416 Elastic Waves at the Surface of Separation of Tivo Solids. By R. Stoneley, M.A., University of Leeds. (Communicated by Prof. H. F. Baker, F.R.S.—Received June 23, 1924.) § 1. Introduction. In considering how the energy of a seismic disturbance is dissipated one is led to enquire into the possibility of the existence of waves, analogous to Rayleigh waves and Love waves, that are propagated in the interior of the earth along the junction of strata, or chiefly within a certain stratum, so that the energy is dissipated by internal viscosity without the occurrence of any ♦ appreciable surface displacement. Two surfaces of discontinuity of density and elastic properties are commonly believed to exist below that part of the earth’s crust which is accessible to geologists, namely, the junction of the granitic layer with the basic rocks, and the surface of separation of the Wiechert metallic core from the rocky shell. It becomes of interest to examine whether a wave of the Rayleigh type can be propagated along such an interface ; an enquiry may also be made into the circumstances in which a wave of the Love type may exist if a stratum of uniform thickness is bounded on both sides by very deep layers of different materials. It has been pointed out to me by Dr. Harold Jeffreys that the former problem is in some respects a particular case of Prof. Love’s discussion* of the effect of a surface layer on the propagation of Rayleigh waves; the “ layer ” is here taken as of infinite thickness. Whereas, however, Prof. Love’s problem is concerned with a disturbance confined chiefly to the free surface, the present paper deals with a wave motion that is greatest at the surface of separation of the two media, and is not restricted to the case of incompressible solids. Some simplification is effected, however, when the media are taken as incompressible, and several such particular cases have, accordingly, been solved in detail; these throw some light on the general problem, and, in fact, suggested in the first place the investigation of § 3. Generalised R ayleigh Wave. §2. The Wave-Velocity Equation. The two media will be distinguished by suffixes 1 and 2, and will be supposed in “ welded ” contact along an infinite plane face, and otherwise * *Geodynamics,’ p. 163. on June 7, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from
Transcript
416
Elastic Waves at the Surface of Separation of Tivo Solids.By R. Stoneley, M.A., University of Leeds.
(Communicated by Prof. H. F. Baker, F.R.S.—Received June 23, 1924.)
§ 1. Introduction.In considering how the energy of a seismic disturbance is dissipated one is
led to enquire into the possibility of the existence of waves, analogous to Rayleigh waves and Love waves, that are propagated in the interior of the earth along the junction of strata, or chiefly within a certain stratum, so thatthe energy is dissipated by internal viscosity without the occurrence of any
♦appreciable surface displacement.
Two surfaces of discontinuity of density and elastic properties are commonly believed to exist below that part of the earth’s crust which is accessible to geologists, namely, the junction of the granitic layer with the basic rocks, and the surface of separation of the Wiechert metallic core from the rocky shell. I t becomes of interest to examine whether a wave of the Rayleigh type can be propagated along such an interface ; an enquiry may also be made into the circumstances in which a wave of the Love type may exist if a stratum of uniform thickness is bounded on both sides by very deep layers of different materials.
I t has been pointed out to me by Dr. Harold Jeffreys that the former problem is in some respects a particular case of Prof. Love’s discussion* of the effect of a surface layer on the propagation of Rayleigh waves; the “ layer ” is here taken as of infinite thickness. Whereas, however, Prof. Love’s problem is concerned with a disturbance confined chiefly to the free surface, the present paper deals with a wave motion that is greatest at the surface of separation of the two media, and is not restricted to the case of incompressible solids. Some simplification is effected, however, when the media are taken as incompressible, and several such particular cases have, accordingly, been solved in detail; these throw some light on the general problem, and, in fact, suggested in the first place the investigation of § 3.
Generalised R ayleigh Wave.
§2. The Wave-Velocity Equation.The two media will be distinguished by suffixes 1 and 2, and will be
supposed in “ welded ” contact along an infinite plane face, and otherwise
* * Geodynamics,’ p. 163.
