Airy*, by methods of successive approximation. He found that in a progressive wave different parts will travel with different velocities, the wavevelocity corresponding to an elevation ʼn being given approximately by η where c is the velocity corresponding to infinitely small amplitude. A more complete view of the matter can be obtained by the method employed by Riemann in treating the analogous problem in Acoustics, to which reference will be made in Chapter x. The sole assumption on which we are now proceeding is that the vertical acceleration may be neglected. It follows, as explained in Art. 168, that the horizontal velocity may be taken to be uniform over any section of the canal. The dynamical equation is as before, and the equation of continuity, in the case of a rectangular section, is easily seen to be an at' where h is the depth. This may be written .(2) where the function ƒ (n) is as yet at our disposal. If we multiply (3) by f' (n), and add to (1), we get ap ӘР at If we now determine ƒ (n) so that (h + n) {ƒ' (n)}2 = g, .(5) this may be written др at дх where c = (gh). The arbitrary constant has been chosen so as to make P and Q vanish in the parts of the canal which are free from disturbance, but this is not essential. It appears, therefore, that dP = 0, i.e. P is constant, for a geometrical point whilst Q is constant for a point moving with the velocity Hence any given value of P travels forwards, and any given value of Q travels backwards, with the velocities given by (10) and (11) respectively. The values of P and Q are determined by those of 7 and u, and conversely. η As an example, let us suppose that the initial disturbance is confined to the space for which a < x <b, so that P and Q are initially zero for x<a and x > b. The region within which P differs from zero therefore advances, whilst that within which Q differs from zero recedes, so that after a time these regions separate, and leave between them a space within which P = 0, Q = 0, and the fluid is therefore at rest. The original disturbance has now been resolved into two progressive waves travelling in opposite directions. so that the elevation and the particle-velocity are connected by a definite relation (cf. Art. 171). The wave-velocity is given by (10) and (12), viz. it is To the first order of n/h, this is in agreement with Airy's result. Similar conclusions can be drawn in regard to the receding wave*. (13) Since the wave-velocity increases with the elevation, it appears that in a progressive wave-system the slopes will become continually steeper in front, and more gradual behind, until at length a state of things is reached in *The above results can also be deduced from the equation (3) of Art. 173, to which Riemann's method can readily be adapted. which we are no longer justified in neglecting the vertical acceleration. As to what happens after this point we have at present no guide from theory; observation shews, however, that the crests tend ultimately to curl over and break. The case of a 'bore,' where there is a transition from one uniform level to another, may be investigated by the artifice of steady motion (Art. 175). If Q denote the volume per unit breadth which crosses each section in unit time we have u1h1 = u2h2 = Q, ..(14) where the suffixes refer to the two uniform states, h, and h2 denoting the depths. Considering the mass of fluid which is at a given instant contained between two cross-sections, one on each side of the transition, we see that in unit time it gains momentum to the amount pQ (u2 — u1), the second section being supposed to lie to the right of the first. Since the mean pressures over the sections are gph, and gph2, we have If we impress on everything a velocity - u1 we get the case of a wave invading still water with a velocity of propagation in the negative direction. The particle-velocity in the advancing wave is u1 u2 in the direction of propagation. This is positive or negative according as ha h1, i.e. according as the wave is one of elevation or depression. 2 The equation of energy is however violated, unless the difference of level be regarded as infinitesimal. If, in the steady motion, we consider a particle moving along the surface stream-line, its loss of energy in passing the place of transition is per unit volume. In virtue of (14) and (16) this takes the form .(18) Hence, so far as this investigation goes, a bore of elevation (h2 > h1) can be propagated unchanged on the assumption that dissipation of energy takes place to a suitable extent at the transition. If however h2 <h1, the expression. (19) is negative, and a supply of energy would be necessary. It follows that a negative bore of finite height cannot in any case travel unchanged*. * Rayleigh, "On the Theory of Long Waves and Bores," Proc. Roy. Soc. A, t. xc. p. 324 (1914). 188. In the detailed application of the equations (1) and (3) to tidal phenomena, it is usual to follow the method of successive approximation. As an example, we will take the case of a canal communicating at one end (x = 0) with an open sea, where the elevation is given by For a second approximation we substitute these values of ŋ and u in (1) and (3), and obtain Integrating these by the usual methods, we find, as the solution consistent with (20), The annexed figure shews, with, of course, exaggerated amplitude, the profile of the waves in a particular case, as determined by the first of these equations. It is to be noted that if we fix our attention on a particular point of the canal, the rise and fall of the water do not take place symmetrically, the fall occupying a longer time than the rise. The occurrence of the factor x outside trigonometrical terms in (24) shews that there is a limit beyond which the approximation breaks down. The condition for the success of the approximation is evidently that goax/c3 should be small. Putting c2=gh, λ=2πc/σ, this fraction becomes equal to 2π (a/h). (x/λ). Hence however small the ratio of the original elevation (a) to the depth, the fraction ceases to be small when x is a sufficient multiple of the wave-length (^). It is to be noticed that the limit here indicated is already being overstepped in the right-hand portions of the figure; and that the peculiar features which are beginning to shew themselves on the rear slope are an indication rather of the imperfections of the analysis than of any actual property of the waves. If we were to trace the curve further, we should find a secondary maximum and minimum of elevation developing themselves on the rear slope. In this way Airy attempted to explain the phenomenon of a double high-water which is observed in some rivers; but, for the reason given, the argument cannot be sustained*. The same difficulty does not necessarily present itself in the case of a canal closed by a fixed barrier at a distance from the mouth, or, again, in the case of the forced waves due to L. H. *McCowan, l.c. ante p. 251. 18 For a periodic horizontal force in a canal closed at both ends (Art. 179). Enough has, however, been given to shew the general character of the results to be expected in such cases. further details we must refer to Airy's treatise*. When analysed, as in (24), into a series of simple-harmonic functions of the time, the expression for the elevation of the water at any particular place (x) consists of two terms, of which the second represents an 'over-tide,' or 'tide of the second order,' being proportional to a2; its frequency is double that of the primary disturbance (20). If we were to continue the approximation we should obtain tides of higher orders, whose frequencies are 3, 4, ... times that of the primary. If, in place of (20), the disturbance at the mouth of the canal were given by 5= a cos at + a' cos (σ't +e), it is easily seen that in the second approximation we should in like manner obtain tides of periods 2/(σ +σ′) and 2π/(σ −σ'); these are called 'compound tides.' They are analogous to the 'combination-tones' in Acoustics which were first investigated by Helmholtzt. Propagation in Two Dimensions. 189. Let us suppose, in the first instance, that we have a plane sheet of water of uniform depth h. If the vertical acceleration be neglected, the horizontal motion will as before be the same for all particles in the same vertical line. The axes of x, y being horizontal, let u, v be the component horizontal velocities at the point (x, y), and let be the corresponding elevation of the free surface above the undisturbed level. The equation of continuity may be obtained by calculating the flux of matter into the columnar space which stands on the elementary rectangle dxdy; viz. we have, neglecting terms of the second order, The dynamical equations are, in the absence of disturbing forces, ди др av др .(1) = Pat = дх P де dy' "Tides and Waves," Arts. 198, ... and 308. See also G. H. Darwin, "Tides," Encyc. Britann. (9th ed.) t. xxiii. pp. 362, 363 (1888). † "Ueber Combinationstöne," Berl. Monatsber. May 22, 1856 [Wiss. Abh. t. i. p. 256]; and "Theorie der Luftschwingungen in Röhren mit offenen Enden," Crelle, t. lvii. p. 14 (1859) [Wiss. Abh. t. i. p. 318]. |