148 



DYNAMICAL THEOKY OF SOUND 



We have here s nodal diameters, given by 

 s0 + a= \ir, f?r, ..., 



and accordingly arranged at intervals of TT/S. Again for every 

 value of k after the lowest we have one or more nodal circles 

 whose radii are given by the roots of lower order. In the case 

 s = l, where there is one nodal diameter, we have 



ka/7r= 1-2197, 2'2330, 3'2383, 4'2411, (17) 



the numbers tending to the form ra + J . The characters of 

 the corresponding modes may be gathered from the annexed 



10 



10 



Fig. 52. 



graph of the function J l (z) ; this may be supposed to represent 

 a section through the centre, normal to the nodal diameter. In 

 the second of the above modes, the radius of the nodal circle is 

 given by 



r/a = 1-2197/2-2330 = '546. 



Fig. 53 shews in plan the configuration of the nodal lines 

 in the first three modes of the types s = 0, 5 = 1, s 2, re- 

 spectively. The + and signs distinguish the segments 

 \vhich are at any instant in opposite phases of vibration. 



Whatever the form of the boundary, the value of f in the 

 neighbourhood of any point of a membrane must admit of 

 expression in the form (9), with 



R 8 = A 8 J.(kr), S 8 = B 8 J 8 (kr), (18) 



the factor cos (nt + e) being of course understood. If be on 

 a nodal line we must have f = for r = 0, and therefore A = 0. 

 The form of the membrane near is then given by 



^ = (A 1 cosO + B 1 sme)J l (kr), (19) 



ultimately, and the direction of the nodal line at is accord- 



