CHAPTER X. 



WAVES OF EXPANSION. 



254. A TREATISE on Hydrodynamics would hardly be complete 

 without some reference to this subject, if merely for the reason 

 that all actual fluids are more or less compressible, and that it is 

 only when we recognize this compressibility that we escape such 

 apparently paradoxical results as that of Art. 21, where a change 

 of pressure was found to be propagated instantaneously through a 

 liquid mass. 



We shall accordingly investigate in this Chapter the general 

 laws of propagation of small disturbances, passing over, however, 

 for the most part, such details as belong more properly to the 

 Theory of Sound. 



In most cases which we shall consider, the changes of pressure 

 are small, and may be taken to be proportional to the changes 

 in density, thus 



Ap = *.^ (1), 



where K (=pdp/dp) is a certain coefficient, called the ' elasticity of 

 volume. 1 For a given liquid the value of K varies with the 

 temperature, and (very slightly) with the pressure. For water at 

 15 C., K = 2'22 x 10 10 dynes per square centimetre; for mercury 

 at the same temperature K = 5'42 x 10 11 . The case of gases will 

 be considered presently. 



Plane Waves. 



255. We take first the case of plane waves in a uniform 

 medium. 



The motion being in one dimension (x), the dynamical equation 

 is, in the absence of extraneous forces, 



du du I dp 1 dp dp /1X 



I a i . __ JL__ - ___ * ' / I \ 



dt dx p dx p dp dx'" 



