685 



(. ("i H' 





ISR' 



P' (x) 



-nr /^ P" (x) sin --1 

 g t, -a iR 



P(x) cos — t (1- i) 



(1-4) ax 



The residue of the first square bracket after the limits have been inserted is zero, because P 

 vanishes at the limits. The residue of the next square brack^jt is also seen to be zero by using the fact 

 that the crests occur at values of gtj/'iS given Oy equation (22) and also that the coefficient of the "even" 

 terms in P' (x) is zero, and the "odd* terms vanish by symmetry when the limits are inserted. Hence only 

 the integral remains; apart from high order effects, by which are meant residues in the first two square 

 brackets when higher order terms are retained, the surface elevation at time t, varies like 



4 ~ p\ k > 



the most prominent value of k near the ureatest wave being 2, 



The velocity of the nth wnve, at th£ epoCfi when it is the greatest wave is 



V = / (2 ga/77) (29) 



a result which is independent of n, R and t. Thus the velocity of the nth wavf increases steadily with 

 time, according to (25) and during the period when it is the greatest wave, its velocity is given by (29), 



The ratio of the wavc> height of the mthwave during the period whi;n it is the greatest wave, to 

 the wave height of the nth wave during the period when it is the greatest wave is 



(4n - 3)/(*r, - 3) m > n > 1 



(30) 



The velocity of the point of greatest wave height, i.e., the grouo velocity, is one half of v given 

 by (29) i.e. 



/ (ga/2- 



(31) 



The afiove argument is easily modified to the- discussion of troughs. Thus the pth trough is at 

 its greatest, and is greater at this instant than any other crest or trough, when 



R = (ip - 1) a 



t = (<*p - 1) / (2 7T a/g) 



The pth trough is defined a? the trough following the pth crest, i.e., the leading edge of the 

 disturbance, an extremely shallow trough extending to infinity, is neglected in counting the number of 

 the trough. 



Numerical Examples. 



Consider the explosion jf a I oz. charge on the surface. Then a is about 2 feet. Over the 

 range < R < 6 feet the first 'wive is the greatfsl. At R = 6 feet the magnitudes of the first and 

 second waves as they pass are about equal. The time for thi first wave to reach f> feet is 1.36 seconds. 

 (This may w.-;l 1 be an stiirete on the high side because the first wavo, because of its great height, will 

 travel faster than indicated by equation (2), e.nd rray in fact be a broken wave). The second wave is the 

 gr-.atest at R = 10 feet, its velocity fs then 6.U feet/cfxond and the time is 3.1U seconds. The first 

 wave is then at 32 feet. The third wavf is tne greatest at R = IS f^c-t, and its height is 5/9 of the 

 height of the second wave -.t its greatest; its velocity is then 6.1 feet/second, th.- time is 5.65 seconds, 

 the first crest is at 102 feet, and the s-cond crest is '^t 32.5 feet. The second crest is 0.095 of the 

 height of tne third crest, and the first crest is indv' tec table. 



Cons ider 



