and consists therefore of two parts, one representing a uniform flow in the negative direction with a velocity c, and the other a motion of periodic oscillation. To this order therefore there can be no question that c should be the horizontal velocity in a positive direction which we must superpose on the whole mass of fluid in order to pass to pure wave motion without current. In passing to the higher orders it will be convenient still to regard this constant as the velocity of propagation, and accordingly as representing the velocity which we must superpose, in the positive direction, on the steady motion in order to arrive at the wave motion; but what, in accordance with this definition, may be the mean horizontal velocity of the whole mass of fluid in the residual wave motion, or what may be the mean horizontal velocity at the bottom, &c., or again what is the distance of the origin from the plane of mean level, are questions which we could only answer by working out the approximation, and which it would be of very little interest to answer, as we may just as well suppose the constant h defined by (34) given as suppose the mean depth given, and similarly as regards the flow. = Putting 0 in (31) and (32), we have implicitly for the equation of the surface the pair of equations The ratios of the coefficients of the successive cosines in y or sines in a to what they would have been for an infinite depth, supposing that of cos mo/c the same in the two cases, are for the sines in c. Expressed in terms of D1, the first three ratios become 16 -2 1, 1+3D ̧3, 1+18 D2+6D,~, and increase therefore as the depth diminishes, and consequently D, diminishes. The same is the case with the multipliers D/D, DD, S/D,, &c, and on both accounts therefore the series converge more slowly as the depth diminishes. Thus for D3 the first three ratios are 1, 2, 34. D=3 corresponds to h/λ=0·125, nearly, so that the average depth is about the oneeighth of the length of a wave. The disadvantage of the approximation for the case of a finite as compared with that of an infinite depth is not however quite so great as might at first sight appear. There can be little doubt that in both cases alike the series cease to be convergent when the limiting wave, presenting an edge of 120°, is reached. In the case of an infinite depth, the limit is reached for some determinate ratio of the height of a wave to the length, but clearly the same proportion could not be preserved when the depth is much diminished. In fact, high oscillatory waves in shallow water tend to assume the character of a series of disconnected solitary waves, and the greatest possible height depends mainly on the depth of the fluid, being but little influenced by the length of the waves, that is, the distance from crest to crest. To make the comparison fair therefore between the convergency of the series in the cases of a finite and of an infinite depth, we must not suppose the coefficient of cosmo/c the same in the two cases for the same length of wave, but take it decidedly smaller in the case of the finite depth, such for example as to bear the same proportion to the greatest possible value in the two cases. But with all due allowance to this consideration, it must be confessed that the approximation is slower in the case of a finite depth. That it must be so is seen by considering the character of the developments, in the two cases, of the ordinate of the profile in a harmonic series in terms of the abscissa, or of a quantity having the same period and the same mean value as the abscissa. The flowing outline of the profile in deep water lends itself readily to expansion in such a series. But the approximately isolated and widely separated elevations that represent the profile in very shallow water would require a comparatively large number of terms in their expression in harmonic series in order that the form should be represented with sufficient accuracy. In extreme cases the fact of the waves being in series at all has little to do with the character of the motion in the neighbourhood of the elevations, where alone the motion is considerable, and it is not therefore to be wondered at if an analysis essentially involving the length of a wave should prove inconvenient. INDEX TO VOL. I. Aberration of light, 134, 153; Fresnel's theory respecting, 141 Airy, Sir G. B., on tides and waves, 163, Angular velocities of a fluid, 81, 112 Babinet's result as to non-influence of Ball pendulum, resistance to, 180, 186; Box, motion of fluid within a closed, of Cauchy's proof of a fundamental pro- position in hydrodynamics, 107, 160 Convergency, essential and accidental, Cylindrical surfaces, (circular) motion of perfect fluid between, 30; (elliptic) Determinateness of problems in fluid Discharge of air through small orifices, function expressed by series or inte- Doubly refracting crystals, formula for 148 Earnshaw, S., on solitary waves, 169 Elastic solids, isotropic, equations of equilibrium &c. of, 113; necessity of Fresnel's theory of non-influence of earth's motion on the reflection and Gerstner's investigation of a special case 162; on sound, 178; on the motion of Hydrodynamics, report on, 157 Impulsive motion of fluids, 23 Instability of motion, 53, 311 Kelland, Prof., long waves in canal of Lee-way of a ship, effect of waves on, 208 Lines of motion (see Stream lines) Motion of fluids, some cases of, 17; sup- Newton's solution of velocity in a vortex, Parallelepiped, rectangular, motion of Periodic series, critical values of the Pipe, linear motion of fluid in a, 105; Rankine's investigation of a special case 225 Rectangle, different expressions for the Reflection, principle of, as applied to Saint-Venant and Wantzel, discharge of Sound, intensity and velocity of, theo- given family of curves can be a set of, Thomson, F. D., demonstration of a the- Tides (see Waves) Triangular prism, (equilateral) motion of udx+vdy+wdz an exact differential, Uniqueness of expression for p, q, or ø, 23 Waves and tides, report respecting, 161 CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. |