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Elastic Waves at Surface o f Separation o f Two . 417
-ending to infinity, so that there is no slipping at the interface, in which ..origin and a set of axes of x and y are taken; the axis of z is drawrn .itively into the medium 1. Let p denote the density and A, the usual
elstic constants.f for a wave propagated in the x direction we assume for the displace- nt in medium 1 a solution of the type (Ui, WO where
L, Vl5Wi are functions of z only, we find in the usual way* that a solution i . ding to zero at infinite distance from z 0 is given by
(u „ Vi, WO = (uo , v o , WO) + (U 0'5 v r , w / ') , (i)were
(TV, V,', W,') = - fifr( **> °. - o ) (2)
(TV', V,". W /') = («„ a u »*) Qie- * , (3). which D1? Q1} «i, are constants,
rO == k(1
s 12 = k2 { 1
Pld2 \X —|— 2[juPlC2
(4)
In medium 2 we find in a similar manner, for a disturbance which becomes sensible at a great distance from z = 0,
(U2, V2, W2) = (UO, VO, WO) + (U2", v 2", w 2" ) , (6)’liere
(TV, V2', W2') = ~ * f J c2 2(iK’ °> *•«) EeW>
(U2", V2", W2") = (*,, a „ « ) Q ^ ,which E, Q2, a 2 are constants,
-« = * * ( l------ £ £ .r2
•V
A2 + 2/̂ 2
\ tl 9 /
(7)
(8)
( 10)\ ^2
At the bounding surface the displacements and stresses are supposed ontinuous. Thus, if (wl5 v1} u f , ( u2, are the displacements in the wo media, we have
(dui | dwP(du9 , dun2 j11 dz dx ) V 02 1 0a;
,'3% , 0 w O _
AjA, + 2a = A2A2 + 2^2^
(ID
( 12)
(13)
* Love: ‘ Elasticity,’ 3rd Edition, p. 311, etseq. The notation is based on that ofJeffreys, * The Earth,’ p. 157.
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where A is tlie dilatation,ul — u2 (14vi = v2 (15;wx = w2 (16
at the boundary, which to the first order of small quantities may be taken as j 2 = 0 .
The equations (15) and (12) give respectively«iQi = «2Q2; — /^iqiQiSi = ,w2a2Q26'2,w hence a1= a 2 = 0,
and therefore 0i == v2 = 0. (17]Write
(Ai + 2^])/pi = ay2; Pi2lr (18'
(A2 + 2m2)/p2 = a22; ^ 2/P2 ~ P22 Jthen, on substituting in (11), (13), (14), (16) the values of the displacements given by (1) and (6), we obtain
c2 \®ai2T-v , n . l t c2\®a9!2T-, . !n c22/^(1 — ^-2) X~2D + 2/^2(1 — — ) ^ - E + ^ i (2a22 c2 ix2 Qi
/ . 2(B-2 '(Ax + 2qx) ( l —^ ~ ) D — (A2 + (1
c2 D - 24 E + ( 1 - — r !;*2Qi - (1 - Z - J Q2 = o,c2
C2
+ ^ 2i2 - = ° (191
2- 2- ‘ E — 2,Ux f 1 — i«2Qx
( 21)
c2 / ' A\ (B12,
+ 2.«2 ( l ~ - 5 L ) ' « 2Q2 = 0. (20;
c2 \s / r2 \i—J i/c2Qx — 1 --- — '$12 ' V l \ P22/
1 s) E + (1 — fa ,2/
E -j- 2*«2Qx + 2*:2Q2 = 0. ( 22)
Eliminating D, E, i«2Qx, ^ 2Q2 from these four equations, and substituting foi ft 1, ^ , (Ax + 2«x)5 (A2 + 2jW2), we obtain as an equation for the wave- velocity c
* * * { ' - 3 s p . w ( i - 4 y :a2 ' * & [ * - £ ) ’ P .p .-(s— g i)
Pi (<*-2Pi!) ; - p 2 (c2 -2 f i , ,2) ; - 2 p lPl* ( l - £ , ) * ;/ C2 \*
2p2^22( 1 pj)
1 ; 1 - ( l - — fV (V?
V a,*/ ( - 5 / ; 1 ; 1
= 0 . (23)
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Elastic Waves at Surface of Separation o f Two Solids. 419
On writing
§3. Compressible Media.
