling the region x3 < ((.,~, x,.) [11, 12]. Following Taylor and Crampin [13], a four-component field u(x, t) is introduced, the first three components of which are the Cartesian components of the mechanical displacement field, and the fourth component corresponds to the e~ectrostatic potential. The position vector x is referred to the material frame, and t denotes time. Furthermore, a generalised Piola Kirchhoff stress tensor Tsj ( J = 1 . . . . . 3 ", j = 1 . . . . . 4) is defined. The components T j j for j = 1. . . . . 3 are the components of the mechanical Piola-Kirchhoff stress tensor, whereas Tsa are the three components of the electric displacement field in the material frame. The nonlinear constitutive laws relating Ts~ to the gradients of the u~ may then be written in the compact form (cf. [14, 15]) I
T~, = Sis,,.,tu,,,..u + ~.S,j,,,.,t,,xu,,.,uu,.x
I
~
4
+ ~ S~,,,;u, xe.~u,,,..,~ ..... vup.e + O(u,,..u).
(2.1)
The components of the S-tensors in (2.1) are material coefficients [12]. We adopt the usual convention of implicit summation over repeated indices and .M denotes the derivative with respect to x.u. The field u obeys a generalised equation of motion. P~l,~ ~7~ = -i), j ,
(2.2)
~i~crc ibm.= d,k -- dj4d,~ and p is the mass density in the material frame. This equation of motion is supplemented by bounda~' conditions at the surface and at interfaces, and by conditions at infinity. For example, the quantities uj(x, t) and h s ( x ) T j , , ( x , t), j = 1. . . . . 4; must be continuous at interfaces between dielectric materials (where h(x) is a material unit normal to the interface at position x). At the traction-free surface x~ = ~, rl~tbTs~ must vanish. In addRlon to this mechanical boundary condition, an electrical condition has to be imposed at the surface. Among various possibilities, the physically most relevant ones are an electr. ically isolated surface and a metallised (electrically earthed) surface. For simplicity, we restric t our considf rations here to the latter case. requiring u4 = 0 at the surface. The detailed form of the boundary conditions in the fe,'mer case has been derived earlier (see e.g. [16, ~7]) up to second order in the field components uj and their gradients. The following derivations of evolution equations can easily be modified to apply also , , ~ case, ~.c. a mat~, ~a~-~u~t'u:~_~micr~ ace Since we consider the propagation of surface waves, we require that ~ vanishes as x~ ~ends to -- ~'~. or. ;n special circumstances, that the 3-cemponent of the energy flux is r~e~.~:~ive for v, <<0. [ h e propag?.ti~n dircctien of the ,:ur~kcc ..va..:: i.: i:Je,-tifi,..-d ~ith the x~-direction, and, for simplicity of presentation, we treat fields L¢which are periodic in x~ of period La. By letting L, ~ ~.., we obtain restdls for general wa~eforms. 3"he linearised equation of motion and boundary copditions admit solutions of the form
D.F. Parker et al. / Projection method for nonlinear waves
~53
with w ~ 0 as x3 ~ - ~ , with ~ ( - k ) = -o2(k) and with w ( x 3 l - k ) = w*(x3[k), where * denotes the complex conjugate. These correspond to time harmonic surface waves with wave vector k = (k~, k2. O~ and corresponding frequency og(k). For waves propagating in the x~-direction we take k, = 0 and write k~ = k. If the c o m p o n e n t s of the S-tensors in (2.1), the surface profile function ~ or the density p depend on x~ a n d / o r xz, w depends also on these coordinates. In the following section the material properties and surface profile, and hence w, are independent of x~ and x:. In Section 4, we shall allow the material parameters, like the surface profile function, to depend periodically on x~ with period a. In this case, the solutions of the linearised equation o f m o t i o n and b o u n d a r y conditions can be chosen to be of the Bloch-Floquet type, i.e. w is then a periodic function of x~ with period a.
