280 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390
in which & = 27r/X, and X is the wave-length; and it appeared that the potential of the secondary waves diverging from the resonator is
.(2)
~ iko r so that 47n/2 Mod2 i/r = 47r/&2c2 ......................... (3)
The left-hand member of (3) may be considered to represent the energy dispersed. At the distance of the resonator
If we inquire what area S of primary wave-front propagates the same energy as is dispersed by the resonator, we have
or
Equation (4) applies of course to plane primary waves, and is then u particular case of a more general theorem established by Lamb*.
It will be convenient for our present purpose to start de novo with piano primary waves, still supposing that the resonator is simple, so that wo are concerned only with symmetrical terms, of zero order in spherical harmonics.
Taking the place of the resonator as origin and the direction of propagation as initial line, we may represent the primary potential by
cos _
The potential of the symmetrical waves issuing from the resonator may be taken to be
. ae~ikr a,, ., , , //n
fy — - = _n —i/cr + ...) ......................... (6)
T
r
Since the resonator is supposed to be an ideal resonator, concentrated in u point, r is to be treated as infinitesimal in considering the conditions to be there satisfied. The first of these is that no work shall be done at the resonator, and it requires that total pressure and total radial velocity shall be in quadrature. The total pressure is proportional to d (<f> + ty)/dt, or to i(<f) + Ajr), and the total radial velocity is d(<j> + ^fdr. Thus (<£ + -vjr) and d (<£ + ty}fdr must be in the same (or opposite) phases, in other words their ratio must be real. Now, with sufficient approximation,
so that .a~l ~ ik = real
* Camb. Trans. Vol. xvm. p. 348 (1899) ; Proc. Math. Soc. Vol. xxxn. p. 11 (1900). The resonator is no longer limited to be simple. See also Rayleigh, Phil. Mag. Vol. m. p. 97 (1902) ; Scientific Papers, Vol. v. p. 8.osition of perfectly smooth walls ; but such failure seems probable. We know from the work of Keynolds, Lorentz, and Orr that no failure of stability can occur unless @D*jv > 177, where D is the distance between, the walls, so that j3D represents their relative motion.