258-260] WAVES OF FINITE AMPLITUDE. 471 



It appears from (1) that the equation (6) of Art. 257 could be regarded as 

 exact if the relation between p and p were such that 



7 



p dp 



Hence plane waves of finite amplitude can be propagated without change of 

 type if, and only if, 



(ii). 



A relation of this form does not hold for any known substance, whether at 

 constant temperature or when free from gain or loss of heat by conduction 

 and radiation. Hence sound-waves of finite amplitude must inevitably under- 

 go a change of type as they proceed. 



260. The laws of propagation of waves of finite amplitude 

 have been investigated, independently and by different methods, 

 by Earnshaw and Riemann. It is proposed to give here a brief 

 account of these investigations, referring for further details to the 

 original papers, and to the very full discussion of the matter 

 by Lord Rayleigh *. 



Riemann's method ( has already been applied in this treatise 

 to the discussion of the corresponding question in the theory 

 of 'long' gravity-waves on liquids (Art. 183). He starts from 

 the Eulerian equations of Art. 255, which may be written 



du du _ 1 dp dp 

 ~dt* U dx~~ pdpdn 



dp , dp du , . 



dt +w = - p &* ........................ (2) - 



If we put 



P=f(p) + u, Q=f(p)- u ............... (3), 



where f(p) is as yet undetermined, we find, multiplying (2) by 

 /' (p), and adding to (1), 



dp + u dp - 1**_^'/^ 



W U dx~~ pdpdx PJ (p) dx' 

 If we now determine f(p) so that 



* Theory of Sound, c. xi. 



f " Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite," 

 Gott. Abh., t. viii. (1860) ; Werke, Leipzig, 1876, p. 145. 



