236 DYNAMICAL THEORY OF SOUND 



79. Waves generated by Vibrating Solid. 



We return to the problem of investigating the waves 

 generated by a vibrating body. In order not to complicate the 

 question too much we will assume that the body has some sort 

 of symmetry with respect ta an axis ; thus it may be a form 

 of revolution about this axis, or it may have two mutually 

 perpendicular planes of symmetry meeting in this axis, or 

 (again) a single plane of symmetry perpendicular to the axis. 

 In any case this axis is taken to be the direction (Ox) of 

 vibration. 



The dimensions of the solid being supposed small in com- 

 parison with \/2-7r, the motion of the fluid in the immediate 

 neighbourhood will be sensibly the same as in the case of 

 incompressibility, and the principal effect on the body will there- 

 fore be equivalent to an increase of inertia. To establish this 

 latter point in a general manner, we note that the (irrotational) 

 motion of a frictionless liquid due to the motion of a solid in it 

 will have the velocity at every point in a determinate ratio 

 to the velocity U of the solid, and that the total kinetic energy 

 of the fluid may therefore be expressed by ^pQ'U 2 , where Q' is 

 a constant, of the nature of a volume, depending only on the 

 size and shape of the solid and the direction of its vibration. 

 Hence if M be the mass of the body, the equation of energy 

 takes the form 



(l) 



where the right-hand member represents the rate at which 

 work is being done by the extraneous force X. Thus 



X, .................. (2) 



which shews that the inertia of the body is apparently increased 

 by the amount pQ'. An equivalent statement is that the 

 reaction of the liquid is equivalent to a force pQ'dU/dt. 



In the actual case of the gaseous medium, it is plain that if 

 the solid were removed, and its place supplied by fluid, the 

 motion at a distance would be very approximately the same as 

 would be produced by a suitable periodic force from without, 



