1914] ON THE THEOEY OF LONG WAVES AND BORES 251
Unless A2 can be neglected, it is impossible to satisfy the condition of a free surface for a stationary long wave — which is the same as saying that it is impossible for a long wave of finite height to be propagated in still water without change of type.
Although a constant gravity is not adequate to compensate the changes of pressure due to acceleration and retardation in a long wave of finite height, it is evident that complete compensation is attainable if gravity be made a suitable function of height ; and it is worth while to enquire what the law of force must be in order that long waves of unlimited height may travel with type unchanged. If/ be the force at height h, the condition of constant surface pressure is
..................... (5)
II 2 fj Tl /2
i f <*n t" v " ,n*
whence /_ _ _ . ^ ^^. u,' ^^, .................. (6)
which shows that the force must vary inversely as the cube of the distance from the bottom of the canal. Under this law the waves may be of any height, and they will be propagated unchanged with the velocity V(/iO» where /i is the force at the undisturbed level *.
It may be remarked that we are concerned only with the values of / at water-levels which actually occur. A change in / below the lowest water-level would have no effect upon the motion, and thus no difficulty arises from the law of inverse cube making the force infinite at the bottom of the canal.
When a wave is limited in length, we may speak of its velocity relatively to the undisturbed water lying beyond it on the two sides, and it is implied that the uniform levels on the two sides are the same. But the theory of long waves is not thus limited, and we may apply it to the case where the uniform levels on the two sides of the variable region are different, as, for example, to bores. This is a problem which I considered briefly on a former occasionf, when it appeared that the condition of conservation of energy could not be satisfied with a constant gravity. But in the calculation of the loss of energy a term was omitted, rendering the result erroneous, although the general conclusions 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.
* Phil. Mag. Vol. i. p. 257 (1876); Scientific Papers, Vol. i. p. 264.
t Roy. Soc. Proc. A, Vol. LXXXI. p. 448 (1908) ; Scientific Papers, Vol. v. p. 495.
{ Compare Scientific Papers, Vol. i. p. 258 (1899).