458 HYDROMECHANICS [HYDRODYNAMICS.. Denoting the velocity functions by <p and <(> , $ = U cos a . a + U sin a . y+ A cosh m(z + h) cos (mx-nt) $ = U cos a . x + U sin a . y + A cosh m(z - h ) cos (mx - nt) ; then < and <f> satisfy the equations of continuity, and the conditions that = when z = - h, and -y- = when z = h . Supposing the equation of the moving surface of separation to be 2 = b sin (mx - nt), then the direction of motion of each liquid, relative to the moving surface of separation, must be a tangent to the surface, and therefore, when 3 = 0, d<f> d<p dz dz dz or, neglecting A 2 and A 2 , A sinh mh dx - A sinh mh U cos o - V U cos a - V dividing out by the common factor m cos (mx - nt), and therefore = U cos o.x + Usiuo. 2/ + (U cos a - V)6 . sinh m/z. = U cos a .x + U sin a .y - (U cosa - The dynamical equations are d<l> sinh - nt}. and at the surface of separation, where 2=0, we must have - = -T - dx 2 = H - H - g(p - p )b sin (mx - nt) - p(U cos a - V) coth mh . nb sin (mx - nt) - p (U cos a - V) coth mh . nb sin (mx - nt) - 4 p{ U cos a - (U cos a - V) coth mh . mb sin (mx - nt)}% - pU 2 sin 2 a - 4 p (U cos a - V) 2 ?n 2 6 2 cos 2 (mx - nt) + 4 P { U cos o + ( U cos a - V) coth mh . mb sin (mx -nt)} 2 + 4 p U 2 sin 2 o + 1 p (U cos a - ~V) 2 m 2 b 2 cos 2 (?a; - ?i<) ; and neglecting & 2 and equating to zero the coefficient of sin (mx - nt), m 2 T + g(p - p ) - (U cos a - V)(mU cos a - n)p coth mh - (U Cos o - V)(??iU cos a - n)p coth mh = 0, which, since = V, reduces to m m 2 T + g(p -p )-m(J cos o - V) 2 p coth mh - m(U cos a - V) 2 p coth mh ^Q, 2TT or, since m = ,
- 2f T = j (U cos a- V) 2 p coth -(.(U cosa - V) 2 p coth?^ j
A. A. ( A A ) -g(p-p )-o. If U = 0, U = 0, p = 0, we find V 2 = ( + | tanh as at first, if T = 0. A discussion of the different cases that can arise is given by Lord Rayleigh in his papers on the " Instability of Jets" published in the Proceedings of tJie Royal Society and of the London Mathe matical Society ; also in a paper by Sir W. Thomson in the Phil. Mag., 1871. In the last-mentioned paper an interesting application of the above equations is made to determine the ripples produced by wind blowing over the surface of still water. Put U = , 7t=oo , 7t =oo ; then m 2 T + rj(p - p ) - ?V 2 p - m(U r - V) 2 p = 0. If W be the velocity of propagation of waves of the same length with no wind, then m 2 T + g( P - p ) - mW 2 (p + p ) = ; m p + p p + p the minimum value of which for different values of m is given by and then But and therefore V -^-,U . / I p + p V I giving the velocities of propagation of waves with and against the wind. The least value of U 2 is less than ^ ^- times the least value PP of W 2 , and is therefore PP If the wind be blowing with a velocity greater than this mini mum value of U , the surface of the water as a plane level surface becomes unstable, and ripples are produced. With C. G. S. units, = 981, T = 81, p = l, p = "0012759, and then the minimum value of U is about 664, equivalent to about 14 "8 miles an hour. This velocity is of course much greater than what is required to ruffle the surface of water in reality, the dis crepancy being due to the viscosity of the air. In the case of standing waves in a circular tank, cylindrical co ordinates r, 0, z being used, where x = r cos 0, y = r sin B, the equation of continuity becomes dr 2 J_ r dr __ . If the liquid be of depth h, we must put <p = <f>i cosh k(z + h) cos limt, where n is the number of oscillations per second, and then d 2 ^ 1 d<f>i 1 d 2 <j> l , o , _ n dr 2 r dr r~ dd~ If we put <(>i = fy cos md, then dr 2 r dr Bessel s differential equation ; and therefore and </> = AJ m (r) cos mB cosh k(z + h) cos and k must be determined from the condition that ? = 0, when r = a : dr At the free surface ^0 + dV = 0) or gk sinh kh 47r% 2 cosh kh 0, or 2 = 1-5 47T- For circular waves, ?n = 0, and the roots of &a-3"832, 7-016, 10 173, 13 323, .... (Rayleigh, Sound, p. 274). When the tank is limited by the radial plane 6 = 0, then the- slowest oscillation corresponds to m = ^, and then T ,, v sin kr
vkr and gives and therefore tin / k / sin ka A J hi). A ^_^ eoita ._ jEr j-o, tan ka, ^-1-4808 (Rayleigh, Sound, p. 279). When the tank is bounded by the radial planes = 0, 8=- 1 it, the slowest oscillation corresponds to m = |, and then 1 /sin kr , = . . , cos kr v kr kr J and the equation J (ka) = Q leads to tan ka Ska, For the discussion of the free oscillations of an ocean of uniform depth, covering a central nucleus, under the gravitation of the parts, and the surface tension at this free surface, consult Lamb s Motion of Fluids, p. 196, and Lord Rayleigh s papers in the Pro ceedings of the London Mathematical Society. The propagation of plane waves of longitudinal displacement in air, and the notes produced by open and closed pipes, have been con sidered under the heading ACOUSTICS. When the air is limited by special surfaces, the problem of its vibrations is worked out by Lord Rayleigh in the Proceedings of the London Mathematical Society, 1872. A list of references to the memoirs and treatises on the subject
will be found at the end of Lamb s Motion of Fluids. (A. G. G.)
