160 162.] WAVES IN TWO DIMENSIONS. 195 



where the & s are constants yet to be determined, and &amp;lt;f&amp;gt; does 

 not contain z. Substituting in (37) we find 



gk (4* - &amp;lt;T**) f - 0, 



or (}&amp;gt; = &amp;lt;f&amp;gt; l cos kct + fasin kct .................. (39), 



where c is given by 



_ 

 &quot; 



and (^ 2 are functions of a?, ?/ only, to be determined from 

 (35) which now gives 



with a similar equation for &amp;lt;/&amp;gt; 2 . The solution of (40), when 

 adapted to suit the conditions to be satisfied at the surfaces, if 

 any, which limit the fluid horizontally, gives the possible values 

 of k. By adding together terms of the form (38), in which Jc 

 has the values thus found, we may build up a solution satisfying 

 any arbitrary initial conditions. 



Oscillations in a Rectangular Tank. 



162. We apply the method just explained to the case where 

 the fluid is contained in a rectangular tank whose sides are verti 

 cal. Let the origin be taken in one corner of the tank, and the 

 axes of x, y along two of its sides, and let the equations of the 

 other two sides be x = a, y = b, respectively. The functions &amp;lt; 1} fa 

 must now satisfy the conditions 



p = 0, when x = 0, and when x = a, 

 ax 



and 



- l = 0, when y = 0, and when y = b. 

 ay 



The general value of &amp;lt;/&amp;gt; : subject to these conditions is given by the 

 double Fourier s series 



niTrx niru 



cos cos - T 



a b 



132 



