707 



SURFACE WAVES FROM AN DNDERHATKR EXPL06I0II 



Consider a surface wave generated by an underwater explosion. ^' The effect of the 

 shock wave is assumed to be negligible so that the sole source is the pulsating gas globe, 

 which has em instantaneous volume V(t) at the time t. This gas globe, assumed to be 

 spherically symmetric, as a first approximation, will migrate under the influence of gravity 

 and of the neighboring surfaces. Let Z£(t) designate its instantaneous position above the 

 bottom. It is required to find the Tiow for infinite water of uniform depth h above a rigid 

 bottom. 



Let P be a point (of. Fig. 1) with cylindrical polar coordinates r, 6, z. Then the 

 conservation of mass requires that the divergence of the material velocity q(r, 0, z, t) 

 of an inviscid, incompressible fluid with uniform density a vanish everywhere except 

 at the source (equation of continuity), I.e., 



FiR, ; 



If the flow is further assumed to be irrotational. 



then the velocity q(r, 8, z, t) is the negative T 



gradient of a velocity potential ^ (r, 8, z, t) I 



and Laplace's equation results from the above I T<^!^ — ^ ,^ 



one, i.e., V^ i> = 



(1 



In addition, the function 5 must satisfy two boundary conditions. At the fixed bottom 

 there is no component of the velocity normal to the surface, i.e.. 



If .0 , ... 



At the free surface the pressure p is uniformly atmospheric. The effect of this condition 

 is seen by a consideration of the Eulerian equation of motion, viz., 



where ( ^^ ) is the gravitational potential and TlX. /L, 5,Z,t) is the instantaneous 

 fluid pressure at a point. Integrating with respect to the coordinates and neglecting the 



square of the speed q , we obtain 



% 



at ^ /^ 



(3a 



