CHAPTER XI. VIBRATIONS OF A SPHERE1. 189. THE problem of determining the normal modes and periods of vibration of an isotropic elastic solid sphere or spherical shell whose surface is free was first completely solved by Prof. Lamb. It is a most interesting example of the general theory of the free vibrations of solids explained in arts. 79 and 80. We shall consider, in the first place, the theory of the free vibrations of a solid sphere or spherical shell, and afterwards the problem of forced vibrations in a solid sphere produced by forces derivable from a potential expressible in spherical harmonic series. 190. Differential equations of Free Vibration. We have to find solutions of the equations of displacement 1 The following among other authorities may be consulted: Jaerisch Ueber die elastischen Schwingungen einer isotropen Kugel'. Crelle's Journal, LXXXVIII. 1880. Lamb On the Vibrations of an elastic sphere'. Proc. Lond. Math. Soc. XIII. 1882, and 'On the Vibrations of a spherical shell'. Proc. Lond. Math. Soc. xiv. 1883. Love The free and forced Vibrations of an elastic spherical shell...". Proc. Lond. Math. Soc. xIx. 1888. Chree On the equations of an isotropic elastic solid in cylindrical and polar coordinates'. Camb. Phil. Soc. Trans. xiv. 1887. Rayleigh' On Waves propagated along the plane surface of an elastic solid'. Proc. Lond. Math. Soc. xvII. 1886. which are simple harmonic functions of the time, are finite, continuous, and one-valued within the boundary, and satisfy the condition that the bounding surface is free from stress. Suppose the solid performing free vibrations whose period we may substitute -pu..., and thus the is 2π/p; then for equations (1) become of the type ΟΔ (x + μ) дох + μ▼3u + pp3u = 0 ........................ .............(3). writing we have Differentiating these with respect to x, y, z, adding, and h2 = p2p/(λ + 2μ), к2 = p2p/μ (4), ..(5), where A satisfies (5), and these satisfy (2). Hence the complete solutions of the equations of vibration consist of the sums of these solutions and the general solutions of the equations (▼2 + k2) u = 0, (▼2 + x2) v=0, (▼2 + x2) w = 0, Before proceeding we point out the kind of results to be obtained. According to the theory explained in art. 79, there will be an indefinite number of normal modes of oscillation, and the oscillations of any normal mode can be executed independently. If the system be oscillating in a normal mode then at any instant the displacements can be expressed in the form u = u'A cos (pt +e), v=v'A cos (pte), w = w'A cos (pt + €), where 2π/p is the period, A is a small arbitrary constant, and u, v', w' are functions of x, y, z. These functions are called normal functions, and the determination of the vibrations of any elastic system is effected when the normal functions are known and the frequency-equations have been formed and solved. In what follows we shall first determine the forms of the normal functions; and no confusion ought to arise if we denote them by u, v, w, instead of u', v', w', and write ▲ for du/dx+dv/dy +Əw/əz, where u, v, w are simply normal functions. In strictness each term of the cubical dilatation also contains a factor of the form A cos (pt + €). Among the vibrations of a sphere we shall find that for some modes there are spherical surfaces at which the displacement vanishes, just as in the vibrations of a string there may be one or more nodal points. Such surfaces will be called nodal surfaces, and their number and position are determined by the type of vibration and the frequency, and, conversely, if the number and position of these surfaces be given the type and the frequency are determinate. We shall find also other modes for which there are no surfaces at which the displacement vanishes, but there will then be surfaces at which the radial displacement vanishes, and we shall term such surfaces quasi-nodal. The number of the quasi-nodal surfaces for a particular class of vibrations does not in general determine the frequency or the type. We proceed now to the consideration of the vibrations of an isotropic elastic sphere. 192. Determination of the Dilatation. We have to find a solution of the equation (V2 + h2) ▲ = 0 in a form adapted to satisfy boundary-conditions at the surface of a sphere. We therefore suppose ▲, at the surface of the sphere, expressible in spherical surface-harmonics, and we treat the typical term A = RnSn, where S is a spherical surface-harmonic, and R is a function of r, defined by the equation This is the case of Riccati's equation which is integrable in terms of circular functions, and the solution which remains finite in space containing the origin is This function can be expanded in a convergent series of powers of r, beginning with r", and, if we take such a multiplier as will make the first coefficient unity, and write "Sn = wn, we shall have as the general form of A where on is a spherical solid harmonic of order n, and Θη .(11), ¥n (~) = (−)1 1 . 3 . 5 ...( 2n + 1) (1 d)" (sin *) ... (12). We add a few properties of the functions (x) which admit of ready verification : The equations connecting consecutive y's are ¥n(x) = (2n−1){¥n-2(x)—¥n-1(x)}...(13). 2n+1n(x)=(2n-1) The differential equation is d*ln (2) _ 2 (n+1)diện (2) d dx Yn-1 (x) d2n dx + The series for n (x) is Ꮳ dx ... and thus √(x) = † √√(2π) 1.3...(2n + 1) x−(n+1) Jn+ż (x) ... (16), where Jn+ (x) is the Bessel's function of order n + 1. n The function (2) of the complex variable z is a uniform function in all parts of the plane of z which exclude the point at infinity. This point is an essential critical point of the function. 193. Determination of the Displacements. The forms of u, v, w, which satisfy (8), can be written down in the same way; thus u = ΣUn¥n (kr), v=ΣVn¥n(xr), w=ΣWn¥n (Kr)................(17), where Un, Vn, Wn are spherical solid harmonics of order n, and these have to be arranged to satisfy the condition If Xn be a spherical solid harmonic of order n, then the xUn+yVn+zWn= 0, and ǝUn/dx+ƏVn/əy+dWn/dz = 0, and Un, Vn, Wn are spherical solid harmonics of order n. similar expressions with y and z respectively for x, these will be spherical solid harmonics of order n provided on+1 be one of order n+ 1, and we shall have and anr2n&n−1• xUn+yVn+zWn = (n + 1) $n+1 + α222$n−1· Thus the terms contributed to (19) by such functions Un, Wn, will contain +1 multiplied by Ə¥n+2 n+1 dyn +an+2 (n + 2) r d√n+2 + (n + 2) (2n + 5) αn+2¥n+2 r др др By using (13) the multiplier becomes Vn, where v and w are to be derived from this by cyclical interchanges of the letters x, y, z, and X, and n+1 are spherical solid harmonics of orders indicated by the suffixes. This solution contains two |