Tides and K=-k.C.gr+b)(+ax(s.cos (u+bw) t+wx+C'-w.sin (u+bw) t+wx'+C'). At the mouth of the river, x'=0, and this expression becomes -kC.E(+b). (s.cos (u+bw) t+C'-w.sin (u+bw)t+C'). But the law of the rise of the sea is assumed to be A. sin nt+B. Making these expressions coincide, we have r+bs=0, -kCs.cos (C'-B)+kwC. sin (C'—B)=0, u+bw=n, kCs. sin (C'-B)+kCw.cos (C'-B)=A. The two last equations give only the values of C and C'. Combining the two preceding them with the two account found in (341.) we have four equations for the four quantities r, s, u, w; and eliminating r, u, w, we obtain to be remark, that when b is small, a solution may be obtained which will allow a negative value -s= -f 2v-2b taken for s; that then w has a negative value -w = ; and that the elevation of the water is then expressed by A..sin nt-w'x'+B. This elevation is above the mean level of the river at that point. But the mean level there is higher than that of the sea by .sin a. Therefore the surface of the river is higher than the mean level of the sea by 'sin a+A.". sin nt - w'a' + B. α Low water (343.) At low water at any place, sin nt-w'x+B=-1, and therefore the elevation of low water at any place, in the river above the mean level of the sea, is a sin a-A.ɛ-". The elevation of the high tide of the sea above its mean level may be is +A. The low water then at a point up the river will be higher than the high water of the sea if a' sin a-As-** higher than be greater than A. As, by increasing r', 'sin a may be made as great as we please, and Aɛ-** as little high water in the sea. as we please, it is evident that a point may be found where this condition is satisfied. The circumstance that low water on a tidal river may be higher than high water on the sea, paradoxical as it may appear, is therefore a simple consequence of theory. Motion of (344.) We shall conclude with the following Problem. The water being in the state of undulation represented by X-L.cosit-mx, the forces which have maintained it in that state suddenly cease when t=a: to find the subsequent motion of the water. (345.) It is evident that there can be no such multiplier as " in the expression for X, since there is none water sup- such when t=a. Let therefore posing the tidal forces to cease. The motions will diminish rapidly. X=2.C.".cos (ut+wr)+Z.C'.s".sin (ut+wx), It is plain that w must = ±m. This restricts the assumption to We have first to find r and u. X=C.ε".cos (ut+mx)+E.".cos (ut — mx)+C'. ɛ". sin (ut+ma) +E'.e". sin (ut—mx). The general equations become r2-u2— — fr—v3m2; −2rufu. From the second, ƒ3 ƒ3 r=- Substituting in the first, Then the special conditions to be f satisfied are, that, when t=a, X must =L.cos it. cos mx+L. sin it. sin mr, and must- iL.sin it.cos me dt dX +iL.cos it.sin mx; or X must then = L cos ia, cos mx+L sin ia, sin mr, and must then-iL.sin ia.cos mi dt +iL cos ia.sin mx. Comparing these with the quantities deduced from the assumed expression, we have ε" cos ua (C+E)+ɛ"* sin uɑ (C'+E') = L cos ia ε sin ua (E-C) +ε cos ua (C'—E') = L sin ia e̟TMa (r cos ua — u sin ux). (C+E)+ɛTM (r sin ua+u cos ua). (C'+E')=—¿L sin x ε (r sin ux+u cos ua). (E-C) +εTMa (r cos ua—u sin ua). (C'—E')= iL cos ia. From the first and third, C+E and C'+ E' are found; from the second and fourth, E-C and C'E' are found; and from these C, E, C', E', are found. Then the expression for X is E. {C.cos ut+mx+E.cos ut-mx+C'. sin ul+mx+E'.sin ut—mx}. The multiplier shows that the oscillations will diminish rapidly and will therefore soon become insensible. heary of exactly analogous to the simpler expressions which we have used to represent the free tide-wave in (291.) and the articles which follow it. (347.) We shall now point out the form which the investigation assumes when the motion of water in space of Equations three dimensions is considered. Let z be the original horizontal co-ordinate of any particle measured at right angles to r, and Z the displacement of that particle in the direction of z at the time t, y being the vertical ordinate as before. Then, nearly as in (145.), we shall find the following equation of continuity: And, nearly as in (147.), we shall find the two following equations of equal pressure (no external force being supposed to act): for water d'Y (from k to dt We shall not attempt to solve these equations, except in the case where the depth is uniform, and where the (348.) Assuming, then, the same function of y as that which has occurred in the preceding investigations, U and V being functions of r and z only. 