170 DYNAMICAL THEOKY OF SOUND 



These are identical with Fresnel's formulae for the amplitudes 

 of reflected and transmitted light in the case of normal inci- 

 dence on the common boundary of two transparent media. 



62. Vibrations of a Column of Air. 



When we come to the free oscillations of the air contained 

 in a pipe of finite length, the question definitely arises as to the 

 condition to be satisfied at an open end. There is here a 

 transition, more or less rapid, from plane waves in the tube 

 to diverging spherical waves in the external space, which it is 

 difficult to allow for exactly. In the usual rudimentary theory, 

 which dates from D. Bernoulli, Euler, and Lagrange, it is 

 assumed that the variation of pressure in the tube, at the open 

 end, may be neglected. As already stated, this would be 

 accurately the case if the external air were replaced by a 

 substance capable of exerting pressure (p ) but devoid of 

 inertia. There would then be no loss of energy on reflection 

 at the open end ( 61), and the vibrations in the tube, once 

 excited, would be persistent. The hypothesis is obviously 

 an imperfect representation of the facts ; the condition s = 

 can only be approximately fulfilled, and energy must con- 

 tinually be spent in the generation of waves diverging outwards 

 from the mouth, so that the vibrations if left to themselves will 

 be sensible only for a very limited time ; this may however 

 cover hundreds of periods. We shall return to these questions 

 later (Chapter IX) ; at present we content ourselves with 

 tracing out the consequences of the approximate theory. 



The periodic character of the motion in a finite pipe can be 

 inferred from the theory of waves, exactly as in the case of 

 strings ( 24). Suppose for example that a wave ,of limited 

 extent is started in either direction from a point P of a tube 

 AB. After two reflections, at A and B, the wave will pass P 

 again in the same direction as at first. If both ends be closed, 

 the sign of s is unaltered at either reflection, whilst that of j is 

 twice reversed. Hence after the interval 2l/c, where l AB, 

 the initial circumstances are exactly reproduced. The same 

 result holds if both ends be open, since there have now been 



