Bulletin of the Seismological Society of America

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1 Bulletin of the Seismological Society of America Vol. 69 August 1979 No. 4 ELEMENTARY SOLUTIONS TO LAMB'S PROBLEM FOR A POINT SOURCE AND THEIR RELEVANCE TO THREE-DIMENSIONAL STUDIES OF SPONTANEOUS CRACK PROPAGATION BY PAUL G. RICHARDS ABSTRACT Certain exact solutions to Lamb's roblem (the transient resonse of an elastic half-sace to a force alied at a oint) involve the comutation merely of three square roots, and about ten arithmetic oerations (+, -, x, +). They arise when both source and receiver lie on the free surface. It is just these solutions which are needed in a method due to Hamano for obtaining the sli function (dislacement discontinuity), as a function of sace and time, for lanar tension cracks and shear cracks which grow sontaneously with arbitraryshae. The solutions are described here in detail, for an elastic medium with general Poisson's ratio. They include erhas the simlest-ossible examle of the P-wave. INTRODUCTION A thorough understanding of motions in an elastic half-sace, subjected to an alied force, is an essential art of wave-roagation theory needed to interret seismic waves. For this reason, half-sace roblems have been the subject of an enormous literature, beginning with Lamb's (1904) classic study of dislacements set u by forces alied at a oint and along a line on the free surface. Here, I give some new solutions, these being the horizontal motions of the free surface for a horizontal force alied as a ste in time at a oint also in the free surface. (Throughout this aer, the half-sace is oriented with a horizontal free surface. Taking Cartesian axes with x3 as the deth coordinate, into the half-sace, the free surface is x3 = 0.) Whatever the value of Poisson's ratio, the new solutions (which augment the work of Pekeris, 1955; Chao, 1960; Mooney, 1974) are extremely simle to comute. However, these formulas would be only a minor curiosity if it were not for one very imortant alication, in which seed of comutation is essential. This alication is suggested by Hamano's (1974) method for studying sontaneous crack roagation. Since it is the larger roblem of crack roagation which has motivated the resent study, I shall, in what follows, give a brief review of Hamano's method, before giving the simle solutions to Lamb's roblem. MOTIVATION Within an infinite homogeneous elastic medium, initially at rest, suose that a crack nucleates at time t = 0 and subsequently grows within the lane xa Then a useful reresentation of dislacement u -- u(x, t) throughout the medium can be 947

2 948 given as PAUL G. RICHARDS un(x, t) = dr d4~ d~2 Gn(X, t - ~; ~1, ~2, 0, 0)T(~,, $2, ~). (1) f;; ~ c~ Here, Gn(X, t; $, r) is the Green function for the medium, being the n-comonent of dislacement at (x, t) due to a unit imulse alied in the -direction at osition and time ~. For uroses of comuting Gn, it is required that the whole lane x3 = 0 be a traction-free surface. T(~I, ~2, r) is the actual traction occurring on the whole lane x3 = 0 of which the crack is a art. Equation (1) is described further, and roved, by Das and Aki {1977) and Aki and Richards {1980, their equation 2.43). Intuitively, the above reresentation can be understood as relacing the actual (crack) source of radiation by a whole lane, searating the medium into two halfsaces. Into each half-sace, waves are radiated due to the same tractions as those set u by the crack, alied over the half-sace surfaces. From the sace-time element dr d41 d~2 there is an alied imulse of strength d~ d~, d42 T(~l, 42, T) in the -direction. If the dislacement contribution to un(x, t) from this element is to be considered in isolation from tractions acting elsewhere on x3 = 0, then the aroriate Green function Gn~ must be constrained by having zero traction over the surface of the half-sace. Hence, it must be a solution to Lamb's roblem. Hamano (1974) ointed out that for shear cracks and tension cracks, ~ soluble scheme for the