and A, B, C contain a factor e‹(fx+gy+pt) ̧ Hence the expressions for the displacements The boundary-conditions and equation (65) give, by taking ...(67); 2 2 C' (s2 + ƒ2 + g2) h2 + 2r ( ƒ2 + g2) = 0, and the frequency-equation is where 2 2 K'2 = k2/(ƒ2+92), and h’2=h2/(ƒ2+g2) .......................(69). When the solid is incompressible h2 = 0, and the equation for has one real root '91275, while the complex roots make the real parts of r and s have opposite signs, so that they may be rejected. We now have and 2 x2 = 91275 (ƒ2+g2), r2=ƒ2+g2, s2 = '08725 (ƒ2 + g2), h3w = √(ƒ2+g2)(e−rz −1·840e−82) e1(pt+fx+gy)) ..(71). For progressive waves whose fronts are parallel to the axis of y, we have u = U (e−fz — ·5433e-8) sin (pt +ƒœ),\ w = U (e ̄ƒ2 – 1·840e−3) cos (pt + fx) [ where U is a constant; and the velocity of propagation is p/f=9554 √(u/p), .(72), which is slightly less than that of waves of distortion in an unlimited medium. The horizontal motion vanishes at a certain depth. The motion at the surface is given by u = •4567 U sin (pt +fx)\ w=840 U cos (pt+fx) .(73); so that the particles move in elliptic orbits whose axes are nearly in the ratio 2: 1. Lord Rayleigh also considers the cases where λ=μ, or the material fulfils Poisson's condition, where λ=0, or longitudinal extension is unaccompanied by lateral contraction, and where λ=μ, or the bulk-modulus vanishes. For λ =μ he finds x2 = '8453 (ƒ2 + g2), r2 = '7182 (ƒ2 + g3), s2 = ·1547 (ƒ2 + g2). For a progressive wave u = U (e―12 — ·5773e-8) sin (pt +fx), (e-rz w = U (·8475e−rz - 1.4679e-82) cos (pt + fx) .(74), and the ratio of the axes of the elliptic orbit, described by a surface-particle, is reduced to about 3. Lord Rayleigh suggests that these surface-waves may play an important part in earthquakes and in collision, as they diverge from the source of disturbance in two dimensions only, and consequently gain increasing relative importance at a considerable distance. CHAPTER XII. APPLICATIONS OF CONJUGATE FUNCTIONS. 204. So far as I am aware, the only successful attempt hitherto made, to obtain general solutions of the equations of elastic equilibrium in a form adapted to satisfy arbitrary boundary-conditions at any other surface than a sphere or a plane, is that of Herr Wangerin1. He has shewn how to obtain solutions in terms of conjugate functions of the equations of equilibrium, under no bodily forces, for an isotropic body bounded by a surface of revolution for which Laplace's equation can be solved. We shall give a résumé of his results, and shall then proceed to illustrate the application of conjugate functions to problems of elastic equilibrium by solving some questions relating to plane strain. 205. Wangerin's Problem. Consider in the first place cylindrical coordinates, 4, 2, where z is the distance of any point from a fixed plane, ☎ the distance of the point from the axis z, and the angle between the axial plane through the point, and a fixed axial plane through the axis z. In the meridian plane (z, ) suppose two systems of orthogonal curves a = const. and B = const. given by the equation These curves being rotated about the axis z give rise to a system of orthogonal surfaces whose parameters are a, ß, 4, and we may 1 'Ueber das Problem des Gleichgewichts elastischer Rotationskörper', Grunert's Archiv, Lv. 1873. use the formulæ of ch. VII. The h's are h1 = h2 = h say, and h1 = @ ̄ ̄1, and we have In Herr Wangerin's work h is replaced by J-1, and J2 is the Jacobian of z, with respect to a, B, or we have The solution of this equation V is or is not a function of p. that so far as it depends upon it contains a factor es where s is an integer, and we may denote a solution by X, est, where X ̧ is a function of a and B. In the second case we may denote a solution by X.. takes different forms according as In the first case we may suppose 8 The cubical dilatation ▲ may be expressed in the form 0 and it is shewn that A, is a function of the same form as X., and ▲, is a function of the same form as X ̧. We therefore write The three rotations w1, w2, we can be most simply found by putting 2w1J=01, 2w ̧J=0, 2ww = 0............(5). Then 1, 2, 0, can be expressed in the forms 3 and the L's, M's and N's can be written as follows: which is the integral of a complete differential in virtue of the equation satisfied by X., also where Y, is a function of the same form as X,, i.e. a solution inde 0 μ θα μ дв +2_X θα where Y, is a function of the same form as X, i.e. a solution 8 containing es of equation (3). To determine the displacements we have to introduce three functions P, Q, R of a and B, defined as follows: Q+R is a function of the complex variable a + iß, There is no difficulty in determining particular values of Q and R which satisfy the conditions just given, and any values that do so are sufficient for the purpose. The displacements u, v, w in the directions, o, z can be expressed in the forms. |