so that the angular velocity of the element (a, b) is kce-2kb This is greatest at the surface, and diminishes rapidly with increasing depth. Propagation in Two Dimensions. 161. In the cases already considered the propagation of the waves over the surface of the fluid has been supposed to take place in one dimension (x) only. We will now sketch the method to be pursued in treating cases where the propagation is in two dimensions. Let the origin be taken in the undisturbed surface, the axes of x and y horizontal, that of z vertical and upwards; and let h be the (uniform) depth of the fluid. The velocity-potential & must satisfy Further, at the free surface we must have, making the same approximations as in Art. 156 Now (35) is satisfied by the sum of any number of terms of the form {ek (z+h) + e-k (z+1)} f'. .(38), = 0, [z = 0]................(37). where the k's not contain z. or are constants yet to be determined, and p' does Substituting in (37) we find (ekh +e-kh) + gk (ekh − e−kh) p'′ = 0, where c is given by and dt2 12 are functions of x, y only, to be determined from (35) which now gives with a similar equation for 2. The solution of (40), when adapted to suit the conditions to be satisfied at the surfaces, if any, which limit the fluid horizontally, gives the possible values of k. By adding together terms of the form (38), in which k has the values thus found, we may build up a solution satisfying any arbitrary initial conditions. Oscillations in a Rectangular Tank. 162. We apply the method just explained to the case where the fluid is contained in a rectangular tank whose sides are vertical. Let the origin be taken in one corner of the tank, and the axes of x, y along two of its sides, and let the equations of the other two sides be x = a, y = b, respectively. The functions &1, 1⁄2 must now satisfy the conditions The general value of 4, subject to these conditions is given by the double Fourier's series where the summations include all integral values of m, n from 1 to∞. Substituting in (40), we find If a be the greater of the two quantities a, b, the component oscillation of longest period is got by making m = 1, n = 0, whence k = 7; the motion is then parallel to the longer side of the tank, and consists of two systems of waves of the kind considered in Art. 156, travelling in opposite directions, the wave-length being 2a. Circular Tank. 163. In the case of a circular tank, whose axis is vertical, it is convenient to take the origin in the axis, and to transform to polar co-ordinates by writing Now whatever be the value of o' it can be expanded by Fourier's theorem in the form $' Xn n' where the summation embraces all integral values of n, and ↓„, X are functions of r only. Substituting in (41), we have, to determine The solution of (42), subject to the condition that is finite when r = Ψη=ΣΑ Jη (kr)..... n .(43), n n where A is an arbitrary constant, and J (kr) denotes the Bessel's Function of the nth order of the variable kr, viz.* The summation in (43) is supposed to include all admissible values of k, to be determined by the equation the accent denoting the first derived function, and a being the radius of the tank. It may be shewn that the values of k satisfying (44) are infinite in number and all real. For the particular case n = 0, when the motion is symmetrical about the centre, the lowest roots are given by For a discussion of the various kinds of motion represented by the above formulæ the reader is referred to Lord Rayleigh's paper on Waves already cited†, which contains besides a comparison with theory of some experimental measurements of the periods of oscillation in rectangular and circular tanks made by Guthrie. Free oscillations of an Ocean of uniform Depth. 164. We close this chapter with the discussion of the following problem, which is of some interest in connexion with the theory of the tides :-To determine the free oscillations of an ocean. of uniform depth completely enveloping a spherical earth. The method employed is due to Thomson §. Let r denote the distance of any point from the centre of the sphere, and a, b the values of r at the surface of the solid sphere, and at the mean level of the ocean, respectively. The general * Todhunter, Functions of Laplace, &c., Art. 370. + See also Theory of Sound, c. 9, by the same author. The values of the roots of (44) for the case n=0 are taken from this source. Phil. Mag. 1875. § Phil. Trans. 1863, p. 608. solution of the equation of continuity 2 = 0, subject to the n The 2n+1 where S is the general surface-harmonic of order n. arbitrary constants which the general expression for S contains are functions of t, to be determined. The formula for the pressure is, if we neglect squares and products of small quantities, where p is the density of the fluid, V the potential due to the joint attraction of the earth and the sea. If we assume as the equation of the free surface at time t where T is a spherical harmonic of order n, we have*, at this surface, Here E denotes the total mass of the earth and sea, viz. E = {πα3σ + ƒπ (b3 — a3) p, if σ be the mean density of the earth. Substituting in (46), expanding, and omitting as before the squares of small quantities we find, at the free surface, * Pratt, Figure of the Earth, c. 3. It is assumed of course that E, p, σ are all expressed in astronomical units.' 5} T +1) dSn dt .(48). |