174 DYNAMICAL THEOEY OF SOUND 



pressure in the liquid. If c be the theoretical velocity of 

 sound in the liquid, as given by 59 (8), c the actual velocity, 

 it is found that in the case of a tube of small thickness h 



...(12) 



where a is the internal radius, K is the volume-elasticity of 

 the liquid, and E is the value of Young's modulus for the 

 material of the tube. Thus in the case of water (K = 2'22 x 10 10 ) 

 contained in a glass tube (^=6*03 x 10 11 ) whose thickness is 

 one-tenth of the radius, we find c '759c . Even in the other 

 extreme, when the walls are very thick, it is found that 



where /A is the rigidity. The value of /z, for glass is, roughly, 

 about 10 times the value of K for water; this would give 

 a diminution of about 5 per cent, in the wave-velocity. 



63. Waves of Finite Amplitude. 



The laws of sound propagation, as they are investigated in 

 this and succeeding chapters, are subject to some qualifications 

 which may best be considered in relation to plane waves, where 

 the theory is simplest. 



In the first place, it has been assumed that the conden- 

 sation s may be treated as infinitely small. This hypothesis 

 is adequate for most purposes, but there are certain "second 

 order " effects which are of some theoretical importance. 



It is easy to shew that a progressive wave of finite (as 

 distinguished from infinitely small) amplitude cannot be pro- 

 pagated without change of type, except on the hypothesis 

 of a certain special relation between pressure and density. 

 Assuming, for a moment, that a wave of permanent type is 

 in progress, we may in imagination impress on the whole 

 mass of air a velocity equal and opposite to that of the 

 wave. In this way we obtain a condition of " steady motion " 

 as it is called, in which the velocity, pressure, and density at 

 any point of space are constant with respect to the time. 

 For definiteness we may fix our attention on the air contained 

 in a long straight tube of unit sectional area. The velocity u 



