1908. | Groups of Waves in Dispersive Media, etc. 406 
as unlimited, this indicates oscillations succeeding each other with continually 
increasing frequency and amplitude; also if we follow a group of waves with 
given value of « the amplitude varies inversely as #4, or inversely as the . 
square root of x.* 
§ 6. Initial Displacement of Finite Breadth. 
If 7 is the range within which the initial displacement is sensible, the 
previous results hold with //a small; further, as Cauchy showed, gt l | 4a? 
must be small if the function ¢(«) of (26) is to be taken as constant. 
Prof. Burnsidef has obtained approximate equations for the surface form due 
to certain limited initial displacements not confined to an indefinitely 
narrow strip. From the present point of view, such results can be recovered 
simply by selecting from the integrals the more important groups of waves. 
(a) Let the initial displacement be given by 
Sf (&) = ca /(a? +27), (30) 
where « may be supposed small. 
Then o()=| PO des = eae, 
Hence from (25) the surface form is 
n = tee | e-** cos x (w«— Vt) de+4ca | e~** cos « (a+ Vt) dk. (31) 
0 0 
For points at some distance from the range in which the original 
disturbance was sensible, e~* varies slowly compared with the cosine term ; 
thus we may consider the integrals as made up of simple groups. For « 
positive we need only consider the first integral. 
The predominant value of « is thus connected with w and ¢ by the same 
equation (28) as before. Since the greater amplitudes are associated with 
the smaller values of « and these have the greater values of U, it is clear 
that, at a particular point, the disturbance dies away from its maximum at a 
slower rate than its growth up to it. Using the previous results we can 
write down the disturbance involved in the main group form as 
ta 
n = cart ee ¢ cos (E-2) : (32) 
The following results can be deduced. The cosine term varies rapidly 
compared with the other factors, hence we may obtain the maximum by 
considering the latter alone ; it is easily seen that this occurs when 
a[t = »/(29). 
* Lamb, ‘Proc. Lond. Math. Soc.’ (2), vol. 2, p. 371 (1904). 
+ W. Burnside, ‘Proc. Lond. Math. Soc.,’ vol. 20, p. 22 (1888). 
