350.
NOTE ON BESSEL'S FUNCTIONS AS APPLIED TO THE VIBRATIONS OF A CIRCULAR MEMBRANE.
{Philosophical Magazine, Vol. xxi. pp. 53—58, 1911.]
IT often happens that physical considerations point to analytical conclusions not yet formulated. The pure mathematician will admit that arguments of this kind are suggestive, while the physicist may regard them as conclusive.
The first question here to be touched upon relates to the dependence of the roots of the function Jn (z) upon the order n, regarded as susceptible of continuous variation. It will be shown that each root increases continually with n.
Let us contemplate the transverse vibrations of a membrane fixed along the radii 6 = 0 and 9 = ft and also along the circular arc r = 1. A typical simple vibration is expressed by*
w = /n(^V).sinn0.cos(/9)£), ..................(I)
where *r is a finite root of Jn (z) = 0, and n ~ rr/ft. Of these finite roots the lowest z^ gives the principal vibration, i.e. the one without internal circular nodes. For the vibration corresponding to z^ the number of internal nodal circles is s — 1.
As prescribed, the vibration (1) has no internal nodal diameter. It might be generalized by taking n = virjft, where v is an integer; but for our purpose nothing would be gained, since ft is at disposal, and a suitable reduction of ft comes to the same as the introduction of v.
In tracing the effect of a diminishing ft it may suffice to commence at & = TT, or n—\. The frequencies of vibration are then proportional to the roots of the function J^. The reduction of ft is supposed to be effected by
* Theory of Sound, §§ 206, 207. E. VI. 1ad Vol. i. And in Theory of Sound, Vol. i,