"A steel globe of the same dimensions, without mutual gravitation of its parts, could scarcely oscillate so rapidly, since the velocity of plane waves of distortion in steel is only about 10,140 feet per second, at which rate a space equal to the earth's diameter would not be travelled in less than 1 h. 8 m. 40 s.* x When the surface oscillates in the form of a zonal harmonic spheroid of the second order, the equation of the lines of motion is xw2=const., where a denotes the distance of any point from the axis of symmetry, which is taken as axis of x (see Art. 95 (11)). The forms of these lines, for a series of equidistant values of the constant, are shewn in the figure. 263. This problem may also be treated very compactly by the method of 'normal co-ordinates' (Art. 168). The kinetic energy is given by the formula where SS is an element of the surface ra. Hence, when the surface oscillates in the form ra+n, we find, on substitution from (2) and (4), = * Sir W. Thomson, l.c. The exact theory of the vibrations of an elastic sphere gives, for the slowest oscillation of a steel globe of the dimensions of the earth, a period of 1 h. 18 m. See a paper "On the Vibrations of an Elastic Sphere," Proc. Lond. Math. Soc. t. xiii. p. 212 (1882). The vibrations of a sphere of incompressible substance, under the joint influence of gravity and elasticity, have been discussed by Bromwich, Proc. Lond. Math. Soc. t. xxx. p. 98 (1898). The influence of compressibility is examined by Love, Some Problems of Geodynamics (Adams Prize Essay), Cambridge, 1911, p. 126. To find the potential energy, we may suppose that the external surface is constrained to assume in succession the forms r = a + 05n, where 0 varies from 0 to 1. At any stage of this process, the gravitation potential at the surface is, by (6), Hence the work required to add a film of thickness 80 is The results corresponding to the general deformation (1) are obtained by prefixing the sign Σ of summation with respect to n, in (12) and (15); since the terms involving products of surface-harmonics of different orders vanish, by Art. 87. The fact that the general expressions for T and V thus reduce to sums of squares shews that any spherical-harmonic deformation is of a 'normal type.' Also, assuming that n ∞ cos (σnt + €), the consideration that the total energy TV must be constant leads us again to the result (10). In the case of the forced oscillations due to a disturbing potential 'cos (ote) which satisfies the equation V2 = 0 at all points of the fluid, we must suppose Q' to be expanded in a series of solid harmonics. If be the equilibrium-elevation corresponding to the term of order n, we have, by Art. 168 (14), for the forced oscillation, where σ is the imposed speed, and σ, that of the free oscillations of the same type, as given by (10). The numerical results given above for the case n = 2 shew that, in a nonrotating liquid globe of the same dimensions and mean density as the earth, forced oscillations having the characters and periods of the actual lunar and solar tides would practically have the amplitudes assigned by the equilibriumtheory. 264. The investigation is easily extended to the case of an ocean of any uniform depth, covering a symmetrical spherical nucleus. Let b be the radius of the nucleus, a that of the external surface. The surface-form being 0 r = a + Σ1 Śn‚ 1 .(1) The condition that where the coefficients have been adjusted so as to make ĉp/or =0 for r=b. ..(6) For the gravitation-potential at the free surface (1) we have Hence, putting g=πурa, we find 2n + 1 The elimination of S, between (4) and (7) leads to where .(7) If p=po, we have σ1 =0 as we should expect. When p>po the value of σ, is imaginary; the equilibrium configuration in which the external surface of the fluid is concentric with the nucleus is then unstable. (Cf. Art. 200.) If in (9) we put b=0, we reproduce the result of the preceding Art. 