dislacement discontinuity [u], say, across the crack can be set u from equation (1) by considering the x osition itself in the crack lane, x = (x~, x2, 0), and using symmetry roerties of u and T across x3 = 0 to constrain the dislacement and traction on different arts of the lane of the crack. For a general tension crack [u] = [0, 0, u3] and T = (0, 0 T3) so that the only Green function needed is G33. For a general shear crack, [u] -- [ul, u2, 0] and T -- (T, Te, 0) so that the only Green functiohs needed are Gn, G12, Gel, and Gee (see Das and Aki, 1977, for a related study of two-dimensional cracks). The jum in u3 is zero, because oosite faces of the crack remain in contact. T3 is zero, because lanar shearing cannot change the normal stress on the crack lane. Together with G33 for tension cracks, these five different Lamb roblems/green functions need be studied only for the case that both source and receiver lie in the free surface of the half-sace. Haman0's method is imortant in offering the chance to study sontaneous crack growth for comletely general shaes of lanar cracking. This aer contributes to that goal, by showing that just these Green functions are almost trivially simle to comute. For comleteness, a single integral is also given below, in terms of which the remaining four comonents, GI~, G2~, G~I, G32, can efficiently be comuted. FORMAL STATEMENT OF PROBLEM, AND ITS SOLUTION In this section, exlicit formulas are derived for G H (x b x2, 0, t; 41, ~2, 0, 0), this n being the n-comonent of dislacement at osition (x~, x2, 0) and time t, within the free surface (x3 -- 0) of a homogeneous, isotroic elastic half-sace, due to a unitste force in the -direction alied also within the free surface, at osition (~1, ~2, 0), the ste occurring at time 0. Once this solution has been found, Gn for an imulse (as required in the section above) is given by a Gri (x, t - r; ~, O) = =- G~ (x - ~, s; O, O)I,=t-~. 08

3 _ 1 LAMB'S PROBLEM AND SPONTANEOUS CRACK PROPAGATION 949 Since related roblems have had such a wide exosure, I shall abbreviate the descrition of how a solution is obtained. Thus, in general, the solution to Lamb's roblem for a oint source can be obtained as an integral over just one variable. In our case, I(T H(T-1) LPdP } Imag 2)~/2 (~H(xl, X2, O, t; O, O) -- 7T2# r ~..,'1 (2) (T2 - [(A - 2P2) 2 + 4XYP e] where ~ = rigidity, r = (x, e + x22) I/e, H is the unit Heaviside ste function, and caital letters in the integrand denote dimensionless quantities A --- 2/fi2 (~ = P-wave seed, fl = S-wave seed) T -- at/r (T = 1 being the P-wave arrival time) X = (1 - ~)l/2 or -i (C _ 1)1/2, Imag ( ) denotes the imaginary art of ( }, and Y = (A - 2)l/e or -i {2 _ A)l/e, L~ = ((T 2 - Pe)[2Y - 4X + (A - 22)/Y] + AY} cos e ~ - (Te[2Y - 4X + (A - 2Pe)/Y] - AY} sin s L12 = L2~ = (2T ~ - P2)[2Y - 4X + (A - 22)/Y] cos ~ sin L22 = ((T 2 - Pe)[2Y - 4X + (A - 22)/Y] + AY} sin s O - (T212Y - 4X + (A - 2P2)/Y] - AY) cos 2 and Lla/cos ~ = Lea/sin ~ = -La~/cos ~ = -La~/sin ~ = T(A - 2P 2) - 2TXY, where ~ is given by xl - r cos (~, x2 = r sin ~, so that ~ is the azimuth to x. Equation (1) can be written down from Johnson (1974, his equations 26 to 34, but using 2 for his a2t2r-2 - a22). Our variable P is a (dimensionless) horizontal slowness, and equation (1) is essentially a Cagniard solution in the form advocated by Helmberger (1968). Both source and receiver lie in the free surface, so the Cagniard ath lies just above the real P axis in the comlex P lane (see Figure 1, and legend). In fact, the P integrals for G H, G H, G el, ~ Gee, H and G33 H can be given in closed form. This is not ossible for the four remaining entries in G H, but these four are all roortional to just one integral which is still fairly simle to comute. We note first that G H (x~, x2, 0, t; 0, 0) -- [I~(T)cos 2 ~ - I2(T)sin 2 ~]/(~r/zr) G H = G~ =- [I1 + I2]cos ~ sin ~/(~#r) G H = [Ilsin 2 ~ - I2cos 2 ~]/(Tr#r) G3~ = I3( V) / (~tzr) (3) where arguments have been written out exlicitly only for the first of equations (3).

