426 SURFACE WAVES. [CHAP. IX 



results for a rectangular or a circular tank. The values of k t and 

 the forms of the free surface, in the various fundamental modes, 

 are the same as in Arts. 186, 187*, but the amplitude of the 

 oscillation now diminishes with increasing depth below the surface, 

 according to the law (5); whilst the speed of any particular mode 

 is given by (8). 



When kh is small, we have a^ k^gh, as in the Arts, re- 

 ferred to. 



237. The number of cases of motion with a variable depth, of 

 which the solution has been obtained, is very small. 



1. We may notice, first, the two-dimensional oscillations of water across 

 a channel whose section consists of two straight lines inclined at 45 to the 

 vertical t. 



The axes of y, z being respectively horizontal and vertical, in the plane of 

 the cross-section, we assume 



<t> + fy = A (cosh k(y + iz) + cosk(y+iz)} .......(i), 



the time-factor <***"') being understood. This gives 



(f> = A (cosh ky cos kz + cos ky cosh kz\ 1 

 ty = A (sinh ky sin kz - sin ky sinh kz) } ' " 



The latter formula shews at once that the lines y=z constitute the 

 stream-line ^ = 0, and may therefore be taken as fixed boundaries. 



The condition to be satisfied at the free surface is, as in Art. 216, 



<r*<l>=gd(t>/dz..... (iii). 



Substituting from (ii) we find, if h denote the height of the surface above the 

 origin, 



a 2 (cosh ky cos kh + cos ky cosh kh} =gk(- cosh ky sin kh + cos ky sinh kh). 

 This will be satisfied for all values of y, provided 



<r 2 cos kh= gk sin M, ) 

 o- 2 cosh kh =gk sinh kh ) 



whence tanh kh= tanA (v). 



This determines the admissible values of k ; the corresponding values of 

 a- are then given by either of the equations (iv). 



Since (ii) makes <f> an even function of y, the oscillations which it represents 

 are symmetrical with respect to the medial plane y = 0. 



* It may be remarked that either of the two modes figured on pp. 308, 309 may 

 be easily excited by properly-timed horizontal agitation of a tumbler containing 

 water. 



t Kirchhoff, " Ueber stehende Schwingungen einer schweren Fliissigkeit," BerL 

 Monatsber.. May 15, 1879; Ges. Abh., p. 428. Greenhill, I. c. ante p. 388. 