'If, on the other hand, the depth of the ocean be small compared with the radius, we find, putting b=a -h, and neglecting the square of h/a, provided n be small compared with a/h. This agrees with Laplace's result, obtained in a more direct manner in Art. 200. as in Art. 228. Moreover, the expression (2) for the velocity-potential becomes, if we write ra+z, where is a function of the co-ordinates in the surface, which may now be treated as plane. Cf. Art. 257. The formulae for the kinetic and potential energies, in the general case, are easily found by the same method as in the preceding Art. to be The latter result shews, again, that the equilibrium configuration is one of minimum potential energy, and therefore thoroughly stable, provided p<po• In the case where the depth is relatively small, whilst n is finite, we obtain, putting b=a-h, whilst the expression for V is of course unaltered. If the amplitudes of the harmonics ( be regarded as generalized co-ordinates, the formula (15) shews that for relatively small depths the 'inertia-coefficients' vary inversely as the depth. We have had frequent illustrations of this principle in our discussions of tidal waves. Capillarity. 265. The part played by Cohesion in certain cases of fluid motion has long been recognized in a general way, but it is only within comparatively recent years that the question has been subjected to exact mathematical treatment. We proceed to give some account of the remarkable investigations of Kelvin and Rayleigh in this field. It is beyond our province to discuss the physical theory of the matter*. It is sufficient, for our purpose, to know that the free surface of a liquid, or, more generally, the common surface of two fluids which do not mix, behaves as if it were in a state of uniform tension, the stress between two adjacent portions of the surface, estimated at per unit length of the common boundaryline, depending only on the nature of the two fluids and on the temperature. We shall denote this 'surface-tension,' as it is called, by the symbol T1. The 'dimensions' of T1 are MT-2 on the absolute system of measurement. Its value in c.G.S. units (dynes per linear centimetre) appears to be about 74 for a water-air surface at 20° C.t; it diminishes somewhat with rise of temperature. The corresponding value for a mercury-air surface is about 540. 1 *For this, see Maxwell, Encyc. Britann. Art. "Capillary Action" [Papers, Cambridge, 1890, t. ii. p. 541], where references to the older writers are given. Also, Rayleigh, “On the Theory of Surface Forces," Phil. Mag. (5), t. xxx. pp. 285, 456 (1890) [Papers, t. iii. p. 397]. † Rayleigh, "On the Tension of Water-Surfaces, Clean and Contaminated, investigated by the method of Ripples," Phil. Mag. (5), t. xxx. p. 386 (1890) [Papers, t. iii. p. 394]; Pedersen, Phil. Trans. A, t. ccvii. p. 341 (1907); Bohr, Phil. Trans. A, t. ccix. p. 281 (1909). An equivalent statement is that the 'free' energy of any system, of which the surface in question forms part, contains a term proportional to the area of the surface, the amount of this 'superficial energy' (as it is usually termed) per unit area being equal to T1*. Since the condition of stable equilibrium is that the free energy should be a minimum, the surface tends to contract as much as is consistent with the other conditions of the problem. The chief modification which the consideration of surface-tension will introduce into our previous methods is contained in the theorem that the fluid pressure is now discontinuous at a surface of separation, viz. we have 2 where p, p' are the pressures close to the surface on the two sides, and R1, R2 are the principal radii of curvature of the surface, to be reckoned negative when the corresponding centres of curvature lie on the side to which the accent refers. This formula is readily obtained by resolving along the normal the forces acting on a rectangular element of a superficial film, bounded by lines of curvature; but it seems unnecessary to give here the proof, which may be found in most modern treatises on Hydrostatics. 266. The simplest problem we can take, to begin with, is that of waves on a plane surface forming the common boundary of two fluids at rest. If the origin be taken in this plane, and the axis of y normal to it, the velocity-potentials corresponding to a simple-harmonic deformation of the common surface may be assumed to be = Ceky cos kx . cos (σt + €), .(1) side on which y is negative, and For these values satisfy V2 = 0, =, respectively. where the former equation relates to the the latter to that on which y is positive. V2′ = 0, and make the velocity zero for y The corresponding displacement of the surface in the direction of y will be of the type *The distinction between 'free' and 'intrinsic' energy depends on thermo-dynamical principles. In the case of changes made at constant temperature with free communication of heat, it is with the 'free' energy that we are concerned. 29 L. H. |