Bulletinofthe SeismologicalSocietyof America,Vol.79, No. 6, pp. 1995-1999,December1989

A NOTE ON T H E RAYLEIGH H Y P O T H E S I S
AND T H E AKI-LARNER M E T H O D
BY FRANCISCO J. SANCHEZ-SESMA, MICHEL CAMPILLO, AND KOJIRO IRIKURA
The seismic response of laterally irregular stratified media has been studied bymany authors (for recent reviews, see, e.g., S~nchez-Sesma, 1987; Aki, 1988).
Significant advances have been obtained since the pioneering work of Aki andLarner (1970). In their method, the diffracted and refracted fields are represented
by superposition of plane waves, including inhomogeneous waves, propagating inmany directions. The total motion is obtained through integration over horizontal
wavenumber. Under the assumption of horizontal periodicity of the structure, theintegral is replaced by an infinite sum. Truncation of this sum and spatial Fouriertransformation of boundary conditions lead to a system of linear equations for thecomplex coefficients of the horizontal wavenumber expansion. The Aki-Larnerdiscrete wavenumber method has found many applications in seismology, due
mainly to its flexibility to model elastic wave fields (see e.g., Bouchon, 1973; Bouchon
and Aki, 1977a, b; Bouchon, 1979; Bard and Bouchon, 1980a, b; Bard, 1982; Bardand Gariel, 1986). Its major disadvantage, namely the difficulty to model wave fields
near very steep interfaces, has been corrected by Bouchon (1985) and Campillo andBouchon (1985), who used a single-layer expansion of the fields, similar to the oneused by S~nchez-Sesma and Esquivel (1979). Bouchon and Campillo constructed
the full-space Green's function by discrete horizontal wavenumber summation. In
their treatment, the sources are located along the interface and the truncation of
the sums guarantees a regular field representation everywhere. On the other hand,Kawase (1988) used a discrete wavenumber representation for the Green's function
and a rigorous boundary element method formulation. He carried out analytical
integrations in the boundary elements and his results for the surface response of asemicircular canyon on a half-space are in excellent agreement with the analyticalsolution for S H waves (Trifunac, 1973) and with numerical results for P, SV, andRayleigh waves (Wong, 1982; S~nchez-Sesma et al., 1985; Dravinski and Mossessian,1987).
The failure of the original Aki and Larner (1970) method to accurately represent
wave fields close to large-slope interfaces has been attributed to the limitation
imposed by the so-called "Rayleigh hypotesis," which consists in representing thediffracted fields, say, in an irregular surface half-space with an integral in thehorizontal wavenumber which includes downgoing waves and inhomogeneous planewaves. That integral does not include explicitly upgoing waves, therefore this fact
has been traditionally considered the cause of the failure (see, e.g., Aki and Richards,1980).
In this note, we show that the Rayleigh ansatz is quite good and that the reasonsfor the failure lie elsewhere. They are of numerical nature. Our purpose in this note
is to contribute to a better understanding of this powerful technique. In whatfollows, we consider the problem of incident plane S H waves upon a semicircular
canyon on the surface of a half-space and write the Trifunac (1973) exact solution
as an integral in the horizontal wavenumber. The diffracted part of the solution isrepresented with explicit downgoing waves plus inhomogeneous waves. This makes
clear that upgoing diffracted energy admits a representation in terms of inhomogeneous waves only.
1995

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EXACT SOLUTION

Consider the semicircular canyon of radius a on the surface of a homogeneous,isotropic, elastic half-space shown in Figure 1 and assume incidence of harmonicplane S H waves with incidence angle 00. In this case, the equation of motion for
the antiplane displacement v is given by(~2v02V(1)

ax 2 + az 2 + q2v = O

where q = ~/~, co = circular frequency and fl = shear-wave velocity. Free-boundary
conditions imply zero normal derivative at the irregular surface. This and theSommerfeld (1949) radiation condition for the diffracted field are fulfilled by theexact solution (Trifunac, 1973) which is given byv -- v i + v r - 2 ~bmHm(1)(qr)cosmO(2)

m=0w h e r e v i ' r = e x p ( i k o x • i'yoZ) are the incident and reflected waves for the free-fieldsolution (it is assumed here and hereafter t h a t the time dependence is of the forme-i~t), ko = qcos00, 70 = qsin0o, r = (x 2 + z2) 1/2, tan0 = z / x , bm = t m i m c o S m O o J ' m ( q a ) /Hm(1)'(qa), ~rn = N e u m a n n factor (=1 if m = 0; = 2 if m > 0), Jm(" ) = Bessel functionof the first kind and order m and Hm(1)( • ) = Hankel function of the first kind andorder m. T h e primes mean derivative with respect to the argument. This exact
solution has been computed by T r i f u n a c (1973) up to a normalized frequency of qa= 3~. An asymptotic evaluation for higher frequencies (for a related problem ofelectromagnetic waves) is due to Franz (1954) who discovered the so-called creeping
waves. T h e s e waves are included in the diffracted part of the field and have been
recently computed by Kawase (1988).
SOME HORIZONTAL WAVENUMBER INTEGRALS

