Proc. London Math. Soc.-1885-Rayleigh-4-11

' Lord Eayleigli on Waves propagated along [Nov. 12, " Johns Hopkins University Circulars," Vol. iv., No. 42. "Extensions of Certain Theorems of Clifford and of Cay ley in the Geometry of n Dimensions," by E. H. Moore, jun.: (from the Transactions of the Connecticut Academy, Vol. vn., 1885). " Bulletin des Sciences Mathematiques et Astronomiques," T. ix., November, 1886. " Atti della R. Accademia dei Lincei—Rendiconti," Vol. i., Fasc. 21, 22, and 23. "Acta Mathematical' vn., 2. " Beiblatter zu den Annalen der Physik und Chemie," B. ix., St. 9 and 10. " Memorie del R. Istituto Lombardo," Vol. xv., Fasc. 2 and 3. "R. Istituto Lombardo—Rendiconti," Ser. n., Vols. xvi. and xvn. " Jornal de Sciencias Mathematicas e Astronomicas," Vol. vi., No. 3 ; Coimbra. " Keglesnitslaeren i Oldtiden," af. H. G. Zeuthen; 4to, Copenhagen, 1885. (Vidensk. Selsk. Skr. 6t0 Rsekke Naturvidenskabelig og Mathematisk afd. 3d*, Bd. i.) On Waves Propagated along the Plane Surface of an Elastic Solid. By Lord RAYLEIOH, D.O.L., F.R.S. [Head November 12th, 1885.] It is proposed to investigate the behaviour of waves upon the plane free surface of an infinite homogeneous isotropic elastic solid, their character being such that the disturbance is confined to a superficial region, of thickness comparable with the wave-length. The case is thus analogous to tliat of deep-water waves, only that the potential energy here depends upon elastic resilience instead of upon gravity.* Denoting the displacements by a, /3, y, and the dilatation by 0, we have the usual equations =z(X + fl)f+^a *° (1)' in which e = ^ + f.+ p. (2). ax ay dz If a, /i3, y all vary as eip\ equations (1) become +/*V9+Pi>la = 0, &C (3). * The statical problem of the deformation of an elastic solid by a harmonic appli- cation of pressure to its surface has been treated by Prof. G. Darwin, Phil. Mag., Dec, 1882. [Jan. 1886.—See also Camb. Math. Trip. Ex., Jan. 20, 1875, Ques- tion IV.] 1885.] the Plane Surface of an Elastic Solid. 5 Differentiating equations (3) in order with respect to x, y, z, and adding, we get (V3 + fca)0 = O (4), in which fc8 = ppa/(x+2/*) (5)- Again, if we put ft2 = pp2 / f» (6), equations (3) take the form < * + * ) . _ ( l - * ) « fa (7). A particular solution of (7) is* _ 1 d8 n_ 1 dd _ 1 dd ,Q, a -~WTx' P--T**? y-~"WTz W ; in order to complete which it is only necessary to add complementary terms u, v, w satisfying the system of equations = 0 (9), + + 0 dx dy dz For the purposes of the present problem we take the free surface as the plane z = 0, and assume that, as functions of a? and y, the dis- placements are proportional to ef/*, e = - $ . - , y - ^ . - a*). For the complementary terms, which must also contain ev*, e & c (l5) * whence, as before, on the assumption that the disturbance is limited to a superficial stratum, utzAe-*, v = B&"% w = Oe— (16), where * s9 = / 8 +0 9 - f c f (17). In order to satisfy (10), the coefficients in (16) must be subject to the relation ifA+igB-8(J=Q (18). The complete values of a, /3, y may now be written a = - %Le-"+Ae-\ 0 = - | f e'"+Be-', y = i? e-»+Cfe- ' . , . (19), in which A, B, 0 are subject to (18) ; and the next step is to express the boundary conditions for the free surface. The two components of tangential stress must vanish, when z = 0, and these are propor- ,. , , dQ . dy dy . dational to —• + -/-, -f- + -r-,dz dy dx dz respectively. Hence sB^+igO, sA = 2j£+if0 (20). Substituting from (20) in (18), we find Cf(*8+/9+09)fc9+2r(/M-08)=O (21). We have still to introduce the condition that the normal traction is tero at the surface. We have, in general, 1885.] the Plane Surface of an Elastic Solid. or, if we express X in terms of /x, h, h, so that the condition is or, on substitution for r2 of its value from (12), W-2(f+g2)-2h2sO=0 (22). By eliminating 0 between (21) and (22), we obtain the equation by which the time of vibration is determined as a function of the wave- lengths and of the properties of the solid. I t is {V-2 (f+g2)} {s2+f+g2} + 4rS(/a+). Hence h?a = if { — e"n+-5433e""} eipt eVs ein { } eipte*ei9V [ (30). ipt If we suppose the motion to be in two dimensions only, we may put g = 0; so that /3 = 0, and in which k = -9554/, s = -2954/. ; (32). For a progressive wave we may take simply the real parts of (31). Thus ^8a / / = (e-'»- •5433e-1) sin (pt+fx) } / i 8 y / /= (e-A-l'840e-M)cos (pt+Jx) ) 1885.] the Plane Surface of an Elastic Solid. 9 The velocity of propagation is p/f, or <9554v/(/*//o), in which ,/{ft I p) is the velocity of purely transverse plane waves. The sur- face waves now under consideration move, therefore, rather more slowly than these. From (32), (33), we see that a vanishes for all values of x and t when el"-0' = '5433, i.e., when/* = -8659. Thus, if \ ' be the wave- length (2TT//), the horizontal motion vanishes at a depth equal to *1378\'. On the other hand, there is no finite depth at which the vertical motion vanishes. To find the motion at the surface itself, we have only to put z — 0 in (33). "We may drop at the same time the constant multiplier (A3//) which has no present significance. Accordingly, o = '4567 sin (pt+fx)")Kr J J [ (34), y = - ' 840cos (^+ /a j ) ) . showing that the motion takes place in elliptio orbits, whose vertical axis is nearly the double of the horizontal axis. The expressions for stationary vibrations may be obtained from (30) by addition to the similar equations obtained by changing the sign of p, and similar operations with respect to / and g. Dropping an arbitrary multiplier, we may write a = —/ { —e'"4-'5433e"'i>] cos pi sin fx cos gy ") /3 = — g { — e"" +"54336""} cosptcosfx singy f (35), y = r { e"n—l1840e""} cospt cos fx cos gy ) in which r =