PLANE WAVES OF SOUND 



159 



When the pressure and volume vary in any connected 

 manner, the ratio of 

 the increment &p of 

 the pressure to the 

 " compression," i.e. the 

 negative dilatation 

 Sv/v, may be called 

 the " elasticity of 

 volume." Its value 

 will depend not only 

 on the particular state, 

 but on the manner in 

 which the variations 

 from that state are 

 supposed to take place, 

 i.e. on the direction 

 of the corresponding 



p. 



curve on the diagram. 



If the tangent at the 



point P meet the axis of p in U, and NU be the projection of 



P U on this axis, we have 



dv 



.(5) 



this projection therefore represents the elasticity under the 

 particular condition. On the isothermal hypothesis, to which 

 these letters refer in the figure, the elasticity is equal to the 

 pressure p, as follows at once from (1), or from the fact that 

 the tangent to a rectangular hyperbola is bisected at the 

 point of contact. If the variations are subject to the adiabatic 

 law, the elasticity, as deduced from (4), is yp, and so greater 

 than in the former case. This is represented by NU' in the 

 figure. Even in the case of solid and liquid bodies we ought, 

 in strictness, to discriminate between isothermal and adiabatic 

 coefficients of elasticity, but the differences happen not to be 

 very important. 



The work done by unit mass of a gas in expanding between 

 any two adjacent states is easily read off from a diagram as 

 o-v), or Po(v -v) + %(p-p )(v -v), (6) 