It is well known t h a t the H a n k e l function of zero order can be written by meansof (Lamb, 1904)
Ho(1)(qr ) = _1
7re x p ( i k x + i'yz) --,dk~(3)"ywhere ",/-- (q2 _ k2)~/2, I m ( ~ ) >= 0 and z _>-0% This equation represents a cylindricalfunction as a superposition of plane waves in terms of the horizontal wave number.
It is possible, through direct derivations of equation (3), to show t h a t2Oi- //%-zFIG. 1. Geometry of the problem.

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1997

Hm(1)(qr)cosrnO = ~1 f_~ { ( ~ - ~ - ~ ) m + ( 7 7 - i k ) }me x p ( i k x + i ' y z ) - - . ( 4 )d kq

It is also possible to obtain this expression from an integral representation ofHankel function (see equation (8), of Watson, 1958). A similar representation has
been used by Scheidl and Ziegler (1978).
A very interesting aspect arises from this integral representation. It is easy toshow that the left hand side of equation (4) satisfies the reduced wave equation andthe free-boundary condition at 0 = 0, ~r, except at r = 0 because of the singularitiesof Hankel function. That the right-hand side is also a solution is clear, but thefulfillment of zero normal derivative of the integral at z = 0 is not evident. It isconvenient to note that the expression inside brackets in the integrand is an integer
polynomial of order rn in the horizontal wavenumber k. Therefore, the boundary
condition at z = 0 is satisfied in the distribution sense because--in2~r~k n e x p ( i k x ) d k _ d%(x)__dx ~(5)where 5 (.) = Dirac delta function and n is an integer.
THE EXACT SOLUTION AS AN INTEGRAL IN THE HORIZONTAL WAVENUMBER

From equations (2) and (4) it is possible to write the exact solution as~_c~v = exp(ikox - i%z) +A(k)exp(ikx(6)+ i'yz)dk,

where z _->0 ÷, andA ( k ) = 5(k - ko) - - 1

F~ bm
m=0+V -1.

(7)This expression shows that the diffracted part of the solution can be written interms of explicitly downgoing waves plus inhomogeneous waves. This makes clearthat the upgoing diffracted field admits a representation in terms of only inhomogeneous waves. The complete evaluation of this part of the solution requires dealing
with infinite integrals or, if the periodicity is invoked, with infinite sums. It seems
convenient at this point to mention that, for a related problem, Millar (1973)
established the completeness of the set of plane waves and showed that there is "alinear combination of elements of the set that converges on the boundary to theprescribed values, in the mean square sense as N--~ ~." Furthermore, he establishedthat at points not on the surface, "the expansion converges uniformly to the sought
solution whether or not the Rayleigh hypothesis is satisfied."
CONCLUSIONS

The difficulty of the Aki and Larner (1970) discrete wavenumber method toaccurately represent wave fields close to large slope interfaces is due to the very
slow convergence associated to this representation. The Rayleigh ansatz is quite
good. We illustrated this fact with one of the few problems that has an exact

