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216 FURTHER APPLICATIONS OF BESSEL'S FUNCTIONS OF [380
When n is great, the vibration at a distance is extraordinarily small in comparison with what it would have been were lateral motion prevented. As another example, let n^l'lka. Then (27) = e-™n+ V('4587). Here n would need to be about 17 times larger for the same sort of effect.
The extension of Stokes' analysis to large values of n only emphasizes his conclusion as to the insignificance of the effect propagated to a distance when the vibrating segments are decidedly smaller than the wave-length.
We now proceed to the case of the whispering gallery supposed to act by " total reflexion." From the results already given, we may infer that when the refractive index is moderate, the escape of energy must be very small, and accordingly that the vibrations inside have long persistence. There is, however, something to be said upon the other side. On account of the concentration near the reflecting wall, the store of energy to be drawn upon is diminished. At all events the problem is worthy of a more detailed examination.
Outside the surface of transition (r — a) we have the same expression (9) as before for the velocity-potential, k and V having values proper to the outer medium. Inside k and V are different, but the product kV is the same. We will denote the altered k by h. In accordance with our suppositions h > k, and h/k represents the refractive index (//,) of the inside medium relatively to that outside. On account of the damping k and h are complex, though their ratio is real ; but the imaginary part is relatively small. Thus/ omitting the factors eikVt cos n6, we have (?- > a)
(f> = ADn(kr), .............................. (28)
and inside (r < a) (p = BJn(hr) ............................... (29)
The boundary conditions to be satisfied when r = a are easily expressed. The equality of normal motions requires that
kADn'(ka)=>hBJn'(ha); ........................ (30)
and the equality of pressures requires that
a-ADn(ka) = pBJn(ha), .............. ? ......... (31)
a-, p being the densities of the outer and inner media respectively. The equation for determining the values of ha, ka (in addition to h/k = /j,) is accordingly
kDn' (ka) _ hJn' (ha)
<rDn(ka) ~ PJn (ha)
Equation (32) cannot be satisfied exactly by real values of h and k ; for, although JnfJn is then real, Dn'/Dn includes an imaginary part. But since the imaginary part is relatively small, we may conclude that approximately h and k are real, and the first .step is to determine these real values.g. Vol. ni. p. 338, 1902; Scientific Papers, Vol. v. p. 47.