453 
1908-9.] On Group-Velocity and Propagation of Waves. 
Using now equation (8) in this, we may write it in the form 
M*-W} = M*-W}+^{-2/W-V'(W • < 10 )- 
In general, for the evaluation of the integral in equation (1) according 
to the method used by Lord Kelvin in his paper of 1887 our assumption 
is that the dispersion is exceedingly far advanced, and t therefore so great 
that the term { 2/ (Jc 0 ) + k 0 f"(k 0 )}( is very large for the greatest and 
A 
least values of k considered. If the dispersive medium is such that equation 
(10) is satisfied for all values of k, it is unnecessary to assume t to be very 
large, as (k — k 0 ) may be taken as large as we please positively and negatively. 
In either case, therefore, when we transform equation (1) by the 
substitution 
■ • • • (ii), 
A 
where the sign taken is such as to make £ 2 positive, the limits of 0 are 
practically + co and — co , and the value of £ is given by 
+ 00 
2 i I dz cos [k Q {x - tf(k 0 ) } ± z 2 ] t 
c = ~-oo (i2) 
" 2*fi[+{2 /'W + VW}f ’ 
By means of the integrals 
^+oo ^+oo 
j dz cos z 2 = j dz sin z 2 = . . . . ( 13 ) 
-CO ”-00 
the above equation ultimately reduces to 
cos [k Q {x - (/(A'o)}] + sin W\ x ~ (/Go)] 
2ttH\ + { 2f\k Q ) + A’ 0 /"(& 0 ) }]' 
cos 
K{x - tf(k 0 )} ± 
7 T 
2 4 7T H\ + { 2 f\k Q ) + /ro/'iAto)}] 4 
( 11 ), 
which has been verified by Lord Kelvin for water-waves in his paper, and 
proved to be in agreement with Cauchy and Poisson’s result for t very 
great compared with x. It is easy to verify that the result holds for all 
values of t and x in the case of flexural waves in an elastic bar. 
§ 15. In this result the terms depending upon k 0 vary very slowly in 
the neighbourhood of x, and the £ curve is nearly a simple sine curve there. 
Equation (14) therefore proves that in the case which we are considering, 
where the dispersion is fairly well advanced, the resultant displacement curve 
near any point of the medium at any time is a curve of exactly the same 
