provided p = n + n', n' being the angular velocity of the moon in her orbit, and a some constant. The equation (16) then becomes ?? dt2 = c2 +μsin 2 (pt+a)....... The solution of this equation consists of two parts. The first part, or complementary function,' is the solution of (17) with the last term omitted, and expresses the free waves which could exist independently of the moon's action. The second part of the solution, or particular integral,' gives the forced waves or tides produced by the moon, and is The tide is therefore semidiurnal, and is 'direct' or 'inverted,' i. e. there is high or low water beneath the moon, according as c is greater or less than pa. Now in the actual case of the earth. Unless therefore the depth of the canal were much greater than the actual depth of the sea, the tides would be inverted. For the case of a circular canal parallel to the equator in latitude λ, we should find Substituting in (16), and solving as before, we find for the forced If the latitude of the canal be higher than arc cos will be direct. = 155. Next let us take the case of a canal coinciding with a meridian. Let e denote the hour-angle of the moon from this meridian, pt+a, say, and x the distance of any point on the canal from the equator. By an easy application of Spherical Trigonometry, we find for the horizontal disturbing force in the direction of the length of the canal, The equation of motion is easily seen to be of the same form, (16), as before. Substituting then and solving, we find The first term represents a permanent deviation of the surface from the circular form; the equation of the mean level being now The fluctuations above and below this mean level are given by the second term of (20). If, as in the actual case of the earth, c be less than pa, there will be high water in latitudes above 45o, and low water in latitudes below 45°, when the moon is in the meridian of the canal, and vice versa when the moon is 90° from that meridian. The circumstances are all reversed when c is greater than pa. For a further development of the canal theory of the tides, the student is referred to Airy, l. c. ante. Waves in deep water. 156. When we abandon the assumption that the depth h of the fluid is small compared with the length of a wave, the Eulerian method becomes more appropriate. Let the origin be in the undisturbed surface, the axis of x horizontal, that of y vertical and its positive direction upwards. We suppose the motion to take place entirely in these two dimen sions x,y; and to be such as may have been generated from rest, so that there exists a single-valued velocity-potential . We retain, for a first approximation, the assumption that the squares and products of the velocities and relative displacements may be neglected. Our equations then are Now (21) is satisfied by the sum of any number of terms of the form ejx+ky, each multiplied by an arbitrary function of t, provided j2 + k2 = 0. Now j must in our case be wholly imaginary, for otherwise we should have infinite for either x +∞, or x = -∞. Hence аф we must have k real and j = ± ik. We write therefore = {eky (A cos kx + B sin kx) + e-ky (A'cos kx + B′ sin kx)}, the coefficients being functions of t as yet undetermined. The condition (24) gives so that the assumed value of 4 takes the form =Σ{ek (y+h) + e-k(y+h)} (P cos kx + Q sin kx)...... (25). So far our work is rigorous. If we now neglect squares and products of small quantities, (23) becomes on substitution from (22) and in this we may suppose y equated to zero, for the error in the value of y will only introduce an error of the second order in this condition. Substituting the value (25) of p, we find that in order that (26) may hold for all values of x we must have F'(t) = 0, and d'P dt2 (ekh +e-kh) +gk (ekh − e−kh) P=0.........................(27), with an equation of the same form for Q. The solution of (27) is and A, B now denote absolute constants. Combining our results we get for a series of terms of the form a{ek(y+h)+e−k(y+h)}cos k (x ± ct)............(29), where a is a constant. Let us examine the motion represented by one of these terms alone, taking, say, in the last factor the cosine with the minus sign in its argument. The form of the free surface (p= const.) is given by (22), viz. it is where in the last term we suppose y equated to zero, for the reason already given. Hence if the origin be taken at the mean level, the equation of the free surface is The wave-profile is therefore the curve of sines, and it advances without change of form in the direction of a positive, with uniform velocity c given by (28). The wave-length,' i.e. the distance between two successive crests or hollows is It appears that the velocity of propagation is not independent of the wave-length, but that it increases continuously with the value being √gh (as in Art. 148) when this ratio is in finitesimal, and gλ when it is infinite. To any given value of c there corresponds only one value of A, and vice versa. 157. Let us examine the nature of the motion of the individual particles of fluid as a system of waves of the above kind passes over them. If E, n be the component displacements at time t of the particle whose mean position is (x, y), we have We ought, in strictness, to have on the right-hand side of these equations + for x, and y+n for y, but the resulting correction x § would be of the order a which we have agreed to neglect. Integrating then the above equations on the supposition that x, y are constant, and remembering the formulæ (28), (31) for c and a, we find Each particle therefore describes an ellipse whose major axis is horizontal; the law of description being the same as for a particle attracted to a fixed point by a force varying as the distance. The ratio of the minor to the major axis of the ellipse, viz. ek (y+h) — e-k(y+h) diminishes from the surface to the bottom, where it vanishes. |
