254 Mr. J. McCowan on the 



velocity parallel to the length of the channel, great in com- 

 parison with the other two components of the displacement 

 and velocity. It is clear that this imposes a certain limitation 

 on the form of the cross section of the channel : near the 

 surface the inclination of the sides to the vertical must not be 

 so great as to occasion too great a transverse motion, as the 

 surface rises and falls between wave-crest and hollow. In 

 obtaining (4) the exact assumption made is that the vertical 

 component of the acceleration of any particle is very small 

 in comparison with the free acceleration g due to gravity. 



As an example, suppose that the form of the surface is 

 given by a curve of sines or similar curve. Then it is clear 

 that, to satisfy the conditions, the height of the waves, and 

 the variation in the breadth of the channel between wave-crest 

 and trough, must be very small in comparison with the length 

 of the wave ; and it may also be seen without much difficulty 

 that the ratio of the wave-height to the wave-length must be 

 large compared with the square of the ratio of the depth to 

 the wave-length. 



2. The Propagation of Waves in one Direction only : 

 General Solution. 



The most important problems in wave-motion are those in 

 which waves are propagated in one direction only. When 

 this limitation is imposed, the complete solution of the problem 

 of the propagation of long waves may be obtained as follows : — 



Let « and ft be functions of z, as yet undetermined ; 

 assume for a unidirectional solution, 



This gives 



and therefore further 



dt dt dx dx* 

 which will become identical with (5), if we take 



/3= (^|)VW' and «*£=^(*), 



which gives 



The double sign has only reference to the direction of pro- 

 pagation of the waves, and we shall therefore obtain sufficient 



z=¥ (x 



-at). 



dz 

 Jt~~~ 



dz 

 *Tx> 



