243 
1909-10.] On Waves in a Dispersive Medium. 
where a is a constant, has been chosen by Professor Schuster in the above 
paper, and also by Professor Burnside * for deep-water waves ; so that it is 
convenient, for the sake of comparison with their results, to begin by 
considering this form of initial displacement in the general case where 
n> — 1. Keeping, however, in the meantime to any initial form given by 
g=f(x), we proceed first to show that in all cases included in the formula 
V — there is no resolution of the original disturbance by the 
medium depending on the wave-length of the constituent Fourier trains, as 
is understood by dispersion. 
§ 3. The problem to be solved is : — to find the displacement, £ at any 
point x, at time t, having been given the initial displacement of the medium, 
g=f(x), with the additional initial condition ^=0: and the solution, 
obtained by the regular Fourier synthesis, is 
OO +00 
£ = — J dk j dx^fipc^) cos k(x - x j) cos kY t . . . (2). 
0 -00 
This fulfils the initial conditions, provided 
oo +oo 
f(x) = — jdk Jdx 1 jf(x 1 ) cos k(x - aq) . . . . (3), 
0 -00 
which we know to be the case by Fourier’s theorem. If, following 
Professor Schuster, we now take V = a-\-bk~ 1 , and define \jx(x) by the 
equation 
_00 +00 
ij/(x) = — jdkjdx 1 f(x^) sin k(x - aq) . .... (4), 
0 -00 
we easily obtain, by expressing the product of the cosines in (2) as the sum 
of two cosine terms and two sine terms, 
£ = jt{f(x + at) +f(x - at)} cos bt - ^{if/(x + at) - ij/(x - at)} sin bt . (5). 
This equation shows that the displacement at point x at time t may be 
regarded as due to two initial disturbances each of which moves along 
without change of form at the constant velocity a ; half the total energy of 
each disturbance going in the positive direction, and half in the negative. 
If a and b are both finite, the original form of the disturbance reappears at 
regular intervals, 2 tt/ 6, displaced in each from its former position by a 
distance ^ira/h. When a is zero the above equation reduces to 
£ =f(x) cos bt . 
* Proc. Lon. Math. Soc., t. xx., 1888. 
( 6 ), 