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SHORT NOTES

solution and admits a relatively simple integral representation. In this note, wehave shown that this integral contains everything and that the locally upgoingenergy is represented by inhomogeneous waves.
The methods of Bouchon (1985), Campillo and Bouchon (1985), and Kawase(1988) are pointed in the appropriate direction as they explicitly include upgoingwaves and, in practice, solve the problem. However, in the search for more efficienttechniques, it seemed appropriate to point out a rather subtle problem and contribute to a better understanding of a powerful technique.
ACKNOWLEDGMENTS
Thanks are given to K. Aki for his encouragement and criticism and to P.-Y. Bard, M. Bouchon, M.
Horike, H. Kawase, K. Kohketsu, and Y. Shinozaki for their valuable comments. The suggestions andhelp of T. Iwata and K. Yumoto are greatly appreciated.
This work was initiated during our visit to the Department of Geological Sciences of the Universityof Southern California, Los Angeles. It was finished while one of the authors (F. J. S.-S.) was on leavefrom the National University of Mexico at the Disaster Prevention Research Institute of Kyoto
University, Japan.
This research was partially supported by Consejo Nacional de Ciencia y Tecnologia (CONACYT), by
the National Science Foundation under Grant EAR-8610905, and by the Japanese Society for thePromotion of Science.
REFERENCES
Aki, K. (1988). Local site effects on ground motion, in Earthquake Engineering and Soil Dynamics II:
Recent Advances in Ground-Motion Evaluation. J. L. Von Thun, Editor, Geotechnical Special
Publication No. 20, 103-155, Am. Soc. Civil Eng., New York.
Aki, K. and K. Larner (1970). Surface motion of a layered medium having an irregular interface due toincident plane S H waves, J. Geophys. Res. 75,933-954.
Aki, K. and P. G. Richards (1980). Quantitative Seismology: Theory and Methods, W. H. Freeman andCo., San Francisco, California.
Bard. P.-Y. (1982). Diffracted waves and displacement field over two-dimensional elevated topographies,Geophys. J. R. Astr. Soc. 71,731-760.
Bard, P.-Y. and M. Bouchon (1980a). The seismic response of sediment-filled valleys. Part 1. The caseof incident S H waves, Bull. Seism. Soc. Am. 70, 1263-1286.
Bard, P.-Y. and M. Bouchon (1980b). The seismic response of sediment-filled valleys. Part 2. The caseof incident S H waves, Bull. Seism. Soc. Am. 70, 1921-1941.
Bard, P.-Y. and J. C. Gariel (1986). The seismic response of two-dimensional sedimentary deposits withlarge vertical velocity gradients, Bull. Seism. Soc. Am. 76, 343-366.
Bouchon, M. (1973). Effect of topography on surface motion, Bull. Seism. Soc. Am. 63,615-632.
Bouchon, M. (1979). Predictability of ground displacement and velocity near and earthquake fault. An
example: the Parkfield earthquake of 1966, J. Geophys. Res. 8 4 , 6149-6156.
Bouchon, M. (1985). A simple, complete numerical solution to the problem of diffraction of S H waves
by an irregular surface, J. Acoust. Soc. Am. 77, 1-5.
Bouchon, M. and K. Aki (1977a). Discrete wave number representation of seismic source wave fields,Bull. Seism. Soe. Am. 6 7 , 259-277.
Bouchon, M. and K. Aki (1977b). Near-field of seismic source in a layered medium with irregularinterfaces, Geophys. J. R. Astr. Soc. 50, 669-684.
Campillo, M. and M. Bouchon (1985). Synthetic SH-seismograms in a laterally varying medium by thediscrete wave number method, Geophys. J. R. Astr. Soc. 8 3 , 307-317.
Dravinski, M. and T. K. Mossessian (1987). Scattering of plane harmonic P, S V and Rayleigh waves bydipping layers of arbitrary shape, Bull. Seism. Soc. Am. 77,212-235.
Franz, W. (1954). Uber die Greenshen Funktionen des Zylinders und der Kugel, Z. Natur[orsch 9a, 705716 (in German).
Kawase, H. (1988). Time-domain response of a semi-circular canyon for incident SV, P and Rayleighwaves calculated by the discrete wavenumber boundary element method, Bull. Seism. Soc. Am. 78,
1415-1437.
Lamb, H. (1904). On the propagation of tremors over the surface of elastic solid, Phil. Trans. Royal Soc.
Lond, A 2 0 3 , 1-42.

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Millar, R. F. (1973). The Rayleigh hypothesis and a related least-square solution of scattering problems
for periodic surfaces and other scatterers, Radio Science 8, 785-796.
Shnchez-Sesma, F. J. (1987). Site effects in strong ground motion, Soil Dyn. Earthquake Eng. 6, 124132.
S~nchez-Sesma, F. J. and J. A. Esquivel (1979). Ground motion on alluvial valleys under incident plane
SH waves, Bull. Seism. Soc. Am. 69, 1107-1120.
Shnchez-Sesma, F. J., M. A. Bravo, and I. Herrera (1985). Surface motion of topographical irregularities
for incident P, SV, and Rayleigh waves, Bull. Seism. Soc. Am. 75, 263-269.
Scheidl, W. and F. Ziegler (1978). Interaction of a pulsed Rayleigh surface wave and a rigid cylindricalinclusion, in Modern Problems in Elastic Wave Propagation, J. Miklowitz and J. D. Achenbach
Editors, John Wiley and Sons, New York.
Sommerfeld, A. (1949). Partial Differential Equations in Physics, Academic Press, New York.
Trifunac, M. D. (1973). Scattering of plane SH waves by a semi-cylindrical canyon, Int. J. Earthquake
Eng. Struct. Dyn. 1,267-281.
Watson, G. N. (1958). A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press,Cambridge, p. 180.
Wong, H. L. (1982). Effect of surface topography on the diffraction of P, SV, and Rayleigh waves, Bull.
Seism. Soc. Am. 72, 1167-1183.
INSTITUTO DE INGENIER]A,UNAM
CD. UNIVERSITARIA,APDO. 70-472
COYOAC.~N04510, MEXICO D. F.
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OBSERVATOIRE DE GRENOBLE
UNIVERSITI~ JOSEPH-FOURIER
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DISASTER PREVENTION RESEARCHINSTITUTE
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Manuscript received 13 February 1989