407 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
Thus the maximum is propagated out with uniform velocity; and we see 
that in its neighbourhood the predominant wave-length is 4crz. 
(>) Let the initial displacement have a constant value A over a range of 
breadth 2c, and be zero at all other points; then we have 
$ («) =o SID KC 
K 
Hence the surface elevation is 
oc . 
4 sin KC 
co. 
sin Ke 
0 K 
Qos cosk(@+Vt)d«. (33) 
T 
cos x (x— Vt) eee | 
0 T 
With the same argument as before, we consider the value of 7 at a point 
as due to the most important of a succession of simple groups, that one, 
namely, for which the argument is there stationary so that the components 
reinforce over a considerable range of «; and we can write down, from the 
previous results, an expression for this group which is valid at least in the 
vicinity of the travelling maxima of the disturbance. We have 
2 2 
n= = (all sing cos (Fi), (34) 
corresponding to Burnside’s result in the paper already cited. 
Here we have a succession of maxima given by those of (a/g¢t?)t sin (gt?/42?), 
that is, at times given by tan @ = 26, where 0 = gi?c/ 42”. 
The period of the group that is thus cumulative is different for different 
localities, and for different times at the same locality; but the accumulation 
is very prominent only for those times and localities which give a maximum 
value to the amplitude, which has been graphed for the next example in 
fig. 1. 
The maxima here diminish continually in value, and are propagated each 
with uniform velocity, namely, the group-velocity corresponding to the 
predominant wave-length in the neighbourhood. 
§ 7. Limited Train of Simple Oscillations. 
Another interesting example is the case of an initial displacement consisting 
of a limited length of simple harmonic oscillations. If f(x) is symmetrical 
with respect to the origin, and is zero except for a range of (2m + 4) wave- 
lengths within which it is A cos «’x, we have 
(22 +4) 17/n! AL y 
p(x) = 2 | A cosk’ coska da = 2«’A Gos osreD aye (35) 
0 
79 > 
Ke — KP 
Hence, from (25), we have the surface elevation 7, of which we write down 
10 
