681 



According to the ideas outlined above, the explosion of a charge W at less than the optimum depth 

 will produce a wave system exactly the same as does a charge of less weight for which the optimum depth 

 is just provided the charge was not so near to the surface that the explosive gases when they break 

 surface do not deliver an appreciable impulse to the surface. For this to happen, the charge depth is 

 only a few charge radii, and the explosion is then more like a surface explosion. 



The Seal ing Law s 



Referring to Lamb's Hydrodynamics (Sixth Edition, page 130) the fundamental solution of the wave 

 equation for cyl indrical ly expanding infinitesimal gravity waves in water of uniform depth d is seen to be 



sin a t cosh k (z + d) 



(kr) (5) 



a cosh kd 



^ = cos a t J^ (kr) (6) 



where (i is the velocity potential and ^ the surface elevation. The condition that the fluid velocity at 

 the free surface equals the normal velocity of the free surface, leads, in the usual way, to 



cr = g k tanh kd (7) 



Generalizing tnese results by the Fourier double integral theorem, we have that, corresponding 

 with an initial surface elevation - 



Co = '(^' ' % = ° 



d> - gf .ii!!_q_t "_sh_klzjLdl J (Kr) kdk f Fta) J (ka) a.d a (s) 

 cr cosh ka o 



4 = _f^ cos tr t .' (kr) kdk /^ f (a) J (ha) a. a a (9) 



' o 



Similarly, corresponding with an initial surface impulse 



(6 = i /" cosat <:?iLliiJ_£l J (kr) kdk /' F(a) J (ka) a.d a (lO) 



i" cosh kd '^ ° 



t, ^- ~r J^ o-sincT-t J (kr) kdk /"^ F(a) J (ka) a.d a (ll) 



->^ o 



Case I. Initial Surface Impulse . 



We consider the scaling laws in this case first because they are simplest. Imagine a comparison 

 of the wave systems from two charge weights w^ and w^ detonated at heights L, and L, above the water. 

 Write 



n = (Wj/Wj) 1/3 (^2, 



The seal ing laws are 



F^ (ajj/Fj (c'.j) = n 



^2 (nx, t /n ) = v/n l^ (x, t) (13) 



