Explosion-Generated Water Waves 



where g is the gravitational acceleration. 



Making use of Green's formula, Kajiura [ 1963] gives a solu- 

 tion of V ^ = satisfying bottom conditions (^^ = 0» z = -1) and 

 the linear free surface condition: 



#^(x,y,z;T) = l/4Tryy(G.|>„^- ^tG^J^ .^dS^ 

 S ° ° ^°" 



-l/4.yy(G*„^-*,G,J ,dS„ (1) 



s 



where S denotes the source region, initial conditions are denoted 

 by subscript zero and G is the appropriate Green's function; t is 

 a time associated with the generation period. The Green's function 

 is: 



/-»00 _ 



G(xo,yo,Zo;T|x,y,z;t) = j ^^^^ ^^ [sinhkjl - |z-Zo|j 



- sinh k jl + (z +Zo)| 



+ — 7 I 1 - cos u) (t - t) I r-rr cosh k(i +z) cosh k(l +Zq)J dk 



(2) 



where r is the horizontal distance between the source point (xQjy^) 

 ajcid the point under consideration (x,y) i.e. , 



r^ = (x - x/ + (y - Yof (3) 



and 



co^ = k tanh k. (4) 



Clearly, G is symmetric with respect to t and t, and with respect 

 to source and field points. The quantities oj and k are dimension- 

 less frequency and wave number, respectively. 



Applying the given boundary conditions and integrating Eq. (i) 

 with < T < t gives, after some calculation 



^ +P= i^.^y <F| +F2)dS, (5) 



where 



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