444 SURFACE WAVES. [CHAP. IX 



breadth Boo. The fluid pressures on the two sides have a resultant 

 (p p)Sx, and the difference of the tensions parallel to y on 

 the two edges gives S (f^ifty/tfe). We thus get the equation 



to be satisfied when y approximately. This might have been 

 written down at once as a particular case of the general surface- 

 condition (Art. 244 (1)). Substituting in (5) from (2) and (4), we 

 find 



T le* 



&amp;lt;T 2 = ff-7 ........................... (6), 



P + P 



which determines the speed of the oscillations of wave-length 2tr/k. 



The energy of motion, per wave-length, of the fluid included between two 

 planes parallel to xy t at unit distance apart, is 



o .-0 o V .y=o 



If we assume 77 = a cos &e ....................................... (ii), 



where a depends on t only, and therefore, having regard to the kinematical 

 conditions, 



= - 1&amp;lt;r l a(*v cos lex, $ = k~ l ae~*v cos kx ............... (iii), 



we find T=(p+p )k- l &.\ ........................... (iv). 



Again, the energy of extension of the surface of separation is 



Substituting from (ii), this gives 



.\ ................................. (vi). 



To find the mean energy, of either kind, per unit length of the axis of #, 

 we must omit the factor X. 



If we assume that a &amp;lt;x cos (a-t + e), where o- is determined by (6), we verify 

 that the total energy T+ V is constant. 



Conversely, if we assume that 



T) = 2 (a cos kx -f /3 sin kx] ........................... ( vii), 



it is easily seen that the expressions for T and V will reduce to sums of 

 squares of d, /3 and a, 0, respectively, with constant coefficients, so that the 

 quantities a, /3 are normal coordinates. The general theory of Art. 165 

 then leads independently to the formula (6) for the speed. 



