rents everywhere towards the east, these currents being zero at e and w, and reaching a maximum at m. Three lunar hours later, i.e., at r i o'clock, there would be high water everywhere between m and e and low water everywhere between m and w. At 2 o'clock the conditions of 8 o'clock would be reversed, while at 5 o'clock the conditions of i r o'clock would be reversed. The range of tide would increase steadily from zero at m to maxima at e and w, that at e being greater than that at w on account of the convergence of the Channel. Next let us treat the case in which there is no elevation along w so that this line becomes a node. With the barrier at e there would then be no tides at all, but now let us remove this barrier so that the tides will be entirely maintained by the actual currents in the Strait of Dover. Again they will consist of a longitudinal standing oscilla tion but as illustrated in fig 3, 6 hour and 9 hour (two connected half see saws). At 6 o'clock the surface will be level, and the currents will be directed inwards towards m from both sides, being strongest in the Strait of Dover. Three lunar hours later, i.e., at 9 o'clock, there would be high water all over the Channel between e and w, the elevation reaching a maximum near in and being zero at w and near e. At 12 o'clock the con ditions of 6 o'clock would be reversed, while at 3 o'clock the conditions of 9 o'clock would be reversed. This time the line m would be near a loop and e near a node. Now superpose these two standing oscillations. We have the M2 constituent of the Channel-tides as it would be in the ab sence of the dynamical effects of the earth's rotation and the force of fric tion. The effect of the earth's rotation is to cause an acceleration to the left of the current and proportional to the speed of the current, and when the current does not rotate this means that there must be a downward surface-gradient to the left of the current and proportional to the speed of the current. On allowing for this effect we arrive at the six diagrams of fig. 4, the letter H indicating that the water is above mean level and the letter L that the water is below mean level. The lines drawn across the diagrams of fig. 4 separate the regions of II from those of L, and are thus the contour lines at mean level. Three lunar hours after the times indi where K=n/ I (gh), as is easily verified. If Ka is less than ir the eleva tion will everywhere have the same phase as at the open end, and the inward current will have a phase which is 17 in advance; also, the am plitude of the elevation will increase from the sea to the closed end, while the amplitude of the current will decrease from the sea to the end. These
are the conditions of a short deep gulf.
If Ka= Fir the formulae for ?- and u becomes infinite and we have the phenomenon of resonance. These conditions illustrate to some extent the very large tides of the Bay of Fundy, but of course there are many respects in which any actual conditions deviate from those underlying the above equations. For instance, the tides at the mouth of a gulf are not themselves entirely independent of the dimensions of the gulf. The case of resonance corresponds to the coincidence of the nodal line with the place at which a definite amplitude of elevation is prescribed.
When the depth varies as the distance from the closed end we may take h=h0x/a, where is the depth at the open end. On eliminating u be tween the two equations (2) we have— where ) denotes Bessel's function of zero order. The profile of the elevation for a particular time is a wave-form of decreasing amplitude, the greatest height being at x = o, and the succeeding minima and maxima being —0.40, and 0.30 of this greatest value. The effect of de creasing depth towards the head of a gulf is thus to produce an increase in the range of tide at the head. The effect of decreasing breadth is of the same kind and more pronounced, but even when taken together the resulting multiplication is by a very moderate factor. The primary cause of the pronounced amplification observed in some gulfs and es tuaries is resonance.