46 Mr. J. M c Cowan on the Solitary Wave. 



form and the same velocity of propagation as Kussell deduced 

 from his experiments. In 1876 Lord Kayleigh* gave another 

 method of approximation leading to an equation for the sur- 

 face similar to that of Boussinesq and the same velocity of 

 propagation. These theories, however, give little further 

 information regarding the wave, and I am not aware that 

 anything further has been done. 



In the following paper I propose, after briefly discussing 

 the general theory, to proceed to a somewhat detailed ex- 

 amination of the wave based on a simple but close approxima- 

 tion. It will be found that the results are in substantial 

 agreement with Scott Kussell's experiments, and confirm his 

 opinion as to the unique character of the solitary wave of 

 elevation. 



It will be seen further that an approximate account of the 

 phenomena of the breaking of waves on passing into shallower 

 water follows naturally from the results obtained. 



1. General Theory of the Wave. 



Though the possibility of the propagation of a solitary wave 

 without alteration in form and with constant velocity along 

 a straight channel of rectangular cross-section has not been 

 established on theoretical grounds, yet the result of experi- 

 ment is such as to show that a method based on this assump- 

 tion must lead at least to a highly approximate account of the 

 motion. We shall assume, then, the invariability of the wave 

 motion, understanding it of course to be two-dimensional, 

 and shall in the first place suppose it reduced to steady motion 

 by having impressed upon it a velocity equal and opposite to 

 that of the wave propagation. 



Take the axis of x in the horizontal bottom of the channel 

 along the direction in which the wave is propagated, and that 

 of z vertically upwards. Then noting that the motion is 

 essentially irrotational as being propagated into (incompres- 

 sible) liquid at rest, and putting therefore <\> for the velocity 

 potential and yfr for the current function, we must have 

 ijr+ icj) a function of z + lx. {i= s/ — 1). 



At a great distance from the wave the liquid will practically 

 be at rest, and therefore in the corresponding steady motion 

 it will be flowing uniformly : hence for the steady motion it 

 is convenient to take 



f +-t^«^U («■+««) +/(« + «»)* • • • (!) 

 where U is the velocity of propagation of the wave. 



* Phil. Mag-. April 1876. 