K = 2 (pi(3i2 — P2 P22),te equation (23) of the preceding section reduces to
c4 {(Pi P2)2 (Pi-A-2 + P2^-i) (piB2 -f- p2Bx)}
~f" 2Kc2 {piA2B2 p2AiBx pi + P2 }
H-K2(A1B1- 1 ) ( A 2B2- 1 ) = 0 . (1)
It is easy to show that when p2 = 0 this equation reduces to the ordinary juation for Rayleigh waves
On account of the ambiguity introduced by squaring both sides of an quation it is advisable, where practicable, to work with the equation (1) as 5 stands, not attempting to rationalise.
An important particular case of (1) arises when oci = a2 and Pi = p2. Phese conditions appear to be satisfied at the Wiechert surface of dis- ontinuity within the earth, as may be seen from Knott’s table of wave- velocities.* If we write
a i = a2 = a ; Ax — A2 = A ; Pi — P2 = P 5 Bx = B2 = B ; y = p2/a2 ; c2/p2 =
We see at once that /(1) = (px — p2)2, which is positive. /(0) is equal to zero, implying that some power of x is a factor of f{x). If x is small, so that A and B may be expanded as power series in x, we find in fact f (x)Ix2 = — (Pi + p2)2 + y2(pi — p2)2 + terms containing a; as a factor ; thus the limiting value of f(x )/x2 as x tends to zero is — (px -f- p2)2 + y (pi — p2)2. But Pi + pa> |pi — pa|, and y < l , so that (pi + p2)2 >y2 (pi — P2)2> and accordingly { / (x)/x2}x = 0is negative, while is positive. Thus
a root o if(x)/x2 = 0 always exists between x = 0 and 1, in other words,a wave can be propagated along the interface with a velocity less than the common velocity of distortional waves.
We can deduce that a Rayleigh wave must always exist at the free surface of an infinite solid. The direct proof is, however, very simple, for the ordinary equation, obtained by squaring (2) may be written
x3 — 8x2 -f- 24a; — \8yx — 16 -f-16y = 0. (4)
When x — 0 and 1 the left-hand side takes the values 16 — 1) and 1respectively, so that there is a root between 0 and 1. The value so obtained is either a solution of (2) or else a solution of that equation with the right- hand side changed in sign; since the left-hand side is essentially positive the latter possibility is ruled out, and thus a Rayleigh wave always exists.
In Lord Rayleigh’s original paper* the numerical solutions given are found from a rationalised equation, and the fact that these must satisfy an equation equivalent to (2) is not explicitly stated.
As a check on the preceding work we may put pi = p2, so that the two media are the same ; the equation (3) now gives 1 or 1/y,corresponding to c — (3 or c = oc, as would be expected.
Analogy with the incompressible case (§ 4) suggests that in the general problem a wave does not necessarily exist.
I t will be supposed that ocj and do not differ greatly from a2 and (32 respectively ; in these circumstances a wave motion of the assumed type may be shown to exist if the differences are small enough. Put
Elastic Waves at Surface o f Separation o f Two Solids. 421
Whatever the value of a, then, by taking n sufficiently small we can ensure •at /(1) is positive.Taking now the case of x small, we have
[x)/x2 = — 4(7 — (1 — cr)2 (1 — y2)4- (1 — a) [n(1 + a — ya — y — 2 + wy (1 — y + .cr + oy)]
hich can always be made negative by taking m and n sufficiently small, hus, we can definitely assert that when the wave-velocities are not too widely ifferent for the two media, a wave of the Rayleigh type can exist at the iterface. If the wave velocity is not very different from £4, we may use the alue o f /( l) to obtain, for given values of a and y, the approximate limits of ermissible variation of the quantity n ; the variation of velocity of distortional raves thus appears to be of more importance than an equal variation in the elocity of compressional waves.
ncompressible media, when Ax and A2 both become unity. This assumption, tiowever, considerably simplifies the numerical work in actual examples, several of which have been examined.
Equation (1) of §3 now becomes, on squaring and rearranging,
2 ( 1 — ) ( l — tt ) P1P2 (Pi + P2)2 — Ac6 -f" B e4 4 - + D , (1)' Pi / \ P2'
4 - terms in a? ; ... etc.,
§4. Incompressible Media.
