Carrier 



longer, smaller distortions of the wave will be apparent at Xq. 

 Figures la, lb, and ic Illustrate the quantitative aspects of the 

 foregoing statement. In the notation of those figures, 



2 2 



L = 4q'H . 



When the depth of the water Is 3 miles and Xo= 3000 miles, the 

 three cases shown represent ground motions whose half widths are 

 0, 19 and 33 miles. Figure Id indicates the wave which ensues 

 when the ground displacement is given by 



F = Fj^(x,t) - F^(x + 20, t) 



with a = 10. That is, the ground motion has a dlpole character 

 rather than a general subsidence or elevation. The persistent lore 

 that the second or third crest of the Tsunami penetrates more than 

 the first makes It interesting to speculate (in view of Figs. 1) that 

 many initiating ground motions may be of dlpole form. 



III. RUN-UP ON A PLANE BEACH 



When the wave encounters a sloping shelf along which the 

 water depth generally goes to zero, the wave steepens and becomes 

 greater in amplitude. Accordingly one no longer can rely on a linear 

 theory. However, the shelves of real interest are such that the 

 distance along the wave trajectory above such a shelf is short enough 

 so that dispersion in this region is not of any real importance. 



There is a non-iinear, non-dlsperslve shallow water theory 

 which leads to tractable problems when the depth of the basin is 

 linear in one horizontal coordinate and when the entire phenomenon 

 is independent of the other. Thus, we can regard the results re- 

 ferred to in Section 2 as the input information for a study in which we 

 ask how such waves climb up a sloping shelf. The analysis which 

 accompanies such a study Involves only the solution of a linear 

 equation whose interpretation in the non-linear context is explicit 

 and accurate. 



The result of Interest is the ratio of the run-up, r\Q, (the 

 vertical distance above sea level to which water encroaches) to the 

 wave height, r\ , at the edge of the shelf. One interesting result is 

 this: 



For a = 



^ ^ Ae-'^^x^H)-'/^ 