3. Nonlinear surface waves in the absence of dispersion Electro-acoustic waves Iocalised at the surface of a h o m o g e n e o u s medium filling the half space x3 < 0 are non-dispersive, since this system does not define any length scale. If the medium is isotropic, Rayleigh waves are the only surface wave solutions o f the linearised equation of motion and boundary conditions. T h e y are polarised in the sagittal plane. In anisotropic and possibly piezoelectric media, generalised Rayleigh waves exist which, depending on the propagation direction and the symmetry of the surface (the "'cut"), are not of purely sagittal polarisation and m a y be accompanied by an electric field (see e.g. [18]). T o investigate the propagation of generalised Rayleigh waves under the influence of nonlinearity, we expand the generalised displacement field ~ in terms of a small parameter e << 1 in the following way [ 15] :
u(x, t) = sU(O, x~, X~ I~, X~ I~, T ~ ~) + s2d21( 0, .x'~, X~ I~ X~ n~, T ~t~) +O(~3).
(3.1)
Here 0 = x ~ - vt, with v = vR being the phase veloctty of the linear generalised Rayleigh waves. X~ ~= s.v~, X~ ~= cx2 and T q~= st are stretched coordinates. The dependence on X ~ t a k e s accoant of effects due to the finite lateral width o f the wave profile. By inserting (3.1) into the equation of motion and boundary conditions and collecting terms of first order in s, it becomes obvious that U car~ be written in the form
U( O, x3, X ¢1~, T ~l~) =.3-'.eikOw(x3 Ik)A(k, X ~l~, T ~ ) ,
(3.2)
k
where A ( - k ) = A * ( k ) ensures that U is real and where per,odicity requires that the wave numbers k are integer multtpl~s o~ , n / L ~ . (More generally, the summation m a y be ret~laced by an integration over all real k.) T o second order in s, the equation of motion takes the form fro Oim~m,ll -"~ hlmMllm tM - 20.v;
s,.,,,,.,,,,,u,,,.,,~u,,.,.+ 2ovrt,,, ~,~ u,,,.~ + ~s;~,,,:+ s,,~,,o) :-~;!-, ~;,:j,
f3.31
and the corresponding boundary conditions ;~ the surface are 0 ~, = 2', Sj3,, w,,~'~/,,,,,; - c,,.x -r + S;3,,,r g;-,-& -S,~,,,.~.~u,;,..~ U .... j = 1, 2, 3, at x3 = ~.
(3.4a)
u~:"~=O
(~.4bl
atx3=0.
Here a~d in the foii,~wing, Gi-eek subscripts run only over I and 2.
154
D.F. Parker et aL / Projection method for nonlinear waves
For the derivation of an evolution equation for slow variations o f the amplitudes, it is not necessary to determine the field u (') explicitly. The system (3.3) and (3.4) is a linear, inhomogeneous boundary value problem for # ('). To ensure its solvability, compatibility conditions have to be imposed. They may be obtained by projecting (3.3) onto solutions fi(x, t) of the linearised system (i.e. multiplying by fi*, s u m m i n g over j and integrating over the region - c ¢ < x3 < 0, 0 < 0 < L~.) M a k i n g use of Green's theorem then gives
f' f ] "x. lPO"l~lJ'l [ .,_,
f
,,,
l~Jml¢;;;;I -- Ojdmj, fg,lj,jl.~m,M,__2SjJmMnNllj.j~m.MUn, =, ° (2) I -* r ,
+ 2Pv'~*rb., ~T(~) u,,,.,
-
~
~ U., +
"[
=0,
(3.5)
since the boundary terms which arise during application of Green's theorem vanish, in the present case, because of the boundary conditions imposed on the fields u and ft. By applying Green's theorem again, it is readily shown that the first two terms in (3.5) cancel, so yielding an equation which no longer contains the field u °). This form o f the solvability condition then provides the desired evolution equation since, by using the explicit expressions (2.3) for ~ and (3.2~ for U, it m a y be recast as
i{VrO~rt) + ~ }
~(k)= E KR(k,k')A(k')A(k-k').