The results obtained in the preceding section are true in particular of
where
C = 2 K * { + pl(3^ + ?a)} +16K PlP2 (Pl - p2)C = 2K2 | d-lbKpiPafpi— p2),
This equation is rational when (3i = p2 = p, and it is easy to prove in this particular case the general result previously found, that a Rayleigh wave always exists. By putting p2 = 0 we obtain the usual equation for the
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which has a root corresponding toc = 0 - 99287 (3, (4)
so that a wave may be propagated along the bounding surface with a velocity very slightly less than that of distortional waves in either medium.
As pointed out by Rayleigh,* on general dynamical principles we shall not expect the complex roots to correspond to any real wave motion. We do, in fact, find for these complex roots
c2/(32 = 0-75806 ± 1-57720 (5)
On reintroducing s and k, we may write the simplified equation (1) of §3 in the form
{ (Pi - Ps) («• - 2P»)}» + 4 4 (Pi - p2)2 P4fC
“ —{4piP2c4 + (pi — p2)3 (c2 — |S2)2}, (6)Kand substituting s2/*2 from equation (5) of § 2 we have, on reduction
s_ 0-02261 =F 5-49478^k — 10-46019 i 9-27612'i* ■
On bringing this fraction to a real positive denominator it is found that the real part of the numerator is negative whether the upper or the lower sign be taken, and accordingly e~sz and e~KZ cannot both tend to zero at infinity. Thus the complex roots are inadmissible.
It may be noted as a check that when px = p2, (6) can only be satisfied if c = 0, or if s = 0, the latter corresponding to c — ft.
From a consideration of particular cases it appears that solutions of this equation corresponding to real waves may not always exist. I t is essential that c should be positive and less than both and (32, otherwise -sq and s2
* Loc. cit., p. 8; or 4 Sci. Papers,’ 2, p. 441.
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•11 not both be real, and the disturbances will not be insensible at great stance from the surface of separation.A very simple case arises when Pi = 2p2, (322 = 2(3X2,s o that K = 0.
^uation (8) then reduces to
81 (c /^ )4 - 432 (c /^ )2 + 640 = 0, (9)hich has not real roots.
If Pi = 2; p2 = l ; (3X2 = 2(322, so that //x = 4/% equation (8) becomes
-7hich has two solutions between x ==■ 0 and 1, v iz.: a;2 = 0-9042 and 2 = 0-93615, corresponding to the values c = 0-951(3! and c = 0*967(3j. it would therefore appear at first sight that two waves, having different velocities, might exist. I t is to be remembered, however, that the :quation (1) of § 3 has been twice squared in reduction to the form (8). On ictual substitution it is found that neither of these roots will satisfy (1), but shat both satisfy the equation derived from (1) by changing the sign of >ne side.
That there should be an even number of roots of the wave-velocity aquation for these values of px and p2, (3i /(32, between 0 and c — (31? can be seen by reverting to equation (1) of § 3, and writing Ax = A2 = 1. It is then found that { f (x)/x2}x=0 is equal to —25/12, in the notation of (3), §3, and th a t / ( l ) is — (3/16+
It thus appears that a wave of the type under discussion will exist when the velocity of distortional waves in the two media is the same, but not if the wave-velocities are greatly different. The method of the preceding section may be applied to obtain a rough estimate of how widely + and (32 may differ for given values of px and p2. In„the notation of §3, (6), /(1 ) is approximately (1 — a)2 — n4 {4cr2 — 3cr + 1}, and 0 18 approximately — (1 + o)2 + n(1 — a2) ; this may be regarded as a particular case of §3, with y = 0, or it may be obtained more simply from §3, (1) by putting Ai — A2 = 1. I t is seen that by making n sufficiently small, can be
Elastic Waves at Surface o f Separation of Two Solids. 423
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made to change sign between x = 0 and Moreover, if n is very small ni is large compared with n, so that if 4cr2 — 3<r + 1 and 1 — or2 are comparable the range of variation of/(x)/x2 is greater in the neighbourhood of = 1 that at x = 0. If, then, we disregard the possibility that {f{x)lx2}x = 0 raa} change its sign for a small variation of ft, the condition that there shall be a root between x = 0 and x = 1 is that ft4 is not greater than
( l - o ) 2/(4a2- 3cr + l),
for the denominator is always positive. In this way it is possible to obtain roughly the result of the preceding paragraph, but the approximation is too rough to be convincing. In other respects these waves conform to the Rayleigh type. When c is known, expressions may be written down for the displacements of any particle, and by corresponding reasoning it may be shown that particles describe ellipses about their mean positions.
