Hence, writing U for the group-velocity (do/dk), we have where the sign is the opposite to that of d2/dk2. In order that the phase at P, viz. ot - k§±, may be stationary as regards variation in the position of Q we must have, then, This determines k, and the corresponding value of t then follows from (3). If we denote these special values of k and t by к and 7, respectively, the corresponding value of ot - kg will be by Art. 241 (4), we obtain finally, for the disturbance at the point x, the It will be understood that U here denotes the value of the group-velocity corresponding to k K. = It appears from (6) that the wave-length of the progressive wave-train represented by this formula is that of a free wave-train whose velocity of propagation is c. Also, since by (3) the values of x to which the preceding calculation applies will be positive or negative according as U c. If U <c, as is the case of gravity waves on a liquid, the train follows the initiating disturbance, whilst if U> c, as is the case of capillary waves (Art. 266), it precedes it. If there is more than one value of k satisfying (6) there will be a term in 7 corresponding to each of these. η Referring now to Art. 239 (33), we see that to find the elevation n in the case of waves on deep water due to a travelling pressure we must put $ (k) = ioP/gp. .... Since U is now = c, we obtain, on taking the real part, (11) .(12) As the preceding investigation involves a double approximation, it may be worth while to give another method of arriving at the result (9) which will indicate very readily the condition under which it holds. If we introduce the hypothesis of a small frictional force varying as the velocity, the formula (6) of Art. 241, when modified so as to apply to the case of a travelling disturbance, takes the form The quantity μ is by hypothesis small, and will in the limit be made to vanish. The most important part of the result will therefore be due to values of k in the first integral which make σ= kc ...(15) approximately. Writing as before k + k', where x is a root of this equation, we have = nearly, where U denotes the group-velocity corresponding to the wave-length 2π/κ. The important part of (14) is therefore *This method of obtaining the formula (12) was indicated in a footnote on p. 416 of the preceding edition. †The results quoted are equivalent to the familiar formulae (where the upper or lower sign is to be chosen according as x is positive or negative), but can be obtained directly by a contour integration. in the respective cases. If we now make μ → O we obtain our former results. The approximation in (16) is valid only if the quotient is small even when k'x is a moderate multiple of 27. This requires that should be small. Unless U = c, exactly, the condition is always fulfilled if x be sufficiently great. It may be added that the results (20), (21) are accurate, in the sense that they give the leading term in the evaluation of (14) by Cauchy's method of residues. Cf. Art. 242. 249. The preceding results have a bearing on the question of 'waveresistance.' Taking for definiteness the case of U<c, let us imagine a fixed vertical plane to be drawn in the rear of the disturbing agency. If E be the mean energy of the waves, the space in front of this plane gains, per unit time, the additional energy cE, whilst the energy transmitted across the plane is UE, by Art. 237. Hence if R be the resistance experienced by If U >c, the fixed plane must be taken in advance, and the result is (1) .(2) Thus, in the case of a disturbance advancing with velocity c[<√(gh)] over still water of depth h, we find, on reference to Art. 237, 1 gpa2 (1 2kh where a is the amplitude of the waves. (3) As c increases from 0 to (gh), кh diminishes from to 0, so that R diminishes from gpa2 to 0. When c>√(gh), the effect is merely local, and R 0*. It must be remarked, * Cf. Sir W. Thomson, "On Ship Waves," Proc. Inst. Mech. Eng. Aug. 3, 1887 [Popular Lectures and Addresses, London, 1889-94, t. iii. p. 450]. A formula equivalent to (3) was given in a paper by the same author, Phil. Mag. (5), t. xxii. p. 451 [Papers, t. iv. p. 279]. however, that the amplitude a due to a disturbance of given type will also vary with c. For instance, in the case of Art. 244 (43), а ∞ ке-*ь, where K = g/c2, the depth being infinite. Hence An interesting variation of the general question is presented when we have a layer of one fluid on the top of another of somewhat greater density. If p, p' be the densities of the lower and upper fluids, respectively, and if the depth of the upper layer be h', whilst that of the lower fluid is practically infinite, the results of Stokes quoted in Art. 232 shew that two wave-systems may be generated, whose lengths (27/к) are related to the velocity c of the disturbance by the formulae It is easily proved that the value of x determined by the second equation is real only if If c exceeds the critical value thus indicated, only one type of waves will be generated, and if the difference of densities be slight the resistance will be practically the same as in the case of a single fluid. But if c fall below the critical value, a second type of waves may be produced, in which the amplitude at the common boundary greatly exceeds that at the upper surface; and it is to these waves that the 'dead-water resistance' referred to in Art. 232 is attributed*. The problem of the submerged cylinder (Art. 247) furnishes an instance where the wave-resistance to the motion of a solid can be calculated. The mean energy, per unit area of the water surface, of the waves represented by the second term in equation (14) of that Art. is For a given depth (ƒ) of immersion, this is greatest when кf = 1, or .(7) (8) .(9) † Ann. di mat., l.c. The same law of resistance as a function of the velocity c has been obtained by Havelock, in the case of various types of surface disturbance, “Ship Resistance...." Proc. R. S. t. lxxxix. p. 489 (1913). A previous paper by the same author on "The Wave-Making Resistance of Ships, ...," Proc. R. S. t. lxxxii. p. 276 (1909), may also be referred to. Ꭱ (gf) Waves of Finite Amplitude. 250. The restriction to 'infinitely small' motions, in the investigations of Arts. 227, ... implies that the ratio (a/A) of the maximum elevation to the wave-length must be small. The determination of the wave-forms which satisfy the conditions of uniform propagation without change of type, when this restriction is abandoned, forms the subject of a classical research by Stokes *. The problem is most conveniently treated as one of steady motion. If we neglect small quantities of the order a3/23, the solution of the problem in the case of infinite depth is contained in the formulae† y = =0 is found by successive approxi Beky cos kx = ẞ (1 + ky + 1 k2 y2 + ...) cos kx = 1 kß2 + ß (1 + k2ß2) cos kx + 1 kẞ2 cos 2kx + k2ß3 cos 3kx + or, if we put B (1 + 2k282) = a, k2 B2) = a cos kx + ka2 cos 2kx + 3k2 a3 cos 3kx + .... ..(3) * "On the theory of Oscillatory Waves," Camb. Trans. t. viii. (1847) [reprinted, with a "Supplement," Papers, t. i. pp. 197, 314]. The outlines of a more general investigation, including the case of permanent waves on the common surface of two horizontal currents, have been given by Helmholtz, "Zur Theorie von Wind und Wellen," Berl. Monatsber. July 25, 1889 [Wiss. Abh. t. iii. p. 309]. See also Wien, Hydrodynamik, p. 169. Rayleigh, l.c. ante p. 252. |