401 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
expression for the total disturbance, attending only to its prominent features 
and neglecting the rest, provided we assume the change of the amplitude 
factor @(«) to be gradual. On this hypothesis the resulting expression 
contains the amplitude of the component trains simply as a factor; and in 
this way the trains for which it is a maximum show predominantly in the 
formula, which exhibits the main features of the disturbance as they arise 
from place to place through cumulation of synchronous component trains. 
The argument shows that in some respects the integral (6) may be more 
conveniently regarded as a collection of travelling groups instead of simple 
wave-trains ; when ¢(«) is a slowly varying function, the groups will be 
simple groups of the type (3). The limitations within which this is the case 
will appear from the subsequent analysis; one method of procedure would be 
graphical: to take a graph of the fluctuating factor and see that the other 
factor, which is taken constant, does not vary much within the range that is 
important for the integral. 
In the cases we shall examine, the effect is due to a limited initial 
disturbance, and the salient features are due to the circumstance that ¢ («) 
has well-defined maxima; thus the prominent part of the effect can be 
expressed in the form of simple groups belonging to the neighbourhood of the 
maxima. 
Before considering in detail special cases with assigned forms of the velocity 
function V, two illustrations of interest may be mentioned. 
(a) Damped harmonic wave-train—lIf f (x) is a function satisfying the 
conditions for the Fourier transformation, we have 
f (2) =| a AO) eae =a. 
TJ0 —o 
For an even function of z, this gives 
i(@) = =| $(x)cosxadk, where (x) = | J (@) cos kw do. (7) 
T 10 0 
Now let f(x) be an even function of a, defined for all values, and such 
that it is equal to e~“* cos «’a for w positive ; then we find 
(8) 
26(«) = 2 sat —e 
#6) = Eee ea ee) 
Consider this function f(«) as the initial value of a disturbance y which 
occurs in a dispersive medium; then the value of y at any time can be 
expressed, in general, by 
vie [¢ (x) cos « (2— Vt) dk +B [ $ (x) cos x (a+ Vt) dk, (9) 
