ON A PHYSICAL INTERPRETATION OF SCHLOMILCH'S THEOREM IN BESSEL'S FUNCTIONS.
[Philosophical Magazine, Vol. XXI. pp. 567 — 571, 1911.]
THIS theorem teaches that any function /(r) which is finite and continuous for real values of r between the limits r — 0 and r = TT, both inclusive, may be expanded in the form
f(r) = a0 + alJ0(r) + azJ(t(2r')+asJ0(3r) + ..., ......... (1)
Jo being the Bessel's function usually so denoted ; and Schlomilch's demonstration has been reproduced with slight variations in several text-books*. So far as I have observed, it has been treated as a purely analytical development. From this point of view it presents rather an accidental appearance ; and I have thought that a physical interpretation, which is not without interest in itself, may help to elucidate its origin and meaning.
• The application that I have in mind is to the theory of aerial vibrations. Let us consider the most general vibrations in one dimension £ which are periodic in time 2vr and are also symmetrical with respect to the origins of £ and t. The condensation s, for example, may be expressed
s = b0 + 6j cos £cos£+ 62cos 2^ cos 2£ + ..., .............. (2)
where the coefficients &0, &i, &c. are arbitrary. (For simplicity it is supposed that the velocity of propagation is unity.) When t = 0, (2) becomes a function of £ only, and we write
-F(£) = &o + &1cos£ + &2cos2£+..., .................. (3)
in which F(£) maybe considered to be an arbitrary function of £ from 0 to TT. Outside these limits F is determined by the equations
* See, for example, Gray and Mathews' Bessel's Functions, p. 30; Whittaker's Modern Analysis, § 165.th order we obtain four equations from (20), (21) and m — 3 by differentiations of (19), so that all the differential coefficients of the mth order vanish. It follows that every differential coefficient of w with 'respect to x and y vanishes at the origin. I apprehend that the conclusion is valid for all angles « less than STT. That the displacement at a distance r from the corner should diminish rapidly with r is easily intelligible, but that it should diminish more rapidly than any power of r, however high, would, I think, not have been expected without analytical proof.