304 Free Path Phenomena [OH. xvi 



Since x and t enter only through the differential operators - and , 



there will be a solution in which each variable differs from its value in the 

 steady state by a small term proportional to e i( P t ~^ x) , and the solution repre- 

 sents wave motion parallel to the axis of x. Let us then assume a solution 

 of the form 



H = H + H 



K- K + K'e* <**-*>, 



in which the unaccented v, H and K on the right-hand side refer to equi- 

 librium values. 



The forms assumed by equations (725) to (729), on substitution of these 

 solutions and neglect of small quantities of the second order, are as follows : 



2(7 



PVU ' ~ 3m ^ K/ + V/K ) = ..................... ( 73 )' 



pv'-qvu ' = ........................... (731), 



p (vv! + Z>'K) - f quint*' + i&i? Vi< (H' - K') = ............ (732), 



^I/H'-^VK(H'-|K / ) = O ..................... (733). 



Eliminating the accented letters, we arrive at the relation 



P 72 5m 

 ^ 2 K - -^ p 2 



361. If the left-hand member is very great, the relation between q 

 and p is expressed by the vanishing of the denominator on the right- 

 hand, so that 



- ' . ? = ^P* .............. } ............... (735). 



3 3 



Since K = ^ = 27 /) ' 



where p is the pressure in the undisturbed gas, the foregoing equation can 

 be written as 



o - 5 p tf 

 q -lpo P ' 



so that the velocity of propagation is 



