1914] HIGH ORDER TO THE WHISPERING GALLERY 215
Also, the most important part of the real and imaginary terms being retained, D ' M - /smh/3cosh/3y (9,-t , ,vi r2(^
JJn \z)-----I ------n---------- ^^ +16 t................\£v)
X ' \ »^A^«w. / \. ^ ^ '
The application is now simple. From (9) with introduction of an arbitrary coefficient
VC&r) ...................... (21)
If we suppose that the normal velocity of the vibrating cylindrical surface (r = a) is represented by eikm cos w0, we have
kADn'(Jca) = l, .............................. (22)
and thus at distance r
lcDn (lea) or when r is very great
(23)
nT ...........
JcDn(ka)
We may now, following Stokes, compare the actual motion at a distance with that which would ensue were lateral motion prevented, as by the insertion of a large number of thin plane walls radiating outwards along the. lines Q — constant, the normal velocity at r = a being the same in both cases. In the altered problem we have merely in (23) to replace Dn, Dn by DO, D0f. When z is great enough, Dn(z) has the value given in (16), independently of the particular value of n. Accordingly the ratio of velocity-potentials at a distance in the two cases is represented by the symbolic fraction
D0'(ka) .
7 ................................. ( }
in which I>0'(1ca) = -i-e-^+^. ., ................ (26)
We have now to introduce the value of Dn' (Jca), When n is very great, and Jca/n decidedly less than unity, t is negative in (20), and et is negligible in comparison with e~f. The modulus of (25) is therefore
/ n/Jca \* «-«P-*ohfl
' ° ................ • '
For example, if n = 2&a, so that the linear period along the circumference of the vibrating cylinder (^tra/n) is half the wave-length,
cosh/3 = 2, £ = 1-317, sinh£ = 1-7321, tanh ft = -8660, and the numerical value of (27) isnt to fix ideas upon the case of sonorous waves, but the results are of wider application. The general differential equation is of the form