ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 337 



First, it is plain that the terms having or in their denominators 

 will be small compared to the others. Neglecting those terms, or, 

 which is the same thing, considering a infinite, we have the case of 

 an incompressible fluid, and the equation applicable to it is, 



d'<p d(p d*- r( P_ n 



dr' ! ' ' rdr dr* 



The integral of this equation is, 



Hence d -± -*& t and % = -££ + F>(t). 

 dr r* dt r v J 



The known equation which gives the pressure (p) of an incom- 

 pressible fluid is 



d(b v* n 



Hence by substitution, 



•*& - 4 - »» 



As this equation contains two arbitrary functions, two conditions of 

 the motion may be arbitrarily assumed. Let us assume for one con- 

 dition, that the excess of the pressure (jo) above the pressure n, which 

 would exist in the undisturbed state of the fluid, is solely owing to 

 a velocity arbitrarily impressed in the direction of r. Then v and A f{t) 

 being supposed to vanish when p = IT, we must have, 



p — n = - — — — . 



As a second condition, let us suppose that the velocity is impressed 

 at a given distance (r), and is given at any time t by the expression 

 m sin bt. Hence f(t) = mr> sin bt, and f(t) = bmr- cos bt. Consequently 

 by substituting, 



p - n = mbr cos bt — — sin* bt, 



