.915]
THE THEORY OF THE HELMHOLTZ RESONATOR
367
By means of the last, %2, Xs, etc., may be built up in succession from <;„ and XL-From (2)
>r with use of (7)
dtyn/dr = rn~lSn {xn-i — (n + 1) %«}................(9)
Thus if Un be the nth component of the normal velocity at the surface of ihe sphere (r = c)
Un = c*1""1^ (xn-i (c) ~ (n + 1) %n (c)}...................(10)
iVhen n = 0,
TT — C' ^ f si\ — Cf / \ /i T \
The introduction of Sn from (10), (11) into (2) gives tyn in terms of Un upposed known.
When r is very great in comparison with the wave4ength, we get rom (4)
n'n «—ir
v cr\_Le /m
%nVr;— rn+i ' ..............................v1^;
o that -»K = £n——............................(13)
7 ?' x '
We have now to apply these formulae to the particular case where U is ensible over an infinitesimal area dcr, but vanishes over the remainder of he surface of the sphere. If /j. be the cosine of the angle (0) between da ,nd the point at which 27is expressed, Pn(fj.) Legendre's function, we have
?„(/*), ........................(14)
,nd accordingly for the velocity-potential at the surface of the sphere,
When n = 0, Xn-i — (n + I) Xn is to be replaced by —c*xi- Equation (15) jives the value of ^ at a point whose angular distance (0) from da is cos"1/A. f XH has the form given by (4), the result applies to the exterior surface f the sphere.
We have also to consider the corresponding problem for the interior. ?he only change required is to replace Xn as given in (4) by the form ppropriate to the interior. For this purpose we might take simply the maginary part of (4), but since a constant multiplier has no significance, it unices to make
AVESLT.......................(16)
rdrj r .........................v 'we have