4 950 PAUL G. RICHARDS (a) ~" Poles,. (22//92< 3,11,.. (2a/f~2 >3.11 0"~ T 1 oj (2 ct ReaIP /3 )" (b) (2 ± y = R3 2 FIG. 1. The Cagniard integration ath for equations (2), (4), and (6) is shown as a solid heavy line. (a) Singularities of these integrands. They consist of branch cuts trending to the right along the real P axis from oints 1 and a/fl, and a ole at a/t (T being the Rayleigh wave seed). As well as the ole at a/'~ = R31/2, we also show schematically the oles at RI 1/2, Re ~/2 in the right half-lane. R~, R2, and R3 are roots of the Rayleigh cubic in 2, (A - 22) * - 16X2Y2P (1 - A)(P 2 - R1)(P 2 - R2)(P 2 - R3). For A < 3.11, R1 and R2 are real and lie between 0 and 1. But for A > 3.11, R1 and R2 are comlex conjugates. Accurate ~locations for different A are given in Figure 3b. (b) Path of integration in the vicinity of the Rayleigh ole. In the limit, as the semi-circle radius shrinks to zero, integration reduces to a rincial value integration lus -i~r x residue at the Rayleigh ole. The residue is imaginary from the integrands (4) of I1, 12,/3, and hence gives zero net effect when the imaginary art is evaluated after multilication by -iv. But the residue from integrand (6) for L is real, leading to a non-zero contribution from the Rayleigh ole when T > a/t. Just three functions of dimensionless time are needed comonents of G H, namely to evaluate these five II(T) -- Imag (~-~ _-- ~2~/2 ((T2 - P2)[2Y- 4X + (A- 2P2)/Y] + AY) } (A 22) 2 + 4XYP 2 PdP I2(T) = 1 r Imag {fl T H(T- 1) (T 2 _ 2)1/2 ( T212 Y - +-'(~:'~P-~2(A-2P2)/Y]~ 4X" ~ -AY) PdP } (4) and /3(T) -- 1 Imag ~2~/2 (A + (~'E-- _ 22)2 4XYP2 It follows from equation (3) that/1 and Is are dislacements within the vertical lane containing source and receiver, like the dominant motion in P-SV, whereas Is is a dislacement transverse to this lane, like the dominant motion in SH. Analytic exressions are given below for each of these three integrals. A fourth dimensionless

5 LAMB'S PROBLEM AND SPONTANEOUS CRACK PROPAGATION 951 solution to Lamb's roblem is introduced via G~ (x~, x2, 0, t; 0, 0) --/4(T)cos O/(~r#r), G2 H = I4sin O/(~rtLr), G H = -Lcos ~/(Trt~r), and G H -- -I4sin O/(~rt~r). (5) The integral for this solution is (6) which cannot be given in closed form. For a vertical oint force, Pekeris (1955) obtained a closed-form solution for the vertical dislacement (/3, this aer) and a sum of ellitic integrals for the horizontal dislacement (/4, this aer). His solutions were restricted to the case a2/fi 2 = 3 (Poisson's ratio ), and Mooney (1974) indicated how the evaluation of I3 might be carried out for any value of a2/f12 (though Mooney did not ublish the solution formulas). For a horizontal oint force, in a medium with a2/fl 2 = 3, Chao (1960) obtained closed-form solutions for the horizontal dislacements (/1 and /2, this aer) and a sum of ellitic integrals for the vertical dislacement. Solutions themselves have not reviously been given exlicitly, for general a2/fi 2, for any one of the four basic (dimensionless) solutions Ii. In every case, the basic aroach involves writing 1 (A - 22) 2-4XYP 2 (A - 2P2) 2-4XYP 2 (A - 2P2) 2 + 4XYP 2 = (A - 2P2) 4-16X2Y2P 4 ~- Cubic(P 2) (7) so that the new denominator, of sixth order in P, is real throughout the P-axis integration and has no branch cuts. The imaginary arts of the new integrals are easy to identify (together with a semi-residue contribution to/4 from indenting around the Rayleigh ole: see Figure 1 legend). If roots R~, R2, R3 are found for the cubic in 2, integration for I1, /2, 13 becomes ossible using the artial-fraction decomosition 1 a b c (A - 2PC) 4-16X2Y~P ~ 2 _ R, P~ - R _ Ra" (8) The Rayleigh ole lies at P -- R31/2 = a/y (always on the real P axis, just to the right of P = a/fi, since Rayleigh wave seed ~, is a few er cent less than fl). If Poisson's ratio is less than a critical value, aroximately 0.263, then R1 and R2 are real and lie between 0 and 1. But, for greater values of Poisson's ratio, R~ and R2 are comlex conjugates, as are a and b in equation (8), although c is always real. In this case there are oles in the comlex P lane which can be associated with the socalled t5 wave (Gilbert et al., 1962; Chaman, 1972; Aki and Richards, 1980), aearing between P- and S-arrivals. R~ ~/2 and R2 ~/2 lie on a Riemann sheet different from that which contains the Cagniard ath.

