254 ON THE THEORY OF LONG WAVES AND BORES [385
rv ft [l
fdy = fdy- fdy=p0-p, Jo Jo • y
p0 denoting the pressure at the bottom, so that the potential energy is
The difference of potential energies, corresponding to the fifth and sixth terms of (19), is thus
f • 1 I"1 1 (l> 1
lu]p0-p0'-7 I pdy + r, p'dy\ ................ (23)
(. <> Jo >> Jo )
The integrals in (23) compensate those of (22), and we have finally as the loss of energy
It should be remarked that it is only for values of y between I and I' that /is effectively involved.
In the special case where /=/x2/~3, equations (16), (21) give
uaZa=/*, tt'a?a = /A, ........................ (25)
the introduction of which into (24) shows that, in this case, the loss of energy vanishes ; all the conditions can be satisfied, even though there be no dissipation. The reversed motion is then equally admissible.
The formation of bores is illustrated by a very ordinary observation, probably not often thought of in this connection. Something of the kind may usually be seen whenever a stream of water from a tap strikes a horizontal surface [or when water from a can is poured into a flat bath]. The experiment is best made by directing a vertically falling stream into a flat and shallow dish from which the water overflows*. The effective depth may be varied by holding a glass plate in a horizontal position under the water surface. Where the jet strikes, it expands into a thin sheet which diverges for a certain distance, and this distance diminishes as the natural depth of the water over the plate is made greater. The circular boundary where the transition from a small to a greater depth takes place constitutes a bore on a small scale. The flow may be made two-dimensional by limiting it with two battens held in contact with the glass. I have not attempted measures. On the smallest scale surface-tension doubtless plays a considerable part, but this maybe minimised by increasing the stream, and correspondingly the depth of the water over the plate, so far as may be convenient.
* The tap that I employed gives a jet whose diameter is 6 mm. A. much larger tap may need to be fitted with a special nozzle. — May 14, .nclusions are not affected. The error became apparent in applying the method to the case above considered of a gravity varying as the inverse cube of the depth. But, before proceeding to the calculation of energy, it may be well to give the generalised form of the relation between velocity and height which must be satisfied in a progressive wave}, whether or not the type be permanent.