254-255] PLANE WAVES. 465 



whilst the equation of continuity, Art. 8 (4), reduces to 



If we put 



p=p Q (I+s) ........................... (3), 



where p Q is the density in the undisturbed state, s may be called 

 the condensation in the plane x. Substituting in (1) and (2), 

 we find, on the supposition that the motion is infinitely small, 



du K ds ,.. 



Tt = ~p,dx &quot; 



ds du /K , 



and -j- = --j- .......................... (5), 



dt dx 



if K = [pdp/dp] p=p(} ........................ (6), 



as above. Eliminating s we have 



#u = c ,#u 



dt 2 dx 2 &quot; 



where c 2 = K/p = [dp/dp] p=po ..................... (8). 



The equation (7) is of the form treated in Art. 107, and the 

 complete solution is 



u = F(ct-x)+f(ct + x) .................. (9), 



representing two systems of waves travelling with the constant 

 velocity c, one in the positive and the other in the negative 

 direction of x. It appears from (5) that the corresponding value 

 of s is given by 



............... (10). 



For a single wave we have 



u = cs ..................... ...... (11), 



since one or other of the functions F,f is zero. The upper or the 

 lower sign is to be taken according as the wave is travelling in 

 the positive or the negative direction. 



There is an exact correspondence between the above approximate theory 

 and that of long gravity- waves on water. If we write 77 /A for s, and gh for 

 ic/Po, the equations (4) and (5), above, become identical with Art. 166 (3), (5). 



I,. 30 



