HIGH ORDER TO THE WHISPERING GALLERY 213
which we will suppose to be adapted to the region where x is negative. On the right (as positive) F is to be replaced by F1( where Fx > F, and <f> by 0,. In optical notation F1/F=/*) where /A (greater than unity) is the refractive index. We suppose <£ and ^ to be proportional to 6*^+^, b and o being the same in both media. Further, on the left we suppose b and c to be related as they would be for simple plane waves propagated parallel to y. Thus (4) becomes, with omission of ei(bv+et},
of which the solutions are
.A, .#, G denoting constants so far arbitrary. The boundary conditions require that when so=Q, dQ/das = dfa/das and that p<f> = p1$1) p, pi being the densities. Hence discarding the imaginary part, and taking A = l, we get finally
(8) I
It appears that while nothing can escape on the positive side, the amplitude on the negative side increases rapidly as we pass away from the surface of transition.
If fj, < 1, a wave of the ordinary kind is propagated into the second medium, and energy is conveyed away.
In proceeding to consider the effect of curvature it will be convenient to begin with Stokes' problem, taking advantage of formulae relating to B easel's and allied functions of high order developed by Lorenz, Nicholson, and Macdonald*. The motion is supposed to take place in two dimensions, and ideas may be fixed upon the case of aerial vibrations. The velocity-potential <£ is expressed by means of polar coordinates r, 6, and will be assumed to be proportional to cos n6, attention being concentrated upon the case where n is a large integer. The problem is to determine the motion at a distance due to the normal vibration of a cylindrical surface at r = a, and it turns upon the character of the function of r which represents a disturbance propagated outwards. If Dn (kr) denote this function, we have
<t> = eiJcVtcosnd.Dn(kr), ........................ (9)
and Dn (z) satisfies Bessel's equation
(10)
* Compare also Debye, Math. Ann. Vol. LXVII. (1909).tive medium. The problem is that of total reflexion when the incidence is grazing, in which case the usual formulae* become nugatory. It will be convenient to fix ideas upon the case of sonorous waves, but the results are of wider application. The general differential equation is of the form