6 952 PAUL G. RICHARDS Substitution of equation (8) into equations (7) and (4) leads to 24 integrable terms for each of/1 and/2, and six such terms for/3. Extensive cancellation does eventually occur. The solutions, involving real ositive constants cj (j = 1,..., 7) and comlex constants Ck(k -- 1, 2, 3), are as follows For times rior to and including the P arrival, T _- 1, /i =/2 --/3 = 0. (9) For times between P and S arrivals, 1 < t < a/fi, there are two different kinds of elastic media to consider. If Poisson's ratio is less than (corresonding to A = 2~fie < 3.11), 11 = T2[Cl(T 2 - R1) -1/2 - c2(t 2 - R2) -1/2 - c3(r3 - T2) -'/2] 12 = --C4 "4- ci(t 2 - RI) I/2 - c2(t 2 - R2) I/2 4- c3(r3 - T2) I/2 I3 = c4 - cs(t 2 - R1) -1/2 4- c6(t 2 - R2) -1/2 - c7(r3 - T2) -'/2. (10) If Poisson's ratio is greater than 0.263, it is necessary first to define the comlex square root (comlex, because R2 is then comlex), CROOT = [(1 - R2)(T 2 - R2)] 1/2 (11) in which the choice of sign is made such that the comlex number has magnitude less than unity. Then, (1 - R 2 - CROOT)/(T 2-1) / T 2 [Real{C1/CROOT} + c~(r~ - T2) -~/2] /2 = -c4 - Real{C2 CROOT} + c3(r3 - T2) '/2 /3 = c4 + Real(C~/CROOT} - c7(r3 - T2) -'/2, (12) For times between the S arrival and the Rayleigh-wave arrival, a/fl < T < a/-y, /1 = 0.5-2c3T2(R3 - Te) -'/2 Is = -2c4 + 2c3(R3 - T2) 1/2 I~ = 2c4-2c7(R3 - T2) -'/2. (13) For times after the Rayleigh arrival, a/,/< T, /1 = = --2C4 /3 = 2c4. (14) Constants in the above solution need be evaluated just once for a given elastic medium, secified by the ratio c~e/fl 2. An effective aroach is first to find the largest root R3 of the Rayleigh cubic; then to factorise 2 _ R3 from the cubic and solve a quadratic for R~ and Re. Constants a, b, c in equation (8) are given by a -1 = 16(A - 1)(R1 - R2)(R3 - R1) b -1 = 16(A - 1)(R1 - Re)(R2 - R3) c -1 = 16(A - 1)(R8 - R1)(R2 - R3). (15)

7 LAMB'S PROBLEM AND SPONTANEOUS CRACK PROPAGATION 953 Then Cl = -2aA(A - R1)(1 - R1) 1/2 c3 = -2cA(R3 - A)(R3-1) 1/2 c5 = -2aARI(1 - R,)(A - R1) 1/2 c7 = -2cAR3(R3-1) (R3 - A)1/2 C2 = 4bA(A - R2) c2 = 2bA(A - R2)(1 -- R2) 1/2 c4 = A/(8A - 8) c6 = 2bAR2(1 - Re)(A - Re) 1/2 C1 = 4bA(A - R2)(1 - R2) Ca = ba(a - 2R2)2(1 - R2). Solutions given above for I1, 12,/3 require at most the evaluation either of three real square roots, or (deending on Poisson's ratio} the evaluation of one comlex square root and one real square root. These (worst} cases occur only between P and S arrivals. In terms of these closed-form solutions, all the five comonents of G n relevant to Hamano's method for studying sontaneous shear and tension cracks can be raidly comuted via equation (3). Although/4 can be given in terms of ellitic integrals (with comlex arguments when Poisson's ratio is greater than 0.263), it is robably more efficient directly to integrate as follows 0 for T < 1; /4(T) = 2TA (,/2 ~r 3o for 1< where 2 2TA('~/2 ~r 3o (2 1)(A- P2)l/2(A- 2P 2) (x dx T < a/fl, = { T2-1} sin2 X + 1; (2 _ 1)(A - P2){A - 2P 2) d# (T 2 _ 2)1/2[( A _ 22)4 _ 16X 2 y24] H(T- a/'y)cst (T 2 - R3) 1/2 for ~//~ < T, where 2 = (A - 1)sin 2 ~b + 1. (16) Integrals with resect to X and ~ here have well-behaved integrands. Note that, at time T = a/'y = R31/2, a singular Rayleigh wave arrives (see Figure lb legend) with strength roortional to the ositive real constant cs = ½ ca(a - 2R@/R3. (17) Figure 2 shows the time-deendences