684 - * - 



gt^/R = 10 

 or 



R = 3.2t^ (22) 



R = 6.Ut = 3.58S^''^ 



It is perhaps worth pointing out that the three equations just given were obtained on the 

 assumption that the waves are i nf ini tesinel. The first wave for R < 3, will certainly be very large 

 in practice, and may in fact break in this region. Consequently R will be greater than 3.2t . 



At R > 3a, the amplitude of the first wave decreases steadily like fT and a* lesser values of 

 R, the amplituae should be a more slowly varying function of R. Equations (22) should become more and 

 more nearly independent of the finite and even breaking character of the early stages of the wave as time 

 increases and the wave expands. 



The Second and Higher k'aves . 



For simplicity, let us consider the case where the formula (l9) applies, and R is much greater 

 than a. Take co-ordinates (x y) with the x axis joining the origin to the point R. Then in the 

 integration in (la) the dependence of Ct; on y may be nt^glected. Hence 



i {».i) =— tTT r P i(x)(i-iii) 3in 9*^ (1-^) dx. 



2 ^ rrp R -a R ur « 



where 



l(x) = 2 f^U^ - x^) |(r) dy (23) 



We have assumed gt /liR is large compared with unity, and that a/R is small. Clearly, if the 

 product of these two quantities is also large, there will be much cancelling of positive and negative 

 regions in the integration and the value of the integral will be small. Hence, as R decreases, the value 

 of the integral decreases, but the factor P~ increases. The largest value of ^ will occur at the smallest 

 value of R consistent with a minimum of cancelling in the integral. Neglecting the variation with x of 

 all the slowly varying factors, and thus retaining only the sir* terra, we see that the maximum values of 

 C, (ft, t) occurs when 



ii = (4 n - 3) ^, n > 1 (2U) 



UR "^ 



Hence the velocity of thf nth crest at any inctant is 



V = gt/77 (u n - 3) (25) 



The nth crest is the greatest crest of all when in addition to (22), tne following condition also holds 



gt^ 3 , V (26) 



UR " ^ 



Thus, counting wave crests from the outside inwards, the nth crest is at position 



R = gt^/2 (U n - 3), n > 1 (27) 



While the nth will be the greatest of all crests when it is at 



R = (« n - 3) a ^ 



t = (* n - 3) vTilr a/g)J (28) 



This demonstration is not completely satisfactory because it has yot to be shown that for a given 

 value of time, say t,, the factor R~ in (21) does not increase as R gets snaller, more rapidly than the 

 integral decreases. writing the coefficient of the sine term in the integral as P (x) and transforming 

 successively by p;rts, wc have for example after two reductions 



4 



