that the water surface has an initial displacement equal to the seabed 

 displacement, that the initial displacement is not time-dependent, and 

 then propagate the initial displacement outward from the generating area 

 using long-wave equations (Brandsma, Divoky, and Hwang, 1975). Other 

 means of establishing the initial waves, of varying degrees of complexity, 

 are described by Wilson, Webb, and Hendrickson (1962) and in other sources 



Many of the mathematical representations of waves generated from 

 bottom uplifting are based on circular source regions; however, Levy and 

 Keller (1961) present one solution in terms of elliptic coordinates for 

 a source region which is more elongated than circular. This solution has 

 the form 



n(r, 0, 



t) = r" 1 A (k) 



o 



where 



2ttC, 



dC^ 

 dK 



1/2 



ik(r-at)+i{v/h) j~ Q -j 



(22) 



k 1/2 cosh[k(Z + d)] 

 A (k) = 2 e -iTr/k 



(2tt) 3/2 cosh(kd) 



(23) 



and 



I(k, G) 



U) a 2 E (-i)4sefe, cos e) E M 

 W n=o I n \2 I n,ne\2 j 



* So„ fe , cos e) f m] 



n \2 ) n,no\ 2 J\ 



(24) 



The terms Se n and So^ are even and odd Mathieu functions, and 

 ^n 3 ne (ka/2) and Fn 3 no (ka/2) are even and odd Mathieu transforms. The 

 variables are defined as n the wave height, r and 6 the coordinates 

 of a point in polar coordinates, t the time, C the wave celerity, k 

 the wave number, C^ the group velocity, 1 Q the depth of generation 

 (negative downward and equal to -d for bottom uplifting), d the water 

 depth, and a the interfocal distance of the coordinate ellipses. 



Levy and Keller indicate that the velocity of the bottom uplifting is 

 unimportant if the time of uplifting is small in comparison with the period 

 of the generated waves. This is generally true for tsunamis. They also 

 indicate that only the first few terms of I(k,9) may be important in the 

 solution, although a computer solution can sum a relatively large number 

 of terms. The limitations on the solution are that the solution was 

 derived for water of uniform depth, the initial wavelength (or wave period) 

 must be known, and the solution assumes that r is much greater than the 

 dimension (diameter or length) of the source. 



37 



