248 Proceedings of the Royal Society of Edinburgh. [Sess. 
during all the time when it is comparable in magnitude with its maximum 
value is therefore given with sufficient accuracy by the equation 
_1 ,_1 1 n +1 
C. -rjTT Xto 2 f -£.X n 7T ) /OAX 
£=P — € » COS < B— - + - > . . . (20). 
a A- ( ~ 4 ) 
t-in t n 
§ 9. In the case of waves in deep water, we have V = g*k~ h , or A = g* and 
n—— J; these give P == and B=— and the equation to the water 
surface becomes 
i / . jO\ i f 1^) \ 
• • • (21),* 
t 7 r 2 (qt 2 y J™. j at 2 t r ) 
*75(fe)‘ ix2 cos 1 ix — 4 } • 
which is in agreement with the result given by Professor Burnside in his 
paper “On Deep-water Waves resulting from a Limited Original Disturb- 
ance,” where it is obtained by an entirely different method. 
For flexural waves in a thin elastic rod, we have V = icbk. or A = kI> and 
n = 1 ; P = (47t/c 6)”% and B = (4/c6) _1 . The shape of the rod is then given by 
y a 
'€“2^6 1 COS 
f a; 2 7r \ 
\ 4 kU 4 i 
2 ( K bty a 
For capillary waves we have V = or A = T J and n 
and B = (^T) _1 . The water surface is given by 
(22). 
*; p=(|Tx)-‘, 
2^2 4 X 2 a 
-€~9Tt2 COS 
/ 4a; 3 t r ( 
1 27TT 2 " 4 J 
(23). 
* 3D r W ( 27D 2 
§ 10. We can now find by means of equation (20) the time at which the 
amplitude of the disturbance at any chosen point x has its maximum value. 
The amplitude is given by the right-hand member of (20), with the cosine 
term omitted: 
= P JL f)”Ba 
1 
ft* 
• (24). 
By differentiating this with respect to t, we find that the time of 
maximum disturbance at x is given by the equation 
,t J 2(n + l)Ba 
and the amount of this greatest amplitude is then given by 
1 
_ 7T2 
2 at ( nx)i 
(25), 
(26), 
* After this paper had been completed I found this result given in a very comprehen- 
sive paper by Dr T. H. Havelock, “The Propagation of Groups of Waves in Dispersive 
Media,' 5 in Proc. Boy. Soc ., lxxxi., 1908, obtained by a similar application of Lord Kelvin’s 
1887 result to that given in §§ 6, 7 above. The view of group-velocity given in my paper, 
Proc. R.S.E., 1909, is there fully discussed and illustrated. 
