24 Lord Kelvin on the Growth of a Train of Waves. 



(greatly exaggerated in respect to slopes of course), and may 

 be shortly named water-surface curves. Their ordinates 

 have been calculated by an analytical method described in 

 § 151 below. They cannot be calculated continuously for 

 successive values of x by the method of continuous quad- 

 ratures ; if that were the method employed, the value of the 

 ordinate for each value of x would need to be calculated by 



an independent quadrature (I dq\ from to the particular 



value of t for which the water-surface is represented by the 

 curve. The values of t chosen for fig. 39 are respectively 

 it, (z + 1/8)t, (i-\-2/S)r, 0' + 3/8)t, 0' + 4/8)t, where i is any 

 very large integer, and t denotes 2rr/co, the period of the 

 varying surface-pressure to which the fluid motion considered 

 is due, 



In all our illustrations we have taken oi=.^/iT, which 

 makes t = 2 v /7t, and, with g= 4 as in § 105. makes the 

 wave-length \=8. 



§ 138. In figs. 3G and 37, all the curves correspond to 

 cosg>(£ — q) in the formulas. In fig. 38, all the curves 

 correspond to sin co(t — g) in the formulas. 



In fig. 39, the inscriptions of times correspond to cos co(t — q) 

 in the formulas. The same curves, with the inscriptions 

 altered to (t + 2/8)r, (• + 3/8)r, (i + 4/8) r, (t + 5/8)r, (t -f 6/8)t, 

 correspond to sin co(t — q) in the formulas'. 



§ 139. In fig. 36, representing velocity-potentials and a 

 surface displacement, none of the curves shows any perceptible 

 deviation from sinusoidality except within period 1. Towards 

 the end of period 1 the numbers found by the quadratures 

 show deviations from sinusoidality diminishing to about 1/10 

 per cent., and imperceptible in the drawings. This proves 

 that sinusoidality is exact within 1/10 per cent, through all 

 time after the end of the first period. 



It is interesting to see, in period 1, how nearly the rise 

 from the initial zero follows the same law for 8^(0, 1, t) and 

 S^(0, 1, t) : notwithstanding the vast difference in the law 

 of initiating surface-pressure, represented by (188), for these 

 two cases. In fig. 36, the initiating surface-pressure com- 

 mences suddenly at its negative maximum value, — s/2 for 

 case (f>, and — *5 for case ^, of which the former is 2*83 

 times the latter. The semi-amplitudes of the subsequent 

 variations of velocity-potential shown in the first and third 

 curves are "954 for case </> and *318 for case -^, of which the 

 former is 3'00 times the latter. 



§ 140. The first, and third, and fifth, curves of fig. 37 

 show, at a distance of one wave-length from the origin, the 



