405 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
ment given by 7 = cosx«z, without initial velocity, the surface form at any 
subsequent time is given by 
n = cos« Vi cos xx = 4 {cos « (w—Vt) +cos x (w+ Vt)}, 
where V = (g/x)® (24) 
Let f(z) be any even function of « which can be analysed by Fourier’s 
integral theorem. Then, corresponding to an initial surface displacement f(z), 
without initial velocity, there is a surface form given at any subsequent 
time by 
mes e| $ (ic) cos x (e—Vt) de +— | $(«) cos (a+Vé) de, (25) 
47 0 0 
ar 
where $ (x) = {7 (@) cos kw dow. (26) 
If we suppose the initial elevation to be limited practically to a line 
through the origin and assume that | J (@) dz = 1, so that ¢(«) = 1, we can 
—_—oO 
use, as an illustration of the procedure, the form 
n= Za | cos (e— VE) det 5 | cos x (2+ Vt) dk. | (27) 
4/04 0 ATT 0 
We select from these integrals the groups which give the chief regular 
features at large distances from the original disturbance. This cumulative 
group from the first integral is given for a given position and time by the 
value of « for which « (c—Vf) is stationary, where V = ,/(g/«), so that 
Lo Oi J 8; 
i ee ra 
and, similarly; from the second integral by 
Thus there are symmetrical groups of waves proceeding in the two directions 
from the origin; for z positive we need only consider the first integral in 
(27), and for x negative the second integral. Thus the predominant wave- 
length at a point « at time ¢ is given by 
K = gt? 4a. (28) 
Evaluating this predominant group by means of expression (21), we obtain 
the known result 
3 2 
i ~$! eos (7-17). (29) 
~ Qarhad 4x * 
At a given position, far enough from the source for the train to be taken 
