Long Waves in Canals and Standing Waves in Closed Basins 241 

 can be derived from these relations, in which formula 



—MS)- 



Here we have assumed c = c(T, h) and h = h{x). For an observer travelling 

 at a velocity V, the wave period remains constant. It is to be noticed that 

 the bottom slope h{x) does not enter in this formula. V has not only the 

 character, but is identical with the group velocity when the depth is constant. 

 The wave period of any system of waves in which the crests retain their 

 identity must satisfy the "equation of continuity" (VI. 150). A formal solution 

 of (VI. 150) is: 



X 



T=T(t-jydx\, (VI. 151) 



' 



in which the limits of integration remain undetermined because no provisions 



were made to fix the co-ordinate system. It seems convenient to put x = 



for / = 0, which designates the time and the position of the initial wave 



generation; the period at this point would then be T = T(0). 



The application of this solution to the tsunami requires the determination 



of its wave velocity. The velocity of propagation for an infinitely long wave 



train is 



gX Ink 



c 2 =^-tanh-^- . 

 2tc cT 



This equation was originally derived for a wave train of constant period, 

 but is also applicable to a wave system of gradually varying period. When 

 h < 0-054, namely for very shallow water, this equation is reduced to the 

 Laplacian equation. 



d=gh. (VI. 151a) 



As the tsunamis fulfil the condition h < 0-05 A, the Laplacian relation has 

 been successfully applied for computing travel times. This, however, does 

 not explain the secondary phenomenon of the increase of the period, for 

 which (VI. 151a) is no longer a satisfactory approximation. The first two 

 terms in the expansion of a hyperbolic tangent are tanha = a— £a 2 , and 

 to the same degree of approximation (VI. 151a) becomes 



c 2 = cl(l-a 2 ) with «=^, (VI. 1516) 



With (VI. 150) one obtains the interesting relation c — V = 2(c — c), i.e. for 

 any given wave period the group velocity is smaller than the wave veloc- 

 ity by twice the amount that the wave velocity is smaller than the 



16 



