GROWING WAVES DUE TO TRANSVERSE VELOCITIES 117 



internal fields to the external ones. For y > ijo 



V = Voe-'^'-e~^" (2.16) 



and 



Similarlv 



^ + i37 = at 2/ = 2/0 

 dy 



dV 



— - ^V = at y = -7/0 (2.17) 



dy 



The most familiar procedure now would be to look for solutions of 

 (2,15) of the form, e''^. This would give the sextic for c 



(c' - /3')[(WiV + nY + a;/(niV - n')] = (2.18) 



with the roots c = ±|8, ±ci , ±C2 , let us s^y. We could then express \p 

 as a linear combination of these six solutions and adjust the coefficients 

 to satisfy the six boundary equations. In this way a characteristic equa- 

 tion for l3 would be obtained. From the S3anmetry of the problem this 

 has the general form F(l3, Ci) = F(i3, C2), where Ci and Co are found from 

 ; (2.18). The discussion of the problem in these terms is rather laborious 

 and, if we are concerned mainly with examining qualitatively the onset 

 of increasing waves, another approach serves better. 



From the symmetry of the equations and of the boundary conditions 

 we see that there are solutions for \p (and consequently for V and p) 

 which are even in y and again some which are odd in y. Consider first the 

 even solutions. We will assume that there is an even function, ^i(y), 

 periodic in y with period 2yo , which coincides with \l/(y) in the open 

 interval, —yo<y<yo and that \pi(:y) has a Fourier cosine series repre- 

 sentation : 



hiy) = E c„ cos \ny X„ = — n = 0, 1, 2, • • • (2.19) 

 1 yo 



yp inside the interval satisfies (2.15), so we assume that ypiiy) obeys 

 (D^ - ^')[{u,'D' + ^'f + o.,\u,'D' - ^-)^, 



+00 



(2.20) 



= Z) 5(2/ - 2m + lyo) 



where 6 is the familiar 5-function. Since D\p and D^\p are required to vanish 

 at the ends of the interval and \l/, D'^ and Z)V are even it follows that all 



