Hence, for the genesis of the waves by ordinary forces, we require as a foundation an initial horizontal motion, in the direction opposite to that of propagation of the waves ultimately set up, which diminishes rapidly from the surface downwards, according to the law (12), where b is a function of y' determined by It is to be noted that these rotational waves, when established, have zero momentum. 252. Scott Russell, in his interesting experimental investigations*, was led to pay great attention to a particular type which he called the 'solitary wave.' This is a wave consisting of a single elevation, of height not necessarily small compared with the depth of the fluid, which, if properly started, may travel for a considerable distance along a uniform canal, with little or no change of type. Waves of depression, of similar relative amplitude, were found not to possess the same character of permanence, but to break up into series of shorter waves. Russell's 'solitary' type may be regarded as an extreme case of Stokes' oscillatory waves of permanent type, the wave-length being great compared with the depth of the canal, so that the widely separated elevations are practically independent of one another. The methods of approximation employed by Stokes become, however, unsuitable when the wave-length much exceeds the depth; and subsequent investigations of solitary waves of permanent type have proceeded on different lines. The first of these was given independently by Boussinesq† and Rayleigh. The latter writer, treating the problem as one of steady motion, starts virtually from the formula $ + iμ = F (x + iy) = d iv dx F (x), .(1) where F(x) is real. This is especially appropriate to cases, such as the present, where one of the family of stream-lines is straight. We derive from (1) y2 2! y1 ყვ уб 5! 3! where the accents denote differentiations with respect to x. The stream-line = 0 here forms the bed of the canal, whilst at the free surface we have * "Report on Waves," Brit. Ass. Rep. 1844. † Comptes Rendus, June 19, 1871. l.c. ante p. 252. = ch, where c is the uniform velocity, and h the depth, in the parts of the fluid at a distance from the wave, whether in front or behind. The condition of uniform pressure along the free surface gives u2 + v2 = c2 2g (y - h), or, substituting from (2), .(3) F'2 — y2F' F''' + y2F''2 + ... = c2 - 2g (y - h). .......(4) But, from (2) we have, along the same surface, It remains to eliminate F between (4) and (5); the result will be a differential equation to determine the ordinate y of the free surface. If (as we will suppose) the function F' (x) and its differential coefficients vary so slowly with that they change only by a small fraction of their values when x increases by an amount comparable with the depth h, the terms in (4) and (5) will be of gradually diminishing magnitude, and the elimination in question can be carried out by a process of successive approximation. and if we retain only terms up to the order last written, the equation (4) becomes Hence y vanishes only for y = h and y y' = gy c2 .(8) c2/g, and since the last factor must be positive, it appears that c2/g is a maximum value of y. Hence the wave is necessarily one of elevation only, and, denoting by a the maximum height above the undisturbed level, we have which is exactly the empirical formula for the wave-velocity adopted by Russell. The extreme form of the wave must, as in Art. 250, have a sharp crest of 120°; and since the fluid is there at rest we shall have c2 2ga. If the formula (9) were applicable to such an extreme case, it would follow that α = h. If we put, for shortness, = if the origin of x be taken beneath the summit. There is no definite 'length' of the wave, but we may note, as a rough indication of its extent, that the elevation has one-tenth of its maximum value when x/b = 3.636. represents the wave-profile in the case a = h. For lower waves the scale of y must be contracted, and that of x enlarged, as indicated by the annexed table giving the ratio b/h, which determines the horizontal scale, for various values of a/h. It will be found, on reviewing the above investigation, that the approximations consist in neglecting the fourth power of the ratio (ha)/2b. If we impress on the fluid a velocity - c parallel to x we get the case of a progressive wave on still water. It is not difficult to shew that, if the ratio a/h be small, the path of each particle is then an arc of a parabola having its axis vertical and apex upwards*. a/h b/h .2 1.915 1.414 .3 1.202 .4 1.080 123456 1.000 .6 .943 .7 .900 .8 .866 .9 .839 1.0 .816 It might appear, at first sight, that the above theory is inconsistent with the results of Art. 187, where it was argued that a wave of L. H. *Boussinesq, l.c. 27 finite height whose length is great compared with the depth must inevitably suffer a continual change of form as it advances, the changes being the more rapid the greater the elevation above the undisturbed level. The investigation referred to postulates, however, a length so great that the vertical acceleration may be neglected, with the result that the horizontal velocity is sensibly uniform from top to bottom (Art. 169). The numerical table above given shews, on the other hand, that the longer the 'solitary wave' is, the lower it is. In other words, the more nearly it approaches to the character of a 'long' wave, in the sense of Art. 169, the more easily is the change of type averted by a slight adjustment of the particle-velocities*. The motion at the outskirts of the solitary wave can be represented by a very simple formula. Considering a progressive wave travelling in the direction of x-positive, and taking the origin in the bottom of the canal, at a point in the front part of the wave, we This will be found to agree approximately with Rayleigh's investigation if we put m =b-1. The above remark, which was kindly communicated to the author by the late Sir George Stokest, was suggested by an investigation by McCowan‡, who shewed that the formula where a denotes the maximum elevation above the mean level, and a is a subsidiary constant. In a subsequent papers the extreme form of the wave when the crest has a sharp angle of 120° was examined. The limiting value of the ratio a/h was found to be 78, in which case the wave-velocity is given by c2 = 1·56gh. 253. By a slight modification the investigation of Rayleigh and Boussinesq can be made to give the theory of a system of oscillatory waves of finite height in a canal of limited depth. * Stokes, "On the Highest Wave of Uniform Propagation," Proc. Camb. Phil. Soc. t. iv. p. 361 (1883) [Papers, t. v. p. 140]. † Cf. Papers. t. v. p. 162. "On the Solitary Wave," Phil. Mag. (5), t. xxxii. p. 45 (1891). § "On the Highest Wave of Permanent Type," Phil. Mag. (5), t. xxxviii. p. 351 (1894). Korteweg and De Vries, "On the Change of Form of Long Waves advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves," Phil. Mag. (5), t. xxxix. p. 422 (1895). The method adopted by these writers is somewhat different. Moreover, as the title In the steady-motion form of the problem the momentum per wave-length (λ) is represented by where corresponds to the free surface. If h be the mean depth, this momentum may be equated to pchλ, where c denotes (in a sense) the mean velocity of the stream. On this understanding we have, at the surface, Vi -ch, as before. The arbitrary constant in (3), on the other hand, must be left for the moment undetermined, so that we write In equations (25), (28), (30) we have four relations connecting the six quantities h1, h2, l, k, A, B, so that if two of these be assigned the rest are analytically determinate. The wave-velocity c is then given by (22)†. For example, the form of the waves, and their velocity, are determined by the length A, and the height h1 of the crests above the bottom. The solitary wave of Art. 252 is included as a particular case. If we put h2, we have k = 1, and the formulae (28) and (30) then shew that λ = ∞, h2 = h. indicates, the paper includes an examination of the manner in which the wave-profile is changing at any instant, if the conditions for permanency of type are not satisfied. For other modifications of Rayleigh's method reference may be made to Gwyther, Phil. Mag. (5), t. 1. pp. 213, 308, 349 (1900). * The waves represented by (27) are called 'cnoidal waves' by the authors cited. For the method of proceeding to a higher approximation we must refer to the original paper. When the depth is finite, a question arises as to what is meant exactly by the 'velocity of propagation.' The velocity adopted in the text is that of the wave-profile relative to the centre of inertia of the mass of fluid included between two vertical planes at a distance apart equal to the wave-length. Cf. Stokes, Papers, t. i. p. 202. |