Manchester Memoirs, Vol, xUi. (1898), No, 3. 7 



Some caution is necessary in the application of the 

 above results. When ceases to be very small, the 

 disturbance in the loaded medium is no longer of the 

 simple character which we are accustomed to associate 

 with the term "wave-motion," and it becomes necessary 

 to consider which of the terms in (13) is to be taken to 

 represent a disturbance progressing in the direction of 

 ;t:-positive. In its present form the problem of this § is 

 indeed, strictly speaking, indeterminate. A more definite 

 form may be given to it if we introduce a very small 

 frictional force, acting on each of the particles M, and pro- 

 portional to the first power of the velocity. It then appears 

 that the foregoing solution applies when sin ka is positive, 

 but that when sin ka is negative, the disturbance in the 

 loaded medium must be represented by 



4 = ^/'^^'+^'* . . . (3=), 



in place of (23).f The amplitude of the reflected wave is 

 then obtained by reversing the sign of 6 in (30). 



When is imaginary, the formula (23) must be re- 

 placed by 



l.-Be ... (33) 



or 



^ i.={-)'Bi'^*-"' . . . (34) 



according as the upper or lower sign obtains in (14). The 

 analytical work is the same as before with the substitution 

 of u or u — iir for 20, and the results can therefore be 

 written down at once. Thus, we find, in the former case, 



_ g^^^ -e^^ _ _ (g" - cos ka) - /sin ka - 22^ / \ 



where 



, smka , ,, 



d) = tan-^- 7- . . . (36) 



^ e'' - cos ka ^^ ' 



+ It appears unnecessary to go into the proof of these statements, as the 

 more general investigation of the next § is free from the difficulty here 

 indicated. 