'dU dV d2Y n2 1 dU dV g/dUdv -mk + (εk —ɛ-mk). cos nt. m dr dz mk Therefore -gK-a (y to k), having regard to the equation 1a (6**+ɛ ̄m2)=gm (ɛTM1 —¿ ̄*2), becomes n2 ɛ ̄mk), Tides and Waves. Partial differential equation whose solution is necessary. m2V + d (du + dr) = 0; dz Tides Wares equations which are of the most general kind for the determination of U and V, and which are cleared of y and t. Sect. I (349.) If we differentiate the first of these equations with regard to x, and the second with regard to 2, Theory and Wares. Form of so lution which would be desirable. It will be remarked that W is proportional to the factor of cos nt in the expression for K, so that, if we could solve this equation, we should at once obtain the expression for the elevation of the wave (supposed stationary) all the circumstances of the motion of the water would be (350.) There are, however, great difficulties in the solution of this equation. The most convenient form for our purposes would be W=P cos Q, P and Q being functions of x and x. If we could obtain this, we could also obtain another W'P sin Q; and combining the former of these as factor with cos nt and the latter with sin nt, we should have for the value of K and the equation determining the position of the ridge of wave at any time t would be Q=constant. But the general solution of the equation in this form does not appear practicable. (351.) There are two limited solutions (and perhaps others) which may be easily shown to satisfy the equation. The first may be interpreted partially; the second completely. Solution The solution of this equation is the following, in which the letter S is put to denote the definite integral between expressing the limits 0 and 7: annular waves. Solution P(r)=C.S, cos (mr cos v) + C.S, {cos (mr cos v).log (r sin3 v)} where v is a new variable, introduced solely for the purpose of forming a function which is to be integrated, and disappearing entirely from the result, which is the sum of two integrals between definite limits. But the values of the two definite integrals cannot be expressed by means of any usually tabulated quantities, and must be computed numerically. (A table of the values of the first integral, to a small extent, will be found in the Philosophical Magazine for January, 1841, page 7.) Putting S' and S" for the two integrals, corresponding to a given value of r, the most general form for W or (r) will be E.S'. cos (nt+F) +E". S". sin (nt+F"), E', E", F', and F", being arbitrary constants. It is evident that this form of W expresses a series of circular waves converging to or diverging from the point whose co-ordinates are a, b. " (352.) The equation determining W will also be satisfied by the sum of any number of functions 4, (T), (r), &c., where r, =√{(x-a,)2 + (z—b,)2}, r„= √↓ {(x—a„)2 + (z—b,,)2}, &c., and where each of the functions p' (r) PP &c. satisfies the equation m2 † (r)+ +p" (r)=0. That is, there may be any number of systems of such circular waves, each system converging to or diverging from an arbitrary centre. (353.) 2d. Let W=A.cos (ax+bz): on substituting we obtain m2-a-b2=0 as the only condition. The expressing same holds if we assume W'À.sin (ax+bz): combining the former as factor with cos nt and the latter with parallel sin nt, we find for the elevation of any part of the water waves. nt±nt±3T open &c., that is m m origin of co-ordinates we draw a perpendicular upon one of the ridges, its length is found to be IV. The positions of the ridges of waves at the time t are determined by making ax+bz=nt, or =nt+37, or heory of=nt±5, &c. The ridges, therefore, are all parallel to the line whose equation is ax+bz=0. If from the nt ± T √a2+b2' , or &c. The distance, therefore, from one ridge to the next is 2π : and the ve m Tides and (354.) The equation determining W will also be satisfied by the sum of any number of expressions A, cos (nt-ax-b), A, cos (nt-a,,x-b,,), &c., provided that a+b=m2: a,,"+b,,"=m, &c. Each of these denotes a series of parallel waves with the interval between one wave and the next, the waves being parallel to 2π n any arbitrary line. And the circumstance of the equation being satisfied by the algebraic sum of the different solutions indicates that the elevation of the water at the intersection of any ridges will be the algebraic sum of the elevations corresponding to each ridge. The same remark applies to the sum of the solutions representing circular waves, or to the sum of any number of solutions of both these kinds or of any other kinds. (355.) Now suppose the water to be terminated on one side by a straight boundary: let the co-ordinates be SO Reflexion taken that the boundary may be parallel to z; let the corresponding value of r be c; then, whatever be the of parallel value of z while a=c, the motion of the particles of water in the direction of a must at all times be 0. For, all waves from a straight the particles which are once in contact with the boundary, that is, all those for which =c, must remain in boundary. contact with the boundary; that is, they must always have r=c; and, therefore, X must =0. It is plain that this condition cannot be satisfied if we confine the expression for the elevation to the single term A. sin (ar+bz), U'———a A. cos (ax+bz); and the complete value of X=U.cos nt + U'.sin nt = m2 sin (nt-ax-bz); which is not generally =0 when r=c. But it may be made to satisfy the required which =0 whatever be the values of r and t. Thus we find that the existence of one series of waves and the assumption of a rectilinear boundary imply the existence of another system of waves, whose elevation will be represented by substituting in the expression for K the additional terms of W and W', and will therefore be Waves. - Sect. IV Tides and This expression, examined in the same manner as before, represents a series of parallel waves, in which the Tides and equation to the ridge of every one is ar+bz=nt-2ac±, &c., and which are all parallel to the