228 DYNAMICAL THEOEY OF SOUND 



It will be noticed that as the wave-length X is increased, 

 and k accordingly diminished, the fundamental equation (2) 

 tends to assume the form 



W> = ........................ (27) 



which is met with in the dynamics of incompressible fluids, and 

 in the theories of attractions and of electric and thermal con- 

 duction. This assimilation may come about in two ways, either 

 through a diminution in the frequency (n/Zir), or by an increase 

 in the elasticity of the medium and consequently in the wave- 

 velocity. Under the same condition the formula (12) 

 approximates to the form 



which is the expression for the potential of a magnetic pole, or 

 for a source of electricity, and so on ; whilst in the case of the 

 double source (21) the limiting form is (23), which is recognized 

 as the potential of an infinitely small magnet. 



A further remark of great importance is that within any 

 region, free from sources, whose dimensions are small compared 

 with X, the configuration of the equipotential surfaces <= const. 

 is at any instant sensibly the same as if the fluid were incom- 

 pressible. For the value of </> due to an external source differs 

 from its value in the case of incompressibility chiefly by a 

 factor e~ ikr , where r denotes distance from the source. If 

 6 denote the greatest breadth of the region, this factor can at 

 most vary in the ratio e~ ikb , which differs very little from unity 

 when kb is small. 



77. Vibrating Sphere. 



By means of the fiction of a double source, of suitable 

 strength, at the centre, it is possible to calculate the waves 

 generated in the surrounding air by a vibrating solid sphere of 

 any radius. As this is almost the only problem of the kind 

 which can be completely solved we devote some space to it. The 

 work is simple, and the results throw a good deal of light on 

 other cases. 



For reasons just referred to, it is instructive to look first at 

 the case where the fluid is incompressible. We take the origin 