(3.6)
The kernel KR for genera!ised Rayleigh wa~es may be written explicitly as an overlap integral over products of the functions wi(xs ]L) which describe ,he depth-dependence of linear surface waves. It has the ferm
= N S,s,,,,v
I
O
^
!iOA--,~)'.,i(x,l--'"~ ~ " ..... ' x
x [O,(k--t,')~,,,(x,l&-&')]} dx3
k-RR(k'/k)
(3.7)
~vhcn t,h,c fm:.ctions w arc normahscd so that O
N = 4pvk
f
~*)(x3 [ --k) rljmw~(x 3 ]k) dx3
(3.8)
--.c
is independent of k and where the operator {)j(k) is
Q.s(k)= 8suik + 8j~c32-~' O
(3.9)
The vector ( Va. V2, 0) is the group vele,city vector for generalised Rayleigh waves having wave vector in the positive.st-direction. ~,~ . . . . . . . ;,.~i ..... h,.,,;,,,, ,,¢,h~ ~. . . . . . 1 ~- ~ I . . . L ~ .a:._l . . . . . . . ~ ,~ _, . . . . . . . . -_ ~,. poCentiM and phase velocity of ,meat generalised Rayleigh waves need be known. In the. limit of infinite period L~ when the sum in (3.2) is replaced by an imegral, the sum over k' in (3.6) is likewise replaced by an integral. Under certain circumstances, the kernel KR vanishes identically tbr reasons of symmetry. This is the case for Bleustein Gulyaev (BG) waves. Linear BG waves are of shear-horizol.~tal polarisation, i.e. only w., and w~ are non-zero in (2.3) [19, 20]. These surface waves occur if, for example, the x:-axis is a two-fold rotation axis. We shall consider this geometry here. The fi~ st term on the right-hand side of (3.3) and of (3.4a) then
D.F. Parker et al. .1 Projection method for nonlinear waves
155
vanishes for j = 1 and 3, and the projection of (3.3) onto the generalised displacement field of a linear BGwave yields the result
8X~
~T
(3.1o)
J
To derive an evolution equation for slow variations of A(k) incorporating nonlinearity, we have to carry the expansion (3.1) to third order in e (i.e. we consider the term e3d 3)) and we introduce further stretched . (',) ., T(2) = ~:t. Equation (3.10) implies that the amplitudes A(k) depend on coordinates X~') = ~'xt, A~." = ~'x:, the stretched coordinates X] ~), T (t) only via the combination ~ :=X] t) - V~T"k Since |"2 = 0, they may also depend on X ~o. In order to proceed further, the second order field d ') has to be determined explicitly. It may be decomposed into a shear-horizontal/electric part d "x) and a sagittal part d ''s) as d :)= d"x)+ u("">, where
u('-,c)=-~ e~k°i[ ~--w(x~[k)] ~k A(k ) ,
LSk
(3.11a)
"
d'~'") = Z dk°iw( x-~Ik) ~
A (k) + d"'~ L
(3.11b)
k
The functions g,(x3[k) have to be determined from a pair of linear inhomogeneous orcfinary differential equations and bound?+ry conditions at x~ = 0. The part u t''~) of the second order solution may be constructed as
u"-,.+)(x, t ) = ~ e ~° E ~,(x, Ik, k')A(.~-')A(,~ -k'), k
(3.12)
k'
where the functions g do not depend on the stretched coordinates, and are unique, They are convenienfl~ determined by first seeking a particular solution of the equation of motion (3.3) and then adding a solution gU,) of the corresponding homogeneous equation to satisfy the boundary conditions. The latter will be of the form -)
g'"'" " Ik, k')= ~ ; tt,~3
HJ~)(k, k') exp(kfl'~)x3),
(3.13)
(k > 0), where the quantifies fl(') are the roots of the secular equation det(Sj.:.v.:_g_jg,::+ 6j,~,po')= 0
~3.14)
with positive real part. Here, ;~I= i6jn + :36j~ and v is the phase velocity of the linear BG waves. Equatioa~ (3.14) need not necessarily have two solutions with positive real part. One of the two ,B~ may, for example, be purely imaginary. In this case, nonlinear BG waves are not fully localised at the surface. For the same reason, the functions ff(x3!k) are then not fully iocalised at the surface. This has the consequence that if a BG wave is excited with high intensity at the surface of a piezoelectric medium, it will radiate ener~' into the bulk medium via the nonlinear coupling.
D.F. Parker et al. / Projection method.lbr nonlinear waves
156
Having determined the second order field u(2k we now consider the shear-horizontal/electric part of the equation of motion of third order in e,
O
+ 2pvoj,,, ~
u,,,., + (sj.,,, +
sj,.,~) ~
~2
Urn,.
~2
+ S~...,: .~.~z
0 . (2.~)
U . , - {O V~ rt,,,- Sj,.,, } a---5 U . , - 2pv Vt rb.,-~g .,,,.,
+ (&.,.,,+ &~,,,~)~
(2,o)u,,,.., -,- (S,,,,2 + s~,.,,,.,) ~
a
(,,) u,,;::;
(3.15)
and corresponding boundary conditions o
131
__¢",
rr
t2.s)..l_l
-~,,,.,,.,,,,,,,.,,- ,_,,~,,,.,,,,.,.~,,,.,,,,,..,. C3
.(2,e).
