16 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



which is evidently of the type (65), we obtain, on inserting the time-factor, 



^ J - ( 73 ). 



Q 



~ 



f* ff^- 



J , ( (2^ - F)* 



<-M 



# 



+ 



(74). 



This is for x positive ; the corresponding results for x negative would be obtained by 

 changing the sign of x in the exponentials, and reversing the sign of u . 



The solution thus found is made up of waves travelling outwards, right and left, 

 from the origin, and so satisfies all the conditions of the question. 



The first term in u gives, on each side, a train of waves travelling unchanged with 

 the velocity c~ l . The second term gives an aggregate of waves travelling with 

 velocities ranging from lr l to a"" 1 . As x is increased, this term diminishes indefinitely, 

 owing to the more and more rapid fluctuations in the value of e'H 



On the other hand, the part of v which corresponds to the first term of w remains 

 embedded in the first definite integral in (74). To disentangle it we must have 

 recourse to another treatment of the integral |^ () c/. One way of doing this is to take 

 the integral round the pair of contours shown in fig. 2, where a consistent scheme of 



-K -h -h 



f 



a, -ft -0,,-p 



a, 



Fig. 2. 



values to be attributed to the radicals ^(t* A 2 ) and v/( 2 L-) is indicated. For 

 the only parts of the left-hand contour which need be taken into account we find 



