432 SURFACE WAVES. [CHAP. IX 



A general formula for these edge- waves has been given by Stokes*. 

 Taking the origin in one edge, the axis of z vertically upwards, and that of y 

 transverse to the canal, and treating the breadth as relatively infinite, the 

 formula in question is 



= m-^ s 0- ssin 0>cos(.*-cO (i), 



where j9 is the slope of the bank to the horizontal, and 



-$***) (") 



The reader will have no difficulty in verifying this result. 



240. We proceed to the consideration of some special cases. 

 We shall treat the question as one of standing waves in an 

 infinitely long canal, or in a compartment bounded by two 

 transverse partitions whose distance apart is a multiple of half the 

 arbitrary wave-length (27T/&), but the investigations can be easily 

 modified as above so as to apply to progressive waves, and we 

 shall occasionally state results in terms of the wave- velocity. 



1. The solution for the case of a rectangular section, with horizontal bed 

 and vertical sides, could be written down at once from the results of Arts. 

 186, 236. The nodal lines are transverse and longitudinal, except in the case 

 of a coincidence in period between two distinct modes, when more complex 

 forms are possible. This will happen, for instance, in the case of a square 

 tank. 



2. In the case of a canal whose section consists of two straight lines in- 

 clined at 45 to the vertical we have, first, the type discovered by Kellandf ; 

 viz. if the axis of x coincide with the bottom line of the canal, 



<f> = A cosh -^ cosh -jy cos kx . cos (at + c ) (i). 



This evidently satisfies v 2 < = 0, and makes 



for y=z, respectively. The surface-condition (7) then gives 



^z. Z.A 



I (iii), 



where h is the height of the free surface above the bottom line. If we put 

 <r = c, the wave-velocity c is given by 



where =2?r/X, if X be the wave-length. 



* "Report on Recent Researches in Hydrodynamics," Brit. Asa. Rep., 1846; 

 Math, and Phys. Papers, t. i., p. 167. 



t " On Waves," Trans. JR. S, Edin., t. xiv. (1839), 