+ Sj3,,,, a-~ u,, u(4~)=0
.-
~s,,,,.,,,,,,,,.,u,,,..,,u,.Nu,.~+ 0
s,,,,, aXi2 ) u,,,
. (2,s)
-,- S~3,,,2 aXe,-------S),,,, , j = 1,2. 3, at x3=0.
atx~=0.
(3.16a) (3. I ob)
At this stage, we again apply the projection method, using the solution ~* for BG wa'~cs. After appiying Grcca's lacorem twice, the following equation may be derived: f~" i" -'t)
-'
f,,.,
~'
0
0
,
{A
|
+S,~,,,,O,* 02 a (,,) _ _ , a u(_,,) " ax ~' a C;,, + S~...,,.z-~? ~aX._ u,c,'~f. S,M,,,.uj..~-~X~ .,, - ~,Sjj,,,,~,,,~peuj.jU,,,.,~U,,.,~,Up.e dx.~ dO = O.
(3.17)
,.,earn, boundary term~ vanish because of the boundary conditions applying to fi and to the fields occurring in (3.15). By using the explicit expressions (3.2), (2.3) and (3.1 lb, 3.12) for U, ti and g °''', equation (3.17) may bc cast into the desired evolution equation for the amplitudes A l k ) r
C-
..,,,.,.
f
C,
+ Y. K~c~.(k,k',k")A(k')A(k")AIk-k'-k').
(3.18)
Again. fl~r genera] non-periodic waveforms, the single and double summations in (3.18) may be replaced by. sin-le.., and double integrals., respectively. To evaluate the kernels K~mc!~and ,~"(:~n¢~numerically, we require
D.F. Parker et al. ,/Projection method for nonhaear waves
157
functions g(x~l k, k') in addition to the w(x~ Ik). Explicit expressions in terms of these quantities will be given elsewhere. (For K ~ see [21].) Here we state only the scaling properties _ ~(1) , / . l~c;~^, k , ) _-kKn~;(k /k),
K[l)tl.
KO.~,~. nov,, k.,, k , , ) = k 3 ~ ( k , / k , k
{3.19!
"/k), '
(3.20)
if a normalisation of the functions w(x3 Ik) is used that leaves (3.8) independent of k. We also emphasise that ~nGv~"~consists of two parts, one depending linearly on the coupling coefficients of the third order nonlinearity, and one depending quadratically on the coupling coefficients of the second order nonlinearity. This latter arises because nonlinear BG waves necessarily excite a sagittal displacement field. It is interesting to note that the evolution equation (3. ! 8), but without the X [~Lderivatives, has recently been derived for nonlinear wedge waves of uneven symmetry [22], which form another example of nondispersive guid,~d acoustic waves. Like BG waves of the form (2.3), such wedge waves do not couple resonantly to their second harmonic. Wedge waves of even symmetry do excite the second harmonic resonantly. Their evolution due to the influence of nonlinearity may therefore be described by an equation of the form (3.6).
4. Nonlinear surface waves in the presence of dispersion
If the system considered in Section 3 is modified in any way which introduces a length scale, the corresponding surface acoustic waves become dispersive. An important effect of dispersion is to inhibit the growth of higher harmonics of an initially sinusoidal wave. One may therefore seek a solution of the equation of motion and boundary conditions for the generalised displacement field u in the form of the asymptotic expansion (3.1), but with the first order term U containing only one harmonic wave with a slowly varying amplitude:
U=e'Vw(x3 Ik)A(X~ '~, X~ ~b, T"', X~ 2', X~ 2', T ~2~. . . . ) +c.¢. = U'A +c.c.
(4.1)
We have here defined ~ = k . x-coCk)t, and we choose as before k = (k, O, O) tor the wave vector of the carrier wave. The following derivation of an evolution equation for the complex amplitude A is kept sufficiently general to account for various mechanisms that lead to dispersion. These mechanisms include layering stratified parallel to the x~-x_,-plane, media with material properties varying continuously along the x3-direction a n d / o r varying periodically in the xrdirection with periodicity a, as well as periodic corrugation of the surface with profile function ~(x~)=((x~ + a ) . Without loss of gener:dity, we may assame L~ to be an integer multiple of a. For simplicity, we require the system to be independent of the x2-coordinate. In the case in which the material properties or the surface profile function are periodic, w depends not only on x, but also is periodic in x~, with period a. Likewise, the higher order functions u ~''~, n ~>2. in the expansion (3.7) depend periodically on xl, in addition to depending on ~, x2 and the stretched coordhlatcs: u'"~( q/, x~, x,, , . ¥ ~ ' . . . .