Furthermore, this appears to be the only type of wave that can be pro pagated along the surface. I t is easy to verify that transverse wTaves of the Love type cannot exist. The displacements in the two media would be
(0, Al5 0) e~and
(0, A2, 0) e iK{x~ct);
the conditions that the displacement and stress at 2 = 0 must be continuou would then give the incompatible relations
A -i — A2; — = ;m2A26,2 ; ^ iA-i == 2A2.
The geophysical interest of this discussion is that in addition to that portioi of the energy of an earthquake which is dissipated by solid friction befori reaching the surface, a further fraction may be “ trapped ” by surfaces 0 discontinuity, and may involve a correction to estimates* of the energy involved in a seismic disturbance.
Transverse Waves in an Internal Stratum.
§ 5. A Generalised Type of Love Wave.Suppose the medium (1) to extend from 2 = 00 to z 0, the medium (2)
to extend from 2 = 0 to 2 = — T, and the medium (3) from 2 = — T tc 2' = — 00. Then in any layer we may take as the components of displace-
* e.g., see Jeffreys: * M.N.R.A.S.,’ Geophys. vol. I, 2, p. 22 (Jan. 1923).
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Elastic Waves at Surface of Separation of Two Solids. 425
tent (w, v, w) in a plane wave travelling in the direction of x increasing the al parts of
(0, V, 0)eiK(x~ct)(1)
^ere V is a function of 2 only.It is at once verified that the dilatation is everywhere zero, and if p, note respectively the density and the rigidity, the only equation to be
•tisfied is
(2)
t, on substituting from (1),
d2\ KW 0.dz2
For a type of wave in which V is periodic in medium (2) and exponential . (1) and (3) we have
c2 > c22; c2 < ; c2 < c32, (4)
here c1} c2, c3 denote the velocities of distortional waves in the three tedia. We thus have
Vi = De~Sl*
V2 = A cos s2z -f- B sin j> (5)
V3 = Ee8** J .
here A, B, D, E are constants,
C2 f c 2 ,« —2 r > «2 = « 1 —2 — 1Cl J lc 22
and .s3 = k -j 1 — —2T .L c3
The boundary conditions are that the displacement and stress are ontinuous at z = 0 and z = — T. We thus obtain
A = D ; A cos s2T — B sin s2T = Ee~S;iT ;— = s2u2B ; s2iu2 (A sin s2T + B cos s2T) = s3̂ 3E~S3T. (6)
Eliminating A, D, B, E from these equations we have the equation deter- aining the wave velocity
tan s2T = s2iu2 + s— (7)
f s 1 = /cor1; s2 = kg2; s3 = kg3, the equation (7) may be written in the quivalent form
then when c = c2, a2 — 0, g1 and g3 are positive, so that the left-hand sicof (8) is negative and the right-hand side positive when c is slightly great than c2. As cincreases the right-hand side continually diminishes (tan w 2T)/ora/U2 increases. When c = c3 the factor {o22/u22 — a ^ o ^ ) positive, and therefore changes sign between c2 and c3 ; this will happe | when
denote the corresponding value of cby c' Then for c2 < c < c'the left-hand side is negative and the right-hand sic
positive unless kg2T has passed the value 2n, f, etc. (as, for example, whe kg2T varies from 0 to 7tin the range c2 to c'), in which case the left-han
side will have diminished to • oo, and then dwindled from + oo to zer< thus equalling at some point the value of the right-hand side. 0 reaching the value zero for kg2T = the left-hand side either begins t increase (as an exceptional case), or (in general) it continues to diminisl In the latter case, if for c = c' (when the left-hand side vanishes
<C kg.2T<C fzc, then in this interval the left-hand side must have decrease to a minimum and increased to zero ; this is in accordance with the fac that at c= c'the factor cr.22/,22 — changes from negative to positiveIf now kg%T reaches the value f nbefore c — c3, there will be another rocof the equation in this interval. There will thus be a root, or a series c roots, between c2 and c' if /cT is made sufficiently large, either by makin k or T large.
Between c and c3 there will be a root of the equation (8) if tan T i greater than q at c — c3,where
q = u 1
for in this range the factor a2u2 — G^J^G^jG^H continually increases fror zero to «2 (c32 — c22)y'c2, so that the left-hand side takes the sign of tan T and has always a positive gradient.
