Explosion-Generated Water Waves 



where T|Q(ro) is the initial deformation. Noting that 



r^ = r^ + r^f - 2rr. cos (G. - 0) (10) 



the Bessel function Jo(kr) may be rewritten according to Graf's 

 addition theorem as 



JodcT) = Jo(kr)Jo(kro) + 2 ^^ J„(kr) J^lkr^) cos n{% - 9). (11) 



n>l 



Integration with respect to 6^ from zero to Ztt deletes the sum- 

 mation so that 



Ti(r,t) = j k cos cotJo(kr) 1^ r\^{r^)Z^{-kx^)r^ dr^j dk. (12) 



The same result was obtained previously by Kranzer and Keller 

 [ 1959] using integrcd transforms; in the literature dealing with radial 

 dispersive waves, Eq. (12) is generally referred to as the Kranzer- 

 Keller solution. 



Equation (12) is a double integral solution that can be con- 

 siderably simplified by additional approximations. In particular, 

 for large r and t, Jo(kr) may be replaced by an asymptotic cosine 

 function, and the resulting integral approximated by the method of 

 stationary phase (Stoker [ 1965]), to obtain 



where ^ 





00 



^o(ro)Jo(kro)ro dro (14) 



is the zero-order Hankel transform of the initial elevation Tlo(ro) a-^^d 



V(k) = -^ Vk tanh k + ^-^ • ^ (15) 



2 cosh k Vk tanh k 



Is the wave group velocity, and \ is the particular value of k for a 

 given r and t found from: 



79 



