144 



DYNAMICAL THEOEY OF SOUND 



Fig. 50. 



54. Circular Membrane. Normal Modes. 



In the case of the circular membrane we naturally have 

 recourse to polar coordinates, with the origin at the centre. 

 The differential equation may be obtained by transformation 

 of 52 (3), but a more direct process is preferable. 



Take first the case of the symmetrical vibrations where 

 is a function of r, the distance from 0, only. The stress across 

 a circle of radius r has a resultant P . ZTTT . 9f/9r normal to the 

 plane of the undisturbed membrane, and the difference of the 

 stresses on the edges of the annulus whose inner and outer 

 radii are r and r + Sr gives a force 



, 



Equating this to p . 2?rr5r . f, which is the acceleration of 

 momentum of the annulus, we get 



If f varies as cos (nt -f e), this reduces to 



where & 2 = n 2 p/P, as before. 



(2) 