) = ~ " ~ I ~'.:. x~. ~'~ + a, X~ j~ . . . .
).
~4.2',
Another effect that gives rise to dispersion is the discreteness of the underlying atomic structure of the medium (the crystal lattice). As has been demonstrated by one of us (AAM, [23]) for a semi-infinite simple cubic cDstal, such effects can be incorporated into continuum theory to leading order of kao, where a0 is the lattice constant, by including a higher derivative term "n the bouada15, conditions and, in certain cases.
158
D.F. Parker et aL ,' Projection method for nonlinear waves
also in the equation of motion. The following derivatives can easily be extended to account for such terms, and the projection method is equally applicable. Since nonlinear terms in u q2~ need be retained only ~o terms quadratic in expressior~ (4.1), the field u ~2~ may be decomvosed into three parts as U~2~(l/t. . . . ) = U~2.0~(...) + { ¢2~2.1)(...) eiV, + ~2.2~(...) ¢2i~, + c.c. }.
(4.3)
Collecting the terms of second order in e in the equation of motion and making use of the linear independence of the factors e ~"~, n = 0 , 1, 2, yields the following three :ahomogeneous equations:
Oxa
aej.,MU.,.M
-pco rb.,u,,,
- - SjS.,M,,xO*,.M (L, NIAI2, - exj " - -Oxs - o~S,,,MU.,,M = 2~poJ rlj,,, Um ~
(4.4)
A
+ ( S'r'''' ~.,',~ + ~--X~ ~ S,,,,., r) C~,,, ~Xg---~ 5 A,
~..~ - '-X~ SjJ.,MU'.7.';~ ~..~ - - 21 ~X~ j &,.,M.~,O.,,MO.,,vA2" -4pro". rb.,U.T"
(4.5) (4.6)
where u~2'"~=~2'"~e'"~. Boundary conditions have to be appended. In particular, we require that the quantities
,bl S,,,,.,,<[I:~ + S,j,,~.,,,,~,.0,*,,. ,, C",,..,.IA I:
(4.7)
~,, ) arj,,,Mu,,,, w ~
(4.8)
&j,,,r 6.,,
iii A J &V r
fi;', S,,.,,,,,,ul,:.~] + ~:S,j., ,,,,, ~7,.,.,, &,. ,.A:
(4.9)
( j = 1, 2, 3) are continuous across interfaces and vanish at the surface, ck is a vector normal to the corresponding interface or surface.) For piezoelectric media, we require again that u~2)= 0 at the surface. We now project equation (4.5) onto the solution (2.3) of the corresponding linearised equation of motion and boundary conditions. Applying Green's theorem twice and making use of the boundary, conditions, we then obtain: r a e,l iN~ V r - ~ . ~ + ~-~-TF, ( d = 0,
(4.10)
where ~', .... L .~m LnN
r., , o
m
dv, dx 'I
- •
(4.11)
are ~he components of the group velocity vector and where
N=2O, [f' i'""~; ~7,. pr~,,,t~.,,dv, d,., "
(4.12)
Q.F. Parker
eI
ai. / Projecrion merhodfir rlorl~j~earuxmes
159
Equation (4.13) suggests the introduction of variables f l_“” = XF’ -. I/,P’“. can be verified of (2.3) into the linearised equation of motion and boundary conditions an ~~e~~~~~~t~~n wi the wave vector components, the general solution of (4.5) satisfying the correct bo~~d~r~ conditions is given by u(2.1)=
-i ,iw [
-w
a I ad+ _
akr
(4.13)
a@)
The amplitude a depends on the stretched coordinates. To determine the fields #*O)= F’O’IAl”
(4. Ma)
UW) = F(Zj.42,
(4.14b)
the system has to be specified in more detail. The presence of dispersion ensures the existence of unique bounded solutions to the corresponding equations for F”’ and Collecting now, in the equation of motion, all third-order terms having the factor exp(iyr), we obtain
where +j-p = s.,Jn&fnN
f
(2’2” &,,.Md:i? + ~hhwV , + ~sjJn,,,~~,~~'pper;f,.,,u,~,~v,.,.