of/1, I2,/3 for four different values of a2/fl 2. We note the following basic roerties: (a) Dislacements I1 and/3 are continuous across the P-arrival, as are/2 and the article velocity di2/dt. These results follow from equations (10) and (12), and relations between constants aearing in these formulas. (b)/1 and/3 are continuous across the S arrival, but have discontinuous sloes, whereas/2 itself is discontinuous. (c) /2 is continuous across the Rayleighwave arrival time, but I1 and/3 are singular. All three solutions are exactly constant after the Rayleigh singularity: these constants must then be the static solutionsl (d) For the horizontal dislacement due to a horizontal force, the ste (in/2) at the S arrival can be seen from equation (3) as having the orientation of an SH wave,

8 954 PAUL G. RICHARDS (a) 11 (b) 12 ( ) I 3 P R s ~ s time... "" S,7" ~ l t... /"": S f P t "' s I t t i P :/s e ~.. ~R /.,... /,, : (22= I 2 2-~.8 a 2 = 5B 2 a z : 6B z a 2 = lob 2 Fro. 2. The fundamental time-deendences of dislacement for (a) 11, the lonl itudinal horizontal dislacement for a horizontally-alied force; (b) I2, the transverse horizontal dislacement for a horizontally-alied force; and (c) /3, the vertical dislacement for a vertically-alied force. We have chosen to lot values of 13 ositive downward, so that uward values in (c) corresond to -/3, and hence to the convention common in seismology of recording vertical motions as ositive uward. In each case, the time-deendence is worked out for four different values of a2/fl ~. Time for these four cases is scaled so that P arrivals (at T = 1) and S arrivals (at T = a/fl) are aligned. Values are lotted, as heavy solid lines, only between amlitudes +_1. I, and/3, in fact, are singular at the Rayleigh arrival (marked R), thereafter juming immediately to the static value. Dotted lines give values of I1,/2,/3 scaled u by a factor of 15, and hence dislay the detailed time-deendence at low amlitudes. whereas the singularity (in 11) at the Rayleigh arrival occurs as P-SV motion. However, because P-wave motion is not in general exactly longitudinal, the transverse motion (given by/2) does begin at the P-wave arrival. (e) A 15 wave is aarent in/3 at times between and P and S arrivals, becoming more aarent with increasing values of a2/ 2. Since it arises from a single algebraic exression, the term Real(CJ CROOT) in equation (12), detailed roerties of this wave are easy to investigate. In Figure 3, the time-deendence of 14 for four different values of 2/fl2 is shown. Romberg integration was used, requiring occasionally u to 128 intervals for 1 er cent accuracy. There is a discontinuous sloe at the P and S arrivals; a jum to a singularity at the Rayleigh arrival; and thereafter a gradual decay to the static limit. CONCLUSIONS Perhas the main achievement of this aer is the exact form of constants cl,.., cs (ositive real, if used) and comlex constants C1, C2, C3, in terms of which the comlete solution to Lamb's roblem can be given for any orientation of alied force, any dislacement comonent, and any value of Poisson's ratio, rovided both source and receiver lie in the free surface. Four scalar solutions in G, involving the cross-terms (vertical or horizontal dislacements due, resectively, to horizontal or vertical alied force), cannot be given in closed form, but a well-behaved integral solution is ossible in general.

