ON THE THEORY OF LONG WAVES AND BOEES.
[Proceedings of the Royal Society, A, Vol. xc. pp. 324—328, 1914.]
IN the theory of long waves in two dimensions, which we may suppose be reduced to a " steady " motion, it is assumed that the length is so greal proportion to the depth of the water that the velocity in a vertical direct can be neglected, and that the horizontal velocity is uniform across o: section of the canal. This, it should be observed, is perfectly distinct ft any supposition as to the height of the wave. If I be the undisfurl depth, and h the elevation of the water at any point of the wave, WQ, u velocities corresponding to l,l + h respectively, we have, as the equation continuity,
•; — ~
l + h
By the principles of hydrodynamics, the increase of pressure due to retardat will be
On the other hand, the loss of pressure (at the surface) due to height will gph ; and therefore the total gain of pressure over the undisturbed parts is
If, now, the ratio h/l be very small, the coefficient of h becomes
g), ................................. (4;
and we conclude that the condition of a free surface is satisfied, provic uf = gl. This determines the rate of flow ua, in order that a station* wave may be possible, and gives, of course, at the same time the velocity a wave in still water.ars to show that the simple motion is unstable, and we ought to be able to derive this result from theory. But even if we omit viscosity altogether, it does not appear possible to prove instability a priori, at least so long as we regard the walls as mathematically plane. We must confess that at the present we are unable to give a satisfactory account of skin-friction, in order to overcome which millions of horse-power are expended in our ships. Even in the older subjects there are plenty of problems left!ethod would be an accurate one. for the comparison of viscosities. The change in the ratio of head to suction, is rather slow, and the measurement is usually somewhat prejudiced by unsteadiness in the. suction manometer. Possibly better results would be obtained in more elaborate observations by several persons, the head and suction being.recorded separately and referred to a time scale so as to facilitate interpolation. But as they stand the results suffice for my purpose, showing, directly and conclusively the influence of viscosity as compensating a, change'.in otKe velocity. , .ies are small in comparison with that of sound.