We now project equation conditions satisfied by t
which no longer contains 8.” us, the n~ust vanish. to t transforming
(4.16)
and again use Green’s tbe0rem together with the boundary (4.15) on s occurring in 14.15). This
esired evolution equation,
D.F. Parker et aL / Projection method for nonlinear waves
160
Equation (4.18) is a generalisation of the nonlinear Schrrdinger equation which includes effects of gradual lateral variation of a wave profile. By some manipulations of the corresponding terms in (4.17), it can be shown that the matrix (.Ora) is given by ~2 1"2ra(k) = ~ k r ~ka co(&)
(4.19)
apart from an irrelevant antisymmetric part. For the nonlinearity coefficient coc2~ the projection method yields the following expression: L~ N
U.,.,~U.,NUp, e fo'-,fj ~,,~.Sjj,,,M.Npp Uj.j -**-+
~,
~
1o)
/'r*
~(2)1
SSs.,X~.N Us,s[ U,,,.MF..x + ,-, .,,M. ..N~ } dx3 dxl ,
(4.20)
N being defined in (4.12). This expression can be evaluated easily once the generalised displacement field /~ of the linear surface wave with wave vector k and the functions F cm and F ~'~ are known. It consists of three parts: a contribution proportional to the coupling coefficients of the third order nonlinearity, a contribution from the static part of the second order solution and a contribution from the second harmonic. The latter two are linear combinations of products of two coupling coefficients of the second order nonlinearity. In the case of weak dispersion, the part resulting from the second harmonic is dominant if, in the limit of zero dispersion, the surface wave couples resonantly to its second harmonic. An example for this situation is the propagation of Rayleigh waves on a surface of a homogeneous elastic medium covered by a thin film of different material, the film thickness being much smaller than the wavelength. The quantity ~c=[2co(k)-co(2~)]/co(&).
I~'l<<1
(4.21)
may then be used as an additional expansion parameter. Projecting (4.6) on the solution of the linearised equation of motion and boundary conditions u ° ' ( x , t) = w(x312k) e'~2~'' -'°~"~m
(4.22)
we obtain, with the help of Green's theorem. (4co2(k) - co2(2k))
- :~
,fLf °
I Ll I 0 1¢(#)*p~jml4('"¿;,, l| "~"~ a,.u.~3dxt } --&
, jj,,,.~i,,xu?,.j ~,,,.,~U,,.xA" d.v~ dxt.
(4.23)
This relation implies that u12'~'* contains a contribution of order O(~c-~) (unless the right-hand side is incidentally of order O(~')). From the equation cf motion (4.6) and associated boundary conditions, it tbNows that this contribution is of the form bA2g£(2x, 2t). T h e coefficient b can be determined directly from ~4.23L where u ~'-'! may then be reptaced by bA'-~'{2x, 2t). Inserting this result into (4.20) with (4.14b), the following expression is obtained for the contribution to the nonlinearity coefficient e~?'~ to leading order in &-:
~cN2
D.F. Parker et al. / Projection method for nonlinear waues
161
where
n=!
20 o, (x,
'
Lt
_~
"
"
0 dx3 dx,
(4.25)
'
and
N2 = 8N to2(k)
O*(2x. 2t)Ptb.,U,,,(2x, 2t) cL,c3dx,.
(4.26)
gl
It is emphasised that for the evaluation of this leading order contribution, only the frequencies and generalised displacement fields of linear surface waves have to be known. An important implication of the result (4.24) is that the sign of the nonlinearity coefficient co¢2~ in the systems to which (4.24) applies, is determined by the sign of ~c alone. In such systems of weak dispersion, it should be expected that nonlinear terms of higher order in (4.18) will also be important.
5. Conclusions In conclusion, we have demonstrated how the projection method based on Green's theorem in elasticity theory, in conjunction with multiple scales theory, can be used to derive efficiently nonlinear evolution equations for surface acoustic waves in the presence and absence of dispersion. The procedure yields expressions for the coupling coefficient or kernels in these equations ~hat can easily be evaluated. These expressions have the form of overlap integrals and, in ~ystems where nonlinearity of second order dominates, they involve only the displacement fields and electrostatic potentials for linear surface acoustic waves.
