Damping of Long Waves in a Rectangular Trough. 155 



Equating the sum of these terms to the mass-acceleration, we 

 get 



~du dp Cb 2 u ~d 2 u\ , 



To obtain the equation of: continuity, consider the rate at 

 which fluid is entering the space bounded by ,r, x + S.r, and 

 equate it to the rate at which the fluid in this space is in- 

 creasing. Hence 



J„ 3* If 



(4) 



Differentiating (4) with respect to as and combining with 

 (2), we obtain 



s v c k y» , ... 



31^ = ""J, ^ ch {0) 



Differentiating (3) with respect to t and substituting from (5) y 

 we obtain the final equation, writing v for /m/p, 



w- g ) Q w d *- v *Aw 2+ ^r ' ' (G) 



This of course reduces to the ordinary equation for long 

 waves if v = () and we assume 'd' 2 u/'d.r to be taken as inde- 

 pendent of z. 



§ 2. Let us now apply equation (G) to the case of water 

 contained in a rectangular trough, the ends .of which are 

 given by .i' = 0, #=a, the sides by ?/ = 0, y=h, and the bottom 

 by = 0. Then, in order that we may apply (G), f and "dtfdt 

 must initially be functions of a; alone. Let the initial con- 

 ditions be 



*=0. 



The boundary conditions are 



u = for .r = 0, ,v = a, y = 0, y = (>, ~ = 0, 



and 9w/9s=0 for .: = /<, 



the latter condition expressing the fact that there is no tan- 

 gential stress on the free surface. 



