PLANE WAVES OF SOUND 179 



It must be remembered that since the equation of motion 

 (9) is not linear, distinct solutions, such as those representing 

 waves travelling right and left, respectively, which we have 

 just been considering, cannot be superposed by mere addition. 

 It may however be remarked that, as a result of a more 

 complete investigation, Riemann* found (1860) that a localized 

 arbitrary initial disturbance does eventually resolve itself into 

 two waves of the above kinds, travelling in opposite directions. 



To follow exactly the career of waves of finite amplitude 

 generated in any given manner is a problem of considerable 

 difficulty; but some indications may be obtained by methods 

 of approximation. This procedure was adopted by Airy f (1845) 

 in his work on the dynamical theory of the tides, where similar 

 questions arise with respect to tides in shallow seas and 

 estuaries. 



Suppose, for instance, we have a long straight tube in which 

 a piston (at x = 0) is made to move in an arbitrary manner 



?=/(<) ....................... ..(24) 



The equation (9) becomes, if we neglect terms of the third 

 order in the derivatives of f, 



If we omit the last term, we have as in 60 the first 

 approximation 



(26) 



Substituting this value of f in the small term of (25) we 

 obtain 



The solution of this which is consistent with (26) is 



....... <2S> 



as is easily verified. The correction to the first approximation 



* Bernhard Riemann (1826 66), professor of mathematics at Gottingen 

 185766. 



t Sir George Biddell Airy (1801 92), Plumian professor of astronomy at 

 Cambridge 182835, astronomer royal 183581. 



122 



