FURTHER APPLICATIONS OF BESSEL'S FUNCTIONS OF HIGH ORDER TO THE WHISPERING GALLERY AND ALLIED PROBLEMS.
[Philosophical Magazine, Vol. xxvu. pp. 100—109, 1914.]
IN the problem of the Whispering Gallery* waves in two dimensions, of length small in comparison with the circumference, were shown to run round the concave side of a wall with but little tendency to spread themselves inwards. The wall was supposed to be perfectly reflecting for all kinds of waves. But the question presents itself whether the sensibly perfect re-fluxion postulated may nob be attained on the principle of so-called "total reflexion," the wall being merely the transition between two uniform media of which the outer is the less refracting. It is not to be expected that absolutely no energy should penetrate and ultimately escape to an infinite distance. The analogy is rather with the problem treated by Stokesf of the communication of vibrations from a vibrating solid, such as a bell or wire, to a surrounding gas, when the wave-length in the gas is somewhat large compared with the dimensions of the vibrating segments. The energy radiated to a distance may then be extremely small, though not mathematically evanescent.
A comparison with the simple case where the surface of the vibrating body is plane (as = 0) is interesting, especially as showing how the partial
* Phil. Hag. Vol. xx. p. 1001 (1910); Scientific Papers, Vol. v. p. 619. But the numbers there given require some correction owing to a slip in Nicholson's paper from which they w*ere derived, as was first pointed out to me by Prof. Macdonald. Nicholson's table should be interpreted as relating to the values, not of 2-1123 (n - «)/«*, but of 1-3447 (n-«)/«*, see Nicholson, Phil. Mag. Vol. xxv. p. 200 (1913). Accordingly, in my equation (5) M814n* should read l-8558?i*, and in equation (8) -51842n* should read -8065*1*. [1916. Another error should be
noticed. In (8), = I cos n(w-sin w) «ZW/TT must be omitted, the integrand being periodic. See
Watson, Phil. Mag. Vol. xxxn. p. 233, 1916.]
•|- Phil. Trans. 1868. See Theory of Sound, Vol. n. § 324.