683 



If & is large, asymptotic formulae for G and I, are 

 Ga [0) = (t?/Uv/2) sin (^/u) 

 £. = ,1/2, 



n p CO " oj 



(20) 



It is not obvious how good an approximation the asymptotic formula is, and in fact one would not 

 expect even reasonable agreement for the values of d which are needed to sketch the first two or three 

 waves. Therefore, the function G {6] has been computed, and the values compared with GA {6). The 

 values of G (6) are given in the table below, wh-^re the accuracy is to the last figure given, and a 

 comparison of G with Ga ts shown in Figure 2. It appears that the asymptotic formula gives a surprisingly 

 good representation of the function over the whole range, except for values of 6 less than about «. Hence 

 the asymptotic formulas can be used with accuracy at least 10 - 15J over the whole range of c?, except for 

 5 ^ u, which represents the leading parts of the disturbance, a trough extending to Infinity, 



The Functions 0{9) and H(g) . 



Hain Projierties of the ''lave Syatcm . 



It appears from (l8) that it ts not an allowable approximation to assume that the impulse from the 

 surface explosion Is concentrated at a point. If this were so, the wave amplitude at any point would 

 Increase indefinitely with time. 



The wave height at R, t is given by 

 5 (R. t) = / I (r) C M d S, 



(21) 



where 4 is given by (l8) or (l9) as the case may be, the integration is over the circle of radius a over 

 which the impulse is delivered, and oi is the distance of the surface element d S to the point R. 



2'Ae First Vave . 



From Figure 2, it is seen that the first wave corresponds with a point "^ 10. Provided the 

 maximum and minimum distances of the point R to the circle correspond at time t with 4 < ^ < 14, the surface 

 elevation will be positive, and tf the distance to the centre of the circle corresponds with 5 = 10, then 

 the surface elevation will be at its greatest value in the first wave. Hence for a fixed value of R such 

 that R > 3a, the position of the first crest is given by the first noaximum of i^G [0) namely - 



9t' 



