388 



NOTE VIII. 



MOTION OF WAVES ON THE SURFACE OF WATER. 



IT appears that it is not every form of waves which can 

 be freely propagated with a uniform velocity without any 

 change of form. The only case in which this is possible, 

 is when the vertical section by a plane passing through the 

 direction of propagation is, to a first approximation, the 

 curve of sines. A solitary wave* cannot, therefore, be 

 propagated without change of form; a result confirmed by 

 Mr. Russell s experiments. If the form of the waves were 

 accurately the curve of sines, they would be exactly si 

 milar above and below the undisturbed level. This com 

 mon experience shows not to be the case. On carrying our 

 investigation to a second approximation, on the supposi 

 tion that the depth of the water is much greater than the 

 length of the wave, it is found that the altitude of the top 

 above the undisturbed level is greater than the depth of 

 the hollow beneath it. The top of the waves are more 

 pointed than the hollows. Such a wave is propagated 

 without any change in its form. If the depth were much 

 less than the length of the wave, the wave will undergo 

 certain changes, which we shall have occasion to refer to 

 presently. 



The motion of the water during the uniform propagation 

 of the wave is greatest at the surface, and decreases rapidly 

 with the depth ; but the rate of decrease varies very 

 nearly inversely as the length of the wave. Thus if the 

 depth of the water be much less than the length of a wave, 



* Camb. Phil. Trans, vol. viii. 



