444 SURFACE WAVES. [CHAP. IX 



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

 (p'p)x y and the difference of the tensions parallel to y on 

 the two edges gives S^drj/dx). We thus get the equation 



(5), 



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 



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



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

 planes parallel to xy, at unit distance apart, is 



T=& fiV^l d*-y f [V^r-l *" ..... ( { )- 

 2 J o L % Jr=o ; o L fy Jy=o 



If we assume 17 = a cos Tex ....................................... (ii), 



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

 conditions, 



<l>=-k- l a*picn*kx, <j>'=k- l ae-*vcQskx ............... (iii), 



we find T=k(p + p'}Jc-itf.\ ........................... (iv). 



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



Substituting from (ii), this gives 



.X ................................. (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 cc cos (o-tf + e), where a- is determined by (6), we verify 

 that the total energy T+ V is constant. 



Conversely, if we assume that 



rj 2 (a cos kx+ft sin kx) ........................... ( vii), 



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

 squares of d, # and a, /3, 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. 



