Lord Rayleigh on Waves. 263 



der this conclusion probable. Should it be correct, the analy- 

 tical character of the solitary wave remains to be discovered." 

 The theory of the solitary wave has been considered by Earn- 

 shaw (Camb. Trans, vol. viii.), who, distrusting what he calls 

 analytical approximations, bases his calculation on a supposed 

 result of experiment, namely that the horizontal velocity is uni- 

 form over each section. This, as we have seen, is the fundamental 

 assumption in the theory of long waves ; but when the length 

 of the wave is moderate, such a state of things is impossible in 

 a frictionless fluid which has been once at rest ; for it involves 

 molecular rotation. In fact if there be a velocity-potential, 

 the horizontal velocity u satisfies Laplace's equation 



d 2 u cPu 

 dx 2 dy 



i + X3 =0, 



and therefore cannot be a function of x without being also a 

 function of y. The motion investigated by Earnshaw has 

 therefore molecular rotation ; and the rotation remains con- 

 stant for each particle ; otherwise the equations of fluid motion 

 would not be satisfied. This is the explanation of the difficulty 

 with which Earnshaw meets, — that while the necessary condi- 

 tions are satisfied in the wave itself, there is discontinuity in 

 passing from the wave to the undisturbed water. The discon- 

 tinuity arises from the fact that, as there is no rotation outside 

 the wave, it is necessary to suppose finite rotations imparted to 

 the particles as the wave reaches and leaves them. It is evi- 

 dent that, except in the case of very long waves, u must be 

 treated as a function of y as well as of x. 



In considering the theory of long waves (reduced to rest by 

 imparting an opposite motion to the water), we saw that it was 

 impossible to satisfy the condition of a free surface if the height 

 of the wave were finite. It occurred to me to inquire whether 

 there might not be compensation in certain cases between the 

 variation of pressure at the upper surface due to a finiteness of 

 height, and the variation due to a departure from the law of 

 uniform horizontal velocity proper to very long waves. It was 

 conceivable that the surface-condition in the case of a wave of 

 given finite height might be better satisfied by a moderate than 

 by a very great wave-length. In this way I have obtained 

 what seems to be a perfectly satisfactory approximate theory 

 of the solitary wave. 



If u and v be the horizontal and vertical velocities in a 

 stream moving in two dimensions without molecular rotation, 

 and cf>, ty the potential and stream functions, we have 



