688 - 10 - 



to permit the neglect of spray, broken water etc. We assume, as has already been described, that the 

 water system is released from rest at time zero from a configuration, 



3h^ 



' ''' "T'] l^^^W^ ' ~i7r7T^ 



(4U) 



Figure 1 shows the shape of such a configuration. An exact solution of this problem can be 

 obtained by the partial differentiation with respect to h, and subsequent simplification, of some formulae 

 given by Terazawa (Proc. Roy. Soc. AU2, 57, 1922). The result is- 



S^ = h^ + r^ , 7 = h/S (15) 



There would be no difficulty in computing the first two waves at the interesting times, by using 

 (U5) and seven figure tables of Legendre polynomials. Unfortunately no such tables are available to the 

 writer, but some rough calculations have been made with the aid of the four figure tables in Jahnke-Emde. 

 The leading part of the wave system is 2 trough, and when it is at about 1000 feet, is roughly JO feet 

 deep. The analysis given earlier, however, enables one to predict the wave heights within 20 or 301 and 

 this is perhaps sufficiently accurate for present purposes. By studying Figure 1 it becomes clear that 

 the waves at 1OOO-30OO feet are caused mainly by the crater at the centre. Only at very large distances, 

 and possibly at the origin, does the surface elevation outside the crater have much effect compared with the 

 waves from the crater. The volume cf the crater may be taken as 



V = / 2 7T r f(r) dr = 0.385 V (46) 



At this point, an encouraging conclusion may be reached. Up till now, it has not been clear that 

 the scaling laws proposed for an undenvater explosion would be applicable to sitbI 1 charges, because the 

 disintegrated water flung into the air does not fall back aaain for several seconds, and by this time the 

 «aves from the hole are well away from the centre, with a very large explosion, the height reached by 

 the water is only about equal to that reached by a small explosion, and most of the water has returned to 

 the surface before the wave system has had a chance to develop. However, even in the case of a large 

 explosion the big waves near the explosion are due almost entirely to the flow into the crater, which may, 

 as above, be regarded as the difference of the Ouoole volume and the volume of the dome to r ~ h. Since 

 the volume of the criter is ibout 0.« tines the volume of the bubble, we may roughly describe cur conclusions 

 by stating that tn the case of a small explosion, the big waves are caused entirely by the bubble cavity 

 and that in the case of a large explosion the big waves are caused by a cavity roughly equal to one half of 

 the bubble. Hence we expect that the sealing laws will apply over a wide range (lO : l) within 3 factor 

 of about 50*. 



Returning now to the details of the large explosion, we see from (ui) and (U2) that the pth crest 

 is the largest crest when it is at 



R = jt^/8 7T p = U.p.h. («7) 



(Notice that "troughs" and "crests" have been interchanged, since (Ul) anO (U2) refer to the 

 troughs from an initfal elevation). 



Performing the integration (3**) with about 101 accuracy, we find that the maximum wave at time t 

 is the pth wave, which is at R, where R, t -rnd p ire r;lited by (U7) and that the height of the wave is 

 approximately 



H = h/8 p (U8) 



P 



The depth of the qth trough is approximately 



H = h/it (2q - 1) (19) 



