424 SURFACE WAVES. [CHAP. IX 



approximately, where the term of the first order in rj has been omitted, in 

 virtue of (6). 



The kinetic energy, p*x> may be expressed in terms of either x or K. We 

 thus obtain the forms 



The variable part of V- T is 



n 



o 

 and that of 



It is obvious that these are both stationary for 77 = ; and that they will 

 be stationary for any infinitely small values of 77, provided x 2 =#A 3 , or 

 K?=ghl 2 . If we put x=7A, or K=Ul, this condition gives 



U*=gh (vi), 



in agreement with Art. 172. 



It appears, moreover, that rj=0 makes V+K a maximum or a minimum 

 according as U' 2 is greater or less than gh. In other words, the plane 

 form of the surface is secularly stable if, and only if, U<(gfifi. It is to be 

 remarked, however, that the dissipative forces here contemplated are of a special 

 character, viz. they affect the vertical motion of the surface, but not (directly) 

 the flow of the liquid. It is otherwise evident from Art. 172 that if pressures 

 be applied to maintain any given constant form of the surface, then if 

 U z >gh these pressures must be greatest over the elevations and least over 

 the depressions. Hence if the pressures be removed, the inequalities of the 

 surface will tend to increase. 



Standing Waves in Limited Masses of Water. 



236. The problem of wave-motion in two horizontal dimensions 

 (#, y), in the case where the depth is uniform and the fluid is 

 bounded laterally by vertical walls, can be reduced to the same 

 analytical form as in Art. 185*. 



If the origin be taken in the undisturbed surface, and if f 

 denote the elevation at time t above this level, the pressure- 



* For references to the original investigations of Poisson and Lord Kayleigh 

 see p. 310. The problem was also treated by Ostrogradsky, " Memoire sur la 

 propagation des ondes dans im bassin cylindrique," Mem. des Sav. Etrang., 

 t. iii. (1832). 