line whose Waves. equation is ax+bz 0. The ridges of the former waves were found to be parallel to the line whose equation is ax + bz=0. Hence the inclinations of the ridges of the two sets of waves to the boundary are equal, but they are inclined opposite ways. This is the mathematical explanation of the reflexion of waves from a straight War. boundary. (356.) The whole elevation Theory of Subsea 1 Wares of the waves is Reflexion of waves of any kind from a The coefficient, therefore, for the undulation at contact with the boundary, is twice as great as that of an unreflected wave. (357.) In the same manner, if we take the expression for W in its most general state, putting it in the form straight equally well, and which in the expression for X adds the new term +' (2c-x, z) to the former term boundary. —p′(x, z), the sum of which is 0 when r=c. And, as above, the expression for K is ——— — Form of broad chan near the sides. which, when x=c, becomes wave. 2 m m · (ɛmk — ɛ ̃mk).Þ (c, z), or is double that at the same point in an unreflected The additional term for W being (c+c−x, z), and the original term being ø (c+x-c, z), it is evident that the system of waves represented by one expression depends on x-c, in the same manner in which the other depends on c-r, and is, therefore, a reflected system whose form is exactly similar to the form in which the original system would have proceeded if not stopped by the boundary. (358.) Leaving for the present the consideration of the motion of the waves as determined by the differential equations, we shall consider one case in which we seem to derive some assistance from general reasoning. (359.) Suppose that a tide-wave is travelling along a canal of large dimensions, and of variable depth in its the crest of cross section, the depth diminishing gradually to both shores. (We may suppose the dimensions to be such as the wave in those of the English Channel, or any similar arm of the sea.) It is evident that the investigation of (218.) nels which does not apply here: for, on account of the shallowness of the water at the sides, the velocity of flow are shallow towards both sides to produce the elevation of water there must be comparable with, perhaps equal to, the velocity of flow at mid-channel in the direction of the canal's length. Moreover, as the slope of the bottom is exceedingly small, the waves in every part of the channel will be travelling in nearly the same manner as if the extent of sea of the same depth were infinitely great, and will therefore travel with the velocity due to that depth and, therefore, the ridge of wave cannot possibly stretch transversely to the channel, and travel along with uniform velocity lengthways of the channel. The state of things, then, will be this: the central part of the wave will advance rapidly (171.) along the middle of the channel; the lateral parts will not advance so rapidly; and the whole ridge will assume a curved shape, its convex side preceding. When this form is once acquired, it may perhaps proceed with little alteration; for if, as in figure 18, we suppose two such curves exactly similar, but one a little in advance of the other, the space which separates the wings of the two curves, measured perpendicularly to the curves, (the direction in which that part of the wave must really travel,) is much less than the space which separates the centres of the curves, and by proper inclination may be less in any proportion; and, therefore, may represent exactly the space travelled over by the wave at that depth while the wave at the greater depth travels over the greater space. That part of the ridge of the wave which is nearest to the coast will, therefore, assume a position nearly parallel to the line of coast. (360.) Now the wave whose ridge is nearly parallel to the coast, or which advances almost directly towards the coast, will be a wave of the same character as that treated of in (307.). For the slope of the beach adds to the surface of the sea a very insignificant quantity, as compared with the breadth of the tide-wave, and the general effect is the same as if a perpendicular cliff terminated the sea on that side. Therefore, for those parts of the sea which are near to the coasts the law of (307.) holds; namely, the greatest horizontal displacement of the particles occurs at the same time as the greatest vertical displacement; and, therefore, when the sea is rising, the water is, for some distance from the coast, flowing towards the coast, and when it is falling, the water is flowing from the coast. (361.) In mid-channel, the motion of the water will be such as is described in (184.), &c.; that is, the water will be flowing most rapidly up the channel at the time of high water, and its motion upwards will cease when the water has dropped to its mean height. (362.) From this there follows a curious consequence with regard to the currents at an intermediate distance from the shore, where the effects of these two motions may be conceived to be combined. At high water the water is not flowing to or from the shore, but is flowing up the channel. When the water has dropped to its mean elevation, the water is ebbing from the shore, but is stationary with regard to motion up or down the channel. |