9 (a) I4 LAMB'S PROBLEM AND SPONTANEOUS CRACK PROPAGATION (b) P- lane 955 fl 't 1 t R ~..... s ~' $ time _..,,,,... s~.. "... ~,., ': ~..! ~ :..: : P P ' i ' \;'i r t R,oo. o... - R½. a -, R1½ 1 ".. Real P--> = 2~, :" " / R2: :. (2 R~[ ""... ~ R3 ~ Cl 2 3B 2 1 ".../;."[ ~ " _1.:'IRz2..,. : '.L" i23 ~ l ~T R,~ ~-~.: "... : ~.[.:R- 3 ~ 1 $! al a z = 6B ~ a 2 ; I0~ 2 Fit. 3. (a) Values of the fundamental solution/4 as a function of time. A closed-form solution is not ossible in this case. Comutation is for four different ratios of a2/fl 2. Dotted lines show values of (b) Since T can be regarded as a value of P, the dimensionless horizontal slowness, we have here shown the comlex P lane with the same scale as the T axis in (a), and Figure 2, a, b, and c. Values of 15 /4 are reeated from (a). Singularities R1 ~/2, R2~/2, R3 ~/2 here; for different values of a2/fl 2, occur then at times (P values) which are indicative ~f what turn out to be roerties of the/3 and Rayleigh waves. Thus, R21/2 for ~2 _ 2¼ff is almost coincident with the ordinary P-wave arrival, making the latter highly imulsive for/1 and I3 because of the term in (T 2 - R2) -1/2. At larger a2/ff, the occurrence of comlex R21/2 with real values greater than 1 leads to the emergent broad ulse between P and S arrivals in/3 and /4. It is interesting that such a/3 wave is not aarent for/1 and 12. In the case of horizontal dislacements due to a horizontally alied force, the solutions are relevant to a method for studying sontaneous shear cracks. For the case of vertical dislacement due to a vertical force, the solution has relevance to tension cracks. In both these cases, solutions given by (3) and (9) to (14), for a stealied force, are so simle that the following can readily be derived in closed form: (a) solutions for an imulsively alied force; (b) solutions averaged over (r, r + Ar), (~, ~ + A~), and (t, t + At); (c) 14 of the dislacement fields c~vn/o~q = G,,,q due to a single-coule. Secifically, we can use recirocity on Gn so that the derivative is conducted with resect to receiver coordinates. From the five closed-form solutions in G, 10 single-coule dislacement fields can be obtained by differentiating in the 1- and 2-directions, arallel to the free surface. The solutions for (n,, q) = (1, 3, 3), (2, 3, 3), (3, 1, 3), and (3, 2, 3) can also be recovered in closed form, by using the linear strain constraints at a stress-free surface. The simlicity of the five scalar solutions in G, which are associated with Hamano's method for studying cracks, is so remarkable that it gives high hoes of

10 956 PAUL G. RICHARDS successful develoment of a 3-dimensional study of sontaneous shear and tension fractures. ACKNOWLEDGMENTS This aer originated from conversations with Shamita Das, who is now engaged at Lamont-Doherty in alying Hamano's method in three dimensions. I thank both her and Bill Menke for hel with identifying some of the fundamental integrals. Critical reviews from Shamita Das and Klaus Jacob are areciated. Suort for this work came from the Advanced Research Projects Agency of the Deartment of Defense, monitored by the Air Force Office of Scientific Research under Contract F C REFERENCES Aki, K. and P. G. Richards (1980). Quantitative Seismology: Theory and Methods, ~1800., W. H. Freeman and Co., San Francisco (in ress). Chao, C.-C. (1960). Dynamical resonse of an elastic half-sace to tangential surface loadings, J. Al. Mech. 27, Chaman, C. H. (1972). Lamb's roblem and comments on the aer 'On leaking modes' by Usha Guta, Pure Al. Geohys. 94, Das, S. and K. Aki (1977). A numerical study of two-dimensional sontaneous ruture roagation, Geohys. J. 50, Gilbert, F., S. J. Laster, M. M. Backus, and R. Schell (1962). Observations of ulses on an interface, Bull. Seism. Soc. Am. 52, Hamano, Y. (1974). Deendence of ruture time history on the heterogeneous distribution of stress and strength on the fault lane (abstract), EOS, Trans. Am. Geohys. Union 55, 352. Helmberger, D. V. (1968). The crust-mantle transition in the Bering Sea, Bull. Seism. Soc. Am. 58, Johnson, L. R. (1974). Green's function for Lamb's roblem, Geohys. J. 37, Lamb, H. (1904). On the roagation of tremors over the surface of an elastic solid, Phil. Trans. Roy. Soc. London, Ser. A, 203, Mooney, H. M. (1974). Some numerical solutions for Lamb's roblem, Bull. Seism. Soc. A~. 64, Pekeris, C. L. (1955). The seismic surface ulse, Proc. Natl. Acad. Sci., U.S. 41, LAMONT-DOHERTY GEOLOGICAL OBSERVATORY AND DEPARTMENT OF GEOLOGICAL SCIENCES OF COLUMBIA UNIVERSITY PALISADES, NEW YORK CONTRIBUTION NO Manuscrit received November 2, 1978