References [iI [2] [3] [4]
R.W. Lardner, "Nonlinear surface acoustic waves on an elastic solid", Int. d. Eng~g. Sci. 2!, 1331-1342 (1983). R.W. Lardner, "Nonlinear surface acoustic waves on an elastic solid of general anisotropy", J. Elast. 16, 63 73 (1986). D.F. Parker, "Waveform evolution fo,' nonhnear surface acoustic waves", Int. J. Engng. Sci. 26, 59 75 (1988). T. Sakuma and Y. Kawanami, "Theory. of surface acoustic soliton. I. Insulating solid", ~%ys. Rev. B20, 869 879 (1984), "Theory of surface acoustic soliton. II. Semiconducting solid", Phys. Rev. B29, 880-888 (1984). [5] K. Bataille and F. Lund, "Nonlinear waves in elastic media", Physica 16D, 95- 104 (1985). [6] A.A. Mamdudin, "Nonlinear surface acoustic waves and their associated surface acoustic solitons", in Recent Deuelopmems in Surfiice Acoustic Wanes, eds. D.F. Parker and G,A. Maugin, Springer, Heidelberg (1988) pp. 62 71, [7] H. Hadouaj and G.A. Maugin, "Une onde solitaire se propageant sur un substrat ~lastique reconvert d'un film mince", C R. Acad. Sci. Paris 309, 1877 1881 (1989). [8] A.A. Maradudin and A.P. Mayer, "Surface acoustic waves on nonlinear substrates", in Nonlinear Waves in Solid State Physics, eds. A.D. Beardman, M. Berto!otti and T: Twardowski, Plenum. New York (1991~ pp. 113 161. [9] D.F. Parker and F.M. Talbot, "Analysis and computation for nonlinear elastic surface waves of permanent lbrm", 5. Elast. t5. 289 426 (1985). [101 A.G. Eguiluz and A.A. Maradudin. "'Effective boundary conditions for a semi-infinite elastic medium bounded by a rough planar stress-free surface", Phys. Rev. B28, 711 327 (1983). [lll D.F. Nelson, Etectric, Optic, and Acoustic Interactions #, Dielectrics, Wiley, New York (1979}. [12] G.A. Maugin, Nonlinear Etectromechanical Effects and Applications, World Scientific, Singapore (1985). [13] D.B. Taylor and S,G. Crampin, "Surface waves in anisotropic media: Propagation in a homogeneous piezoelectric halfspace", Proc. Roy. Soc. A 364, 161 179 (1978). [ 14] G.E. Tupholme, "Multiple scale techniques applied to nonlinear elastic piezoelectric surface waves", in Recent Deveh~pmems in Surface Acoustic Waves, eds. D.F. Parker and G.A. Maugin, Springer, Heidelberg (19S8) p~,. [4 20.
162
D.F. Parker et al. / Projection method for nonlinear waves
[15] E.A. David and D.F. Parker, "Nonlinear evolution of piezoelectric SAWs", in Recent Development in Surface Acoustic Waves, eds. D.F. Parker and G.A. Maugin, Springer, Heidelberg (1988) pp. 21-29. [16] H.F. Tiersten and J.C. Baumhauer, "'An analysis of second harmonic generation of surface waves in piezoelectric solids", J. Appl. Phys. 58, 1867- 1875 (1985). [t7] D.F. Parker and E.A. Dahid, "Nonlinear piezoelectric surface waves", Int. d. Engng. Sci. 27, 565-581 (1989). [18] E. Dieulesaint and D. Royer, Elastic Waves in Solids, Wiley, Chiehester (1980). [19] Yu. V. Gulyaev, "Electroacoustic surface waves in solids", Zh. Eksp. Teor. Fiz. Pis. 9, 63-65 (1969) (So~. Phys.-JETP Lett. 9, 37-38). [20] J.L. Bleustein, "A new surface wave in piezoelectric materials", ,4ppl. Phys. Lett. 13, 412-413 (1968). [21] A.P. Mayer. "Evolution equation for nonlinear Bleustein-Gulyaev waves", Int. J. Engram. Sci. 29. 999-1004 (1991). [22] V.V. Krylov and D.F. Parker, "'Harmonic generation and parametric mixing in wedge acoustic waves", Wave Motion 15, 185200 (1992). [23] A.A. Maradudin, "Surface acoustic waves", in Nonequilibrium Phonon Dynamics, ed. W.E. Bron, Plenum, New York (1985) pp. 395-599.