1908. | Groups of Waves in Dispersive Media, ete. 412 
where « = x—ct, and represents distance in advance of the present position 
of the impulse. We proceed to obtain now the important regular features of 
the disturbance represented by these integrals. 
With the notation of (19) and (20) we have in the first integral 
fe) =e-V = e-V(g/0), 
# (if (e)} = 4/90) 
Hence the required predominant value of «, which corresponds to a stationary 
argument, is given by 
ain LS 8 pees ae 
¢ an/2 oo ee (44) 
Thus the first integral in (43) gives 
Lorigi le cos wie + tn} du. (45) 
o(a+eu)! 4(s+cu) * 
We choose again the principal groups of oscillations by the condition 
a gut 5 } S —— 
Ph (amt 47 = 0, oF Cu= 2a. 
Now wu must be positive to come within the range of the integral (45); 
hence if w is positive we obtain no contribution towards a regular undulatory 
disturbance. If m is negative we obtain a series of travelling waves which 
we can evaluate from (45). 
We have 
d? 
du? 
Hence, using expression (18), we obtain the value of the chief group from (45), 
_ ge Syeee h = —9 
{; ae} De? when cu 2u. 
namely, 
a) sin 2 
G? ca (46) 
which holds when a is negative. 
As regards the second integral in (43), we easily see by taking the principal 
group in « that o+cw must be negative: thus « must be negative and cu 
between zero and w numerically. Then taking the chief group in u, we have 
cw equal to 2m numerically. Hence there is no resulting group of waves 
falling in the range, and the second integral contributes nothing to the regular 
disturbance. 
We have then the well-known result that in front of the travelling impulse 
there is no regular disturbance, while in the rear there is a train of regular 
waves, proportional to (46), with wave-length suitable to the velocity c. 
15 
