430 SURFACE WAVES. [CHAP. IX 



If the axis of x be parallel to the length, and the origin be 

 taken in one of the ends, the velocity potential in any one of the 

 fundamental modes referred to may, by Fourier s theorem, be 

 supposed expressed in the form 



$ = (P + p l cos kx + P 2 cos 2kx + ... 



+ P g cos skx + . . .) cos (o-t + e) ......... (1), 



where k = ir/l, if I denote the length of the compartment. The 

 coefficients P s are here functions of y, z. If the axis of z be drawn 

 vertically upwards, and that of y be therefore horizontal and 

 transverse to the canal, the forms of these functions, and the 

 admissible values of &amp;lt;r, are to be determined from the equation 

 of continuity 



V = ............................. (2), 



with the conditions that 



d&amp;lt;f&amp;gt;ldn = ........................... (3) 



at the sides, and that 



......................... (4) 



at the free surface. Since dfyjdx must vanish for x = and x = Z, 

 it follows from known principles* that each term in (1) must 

 satisfy the conditions (2), (3), (4) independently ; viz. we must 

 have 



^ 2P -0 (5) 



with dP K /dn = ........................... (6) 



at the lateral boundary, and 



a*P 8 = gdP./dz ......................... (7) 



at the free surface. 



The term P gives purely transverse oscillations such as have 

 been discussed in Art. 237. Any other term P s cos skx gives a 

 series of fundamental modes with s nodal lines transverse to the 

 canal, and 0, 1, 2, 3,... nodal lines parallel to the length. 



It will be sufficient for our purpose to consider the term 

 P! cos kx. It is evident that the assumption 



&amp;lt; = P x cos kx . cos (at + e) .................. (8), 



with a proper form of Pj and the corresponding value of cr deter- 



* See Stokes, &quot;On the Critical Values of the Sums of Periodic Series,&quot; Camb. 

 Trans., t. viii. (1847) ; Math, and Phys. Paper*, t. i., p. 236, 



