158 Long Waves in Canals and Standing Waves in Closed Basins 



and the equation of motion 



d 2 v 



h 2 [-ev+l = 0. 



dz 2 



A solution of the latter equation, when the boundary conditions are fulfilled, is 



1 (cose 1 ! Hz I h) 



1 fcosf 1/2 (r//z) ) 



>& = -{ br*- 1 }' 



e { cose 1 ' 2 ) 



whereas the former gives an equation for £ in the form 



tan W 2 = £ i/2+ ae 5 ' 2 . (VI. 43) 



To each root of this equation e s corresponds a ?' s (z), and the general solution of the two equations 

 (VI. 40), when A s are constant coefficients, is 



xr] = cosjcc £ As s e~ ( - e s v l h ^ t , 

 s 



— 2 u = srnxx ^ A s v s (z)e-( £ i vlh2)t . (VI. 44) 



Most important is the determination of the roots e s of equation (VI. 43), if a is given, which 

 Proudman had done in an elegant manner. First, it can be shown that, if a is considered as a func- 

 tion of e, this equation has a minimum for the values 



£ i/2 = 1-1122 and a = 0-5370. For values a < 0-5370 



there are no real values smaller than \n, but there are two conjugated complex roots. The real 

 values can be computed approximately from \nr (r = 3, 5, 7, ....), except if a is very small, from 

 e 1 ! 2 = \nr— 6 in which 



1/0 = Jra + a(rj7r) 5 

 (r= 3,5,7,....). 



For the complex roots when a < 0-5370 we put e 1 ' 2 = £+n? and a = a+ ib, where £,rj,a 

 and b are real and we then derive 



(t-£-ntT)+i(T-n+sm 



a = 



(l-hjT)(g+irj) B 

 in which 



/ = tan | and T = tanh»? . 



We now must find these correlated values for | and r\, for which b = 0. The numerical determi- 

 nation occurs in such a way that, with an arbitrary value of t], I is being varied until the relation 

 for a gives b = 0. Table 20 gives such correlated values up to I = n, according to Proudman and 

 Doodson; an extension to | = 100 was made by Lettau (1934, p. 13). As e s = (I 2 — r) 2 )+2$r)i 

 and if 



A s - p+iq 



we obtain as a contribution to xy: 

 R cos xx ■ exp 



R = -2 V '(p 2 + q 2 ) 

 tany = qjp 



(£ 2 - rf) - 2 • cos I 2£/? - - y } . (VI. 45) 



