170 Long Waves in Canals and Standing Waves in Closed Basins 



W + W) A ~° (VI.58) 



with 



. _4ji 2 a 2 

 X --gTf 



Equation (VI.58) is to be solved with the boundary conditions ^4(0) and 

 ^4(1) =0. This differential equation, according to the said theorem of the 

 calculus of variation, is equivalent to the relation 

 i 



/ {(ar)' - 5® *} * - -w - Minimum • (VI - 59) 



o 



We now assume a solution A in the form of a finite series 



A = A ip +A 1 ip 1 +A 2 y> 2 + ...+A m ip m , (VI.60) 



in which the m + 1 functions ip t must satisfy the same boundary conditions 

 as A itself: y,-(0) = y,-(l) = for i = 0, 1, 2, ..., m. If we choose yt t (z) 

 — z{\— z)z' then these boundary conditions are fulfilled and A becomes 



m 



A = ^z(l-z)z i A i . (VI. 61) 



1=0 



The determination of the unknown A t results from the fact that after sub- 

 stituting (VI. 61) into (VI. 59) J {A) must be a minimum. This requires that 



8J n 8J a ♦ dJ n 



After a few calculations, we finally obtain for determining A t the equation 



m 



yi( _1 2(i+ 1)0 + 1) (/+2)Q + 2) 



Zj\\i+j+\ z+y + 2 i+j+3 



i=0 



C zH\ — z¥? i+J \ 

 A zyi z)z dz [ A (VI. 62) 



The integral can only be evaluated, if a(z) can be represented as an analytical function of z. In 

 order to avoid singularities at the end of the lake, Hidaka puts 



a(z) = hzQ.-z)<p(z), 



in which h is a constant with the dimensions (L 3 ) and q>{z) represents a function of z which is always 

 positive in the interval ^ z ^ 1. <p(z) must now be approximated as well as possible from the 

 normal curve by a suitable analytical function. Hidaka has done this for the Lake of Yamanaka 

 and then evaluated the integral. If the normal curve is irregular, such a function cannot easily 

 be found ; in that case it is better, as Neumann has done in the case of the Baltic, to compute the 

 integral directly by numerical integration, in which, of course, the limiting values at z = and 

 z = 1 in (VI. 62) require special attention. 