If kT is so small that, throughout the range c2 <( c y c3, /ccr2T remains les than \n,and at c = c.3, tan kg2T is less than q, there will be no root. Thus if the wave length is sufficiently long or the middle layer too thin, no wav motion of the Love type is possible. This accords with the result found h § 4, that when T — 0, no Love wave exists.
If c2 < c3 < c<!Cj , g3 is imaginary ; the right-hand side of equation (7)
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Elastic Waves at Surface of Separation o f Two Solids. 427
ow complex and the left-hand side real. There are thus no real roots in his region.When cx < c3 we have as in the foregoing discussion that roots, if they
:cur (and roots will exist if *T is large enough), must occur for c2 < cv Within the layer (2) the displacement is YelKix~ct), where V is of the form
A {s2/u2 cos s2z — sh sin s2z) /s2/i2,
hich vanishes when tan s2z — s2u2[s1/j,1. Now in this layer 2 is negative ; , then, s2z includes the range \n to n for 2) T, the displacement must anish for a certain value of 2 and a “ nodal plane ” will exist, as it may in le case of the ordinary Love waves in a surface layer. If s2z includes the vnge from 0 to rm, while 2 remains less than T, n nodal planes will exist.A similar argument is applicable to the case c3 > cx > > c 2.If we suppose that in medium (2) c < c2, and therefore that
c2»2 = «2 1 2 ’
e have V2 = A cosh s2z-f- B sinh s22 in place of the harmonic terms. The oundary conditions now give
A = D ; A cosh s2T — B sinh s2T = Ee~S:iT ; — = B ;s.2ju2 {A sinh (—s2T) -f- B cosh (—s2T)} = Ee~S:jT (11)
;ading to(«i^i. s zfi3 -f s22 ,u22) tanh s2T = — s 2lu2(s3lu3 -f s ^ ) (12)
s hich gives no relevant solutions.In the other cases which might arise we should have either cx or c3 less
'nan c, so that in one of these regions the solution would be periodic, and lierefore the displacement would not be inappreciable at great distances from == 0, and, moreover, would require an infinite amount of energy to be resent in a cylinder whose generators are perpendicular to 2 = 0. The hysical interpretation of this result is similar to that given by Dr. Jeffreys 0 the maintenance of Love waves. The velocity of distortional waves in ither of the media adjoining the layer (2) is greater than the velocity in the ryer> and a wheeling of the wave-front, analogous to that which occurs in lie phenomenon of total internal reflection, would cause the wave motion to >e confined mainly to the central layer.
[Added— July8.—The solution of a numerical example in the case where i = a2 = a ; = (32 = (3, illustrates one or two points of difference fromhe corresponding example worked out in § 4 for incompressible solids. The (luations are numbered in continuation of those in §3.
2 h 2
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The corresponding equation for incompressible media is a cubic, obtained by putting y — 0 ; this is in agreement with § 4, equation (1).
As a check on this equation, it may be observed that (8) should reduce to the ordinary Rayleigh wave equation when p2 = 0, when M = 1. It is, in fact, found that when M = 1, (8) becomes
where the second factor, equated to zero, is the same as (2), and the former factor gives the inadmissible solution x — {l-\-y)Jy.
To bring the equation (8) into line with this result it is sufficient to observe that, whatever M and y, the left-hand side of (8) reduces to — 1 when is unity. Since M — y2 is essentially positive, the equation must possess a root greater than unity.
At the Wiechert surface of discontinuity, px = 8*2, p2 = 3*2, y 0*2864.* The corresponding quartic equation has a root = 0*9795 . . . corresponding
to c —0*9897(3, and by actual substitution it is verified that this valuesatisfies the unsquared equation (7). I t may be noted that, as would be expected on general dynamical grounds, this velocity is less than that of a wave at the junction of incompressible media (see §4, (4)).]
In conclusion, I wish to thank Dr. H. Jeffreys, who has kindly read through the manuscript and made several valuable suggestions.
428 Elastic Waves at Surface of Separation of Two Solids.
* Knott: ‘ Roy. Soc. Proc., Edin.,’ p. 170 (1918-1919).
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