228 Long Waves in Canals and Standing Waves in Closed Basins 



Another solution corresponds to the free oscillations in the transverse direction of the canal, 

 which can be expressed in the form 



rj = Z(x)cosot , u= U(x)s'mat , v= V(x)cosot 



By substituting into (VI. 138) we obtain 



— ag dZ —fg dZ 



U = — and V = — ^L _ 



o 2 -f 2 dx a 2 -f 2 dx 



d 2 Z o--P 



+ — Z=0, (VI. 140) 



dx 2 gh 



dZ 

 with the boundary condition — = for x = and 2a. 



dx 



The general solution of the last differential equation gives 



S7l 



Z„ = y4 v cos — x, 

 2a 



in which A s is a free constant and s is an integer. 

 For the frequency of the oscillation s we find: 



ol=p+—gh. (VI. 141) 



Aa 2 



The result of a sudden pressure disturbance r\ = Z s (x) starting at a time / = 0, can be known 

 by combining a disturbance of the equilibrium for r\ = Z s (x), a steady solution Z(x) — — (f 2 /a 2 )Z 4 (jr) 

 and a periodical solution with an adequately selected amplitude. It is shown that the level r) in the 

 transverse direction of the canal executes an oscillating motion varying from zero to2(l— f 2 ja])Z s {x) 

 and high water will occur at a time T = njo s . 



A general solution of the problems will be found by expanding the pressure disturbance rj into 

 a Fourier series, according to cos sn j '2a. If we assume that the atmospheric disturbance consists 

 in the appearance of a linear pressure gradient across the canal with intensity 2H valid for entire 

 width 2a, we get as the principal oscillation on the banks of the canal 



t) = — i/(l--|(l-cos(T 1 r), (VI. 142) 



in which a, is given by (VI. 141) with s = 1. 



The period of the s -nodal transverse oscillation of the canal with the 

 earth at rest is T r = 4ajs] gh and T t = 2rc//= 12h/sin<?9 = * pendulum day 

 being the period of the inertia oscillations in the latitude <p; we obtain from 

 (VI. 141) for the period of the forced transverse oscillations in the canal 



T T 



(VI. 143) 



i [i+(r,/r r ) 2 ] V[i+{TrlTi)F\ 



For narrow and deep canals, T r is generally much smaller than the inertia 

 period 7), so that in a first approximation T = T r , i.e. the effect of the earth's 

 rotation is small and can be neglected in a first approximation. However, 

 the natural period for extensive water-masses with the earth at rest, T r , is 



