- 9 - 687 



Assuming that the dimensions of the initial airturbance are h, then the surface elevation at any 

 point it from the centre at tint t is 



i (R.t) - / f (r) t. Ut) dS (34) 



the integration being over tht circle of radium r = h, while the initial elevation at radius r is f (r). 

 Th.j convention of sign chosen implies that f (r) is negative near the origin; a- is the distance: from the 

 point R to the- element dS of the circle of radius h defining .in initial crater. 



The expanding wave system from the crater is led by ^ trough, the bottom of which moves according to 



gt^/R =5 or R = 6.U t^ (35) 



The nth crest moves according to 



R = 9t^/« (2n - 1) (3«) 



so that the velocity of the nth crest is 



R = gt/2 (2n - l) 



^ (37) 



= / {g R/7T (2n - l)} 



Hence the velocity of any crest or trough increasss indefinitely with time. 



The nth crest is the greatest of al! criSt: or troughs, and is at its jn.atcSt size when 



R = i (2n - 1) h (38) 



The velocity of the crc-st at this instant is 



V = /"(Tgh/70 (39) 



The group velocity, i.e., the vrlocity of the region in which the waves art greatest is just half 

 this quantity, namely - 



v = / (gh/2 77) (no) 

 The ratio of the greatest height of the mth crest to the greatest height of the nth crest is 



H^/H^ = (2n- l)/(2m- l) n > m > 1 (ui) 

 The motion of the pth trough is given by 



R = gt/8 T7 p (ttZ) 

 and the trough is at its greatest size when it is at 



R ' u p.^. (U3) 



Since the length h defining the crater dimensions from a charge W Is practically equal to the 

 distance "a" over whFch the impulse from a surface explosion of w extends, there will be a strong 

 resemblance between the wave systems tn the two cases. The main point of difference will be in the wave 

 heights. 



Cal culation of ''^ave Heights from an Underuater Exjlosi o n . 



The wave system caused by an underwater explosion is to be regarded as the interference pattern of 

 the waves produced by a dow and those (.roduced by a hollow or bubble. In view of the difficulties of 

 scaling accurately from very small chsrges to those of two thousand tons, we have nartc a direct attempt 

 to estimate the waves from a large explosion. In this case, the time scale of the waves is great enough 



to permit 



