164 Long Waves in Canals and Standing Waves in Closed Basins 



(ii) The Japanese Method. Investigating the oscillations in several bays 

 along the Japanese coast, Honda, Terada, Ioshida and Isitani (1908, p. 1, 

 see also Defant, 1911, p. 119) have developed a method to determine the 

 period of the free oscillation of the water-masses of an irregularly shaped 

 basin, which method is based essentially on Rayleigh's theory of air oscill- 

 ations in a tube of variable cross-section. Here is a brief outline of the method: 

 if the kinetic and potential energy of the oscillating system can be expressed 

 by quadratic functions of a co-ordinate (p, in such a way that 



T = \a{^\ and V = hc<? 



then the Lagrangian equations of motion give the equation of oscillation 

 for the system Lamb (1932, p. 251). 



a(d 2 (p/dt 2 ) + c<p = 



and the period of free oscillation of the system simply results from 



f 27t\ 2 _ c_ 



TJ ~a' 



when <p = Acos(wt—e). Therefore if both T and V can be expressed by 

 quadratic functions the period of oscillation of the system can be determined 

 without directly deriving the equations of motion. 



8X 

 Considering an oblong basin, we put S(x)^ = X, and as, according to (VI. 47), b(x)$ = — — - 



ox 



then 



1 } 1 (BXV gQ / 1 lcX\ , 



- p I - — I dx and V = — j - — ) dx 



Now if we assume for X an expression of the same form as is obtained if S were a constant, 

 then 



v-i tin 



X = 2j ^/i sin — x cos io n t . 



If we select as a new co-ordinate <p n = a n coso) n t, then we get after a long computation 



n » ' n m 



and 



y= 2 {C n +AC n }<p 2 „+ XS 4Cn,m'»<Pn-<Pm 

 n n m 



in which 



p r \ tin 1 n 2 n 2 gQ r \ nn 



A n +AA n = | J - sin 2 — xdx and C„+ AC n = - — ^— J - cos 2 — xdx . 



o 



The summation has to be done over all n # m. When the lake approaches a rectangular basis of 

 constant cross-section AA„ and AC n become small. If both S(x) and b(x) do not vary too rapidly 



