KESONATOKS EXPOSED TO PRIMAEY PLANE WAVES 289 msider free vibrations. These are of necessity subject to damping, owing • the communication of energy to the medium, forthwith propagated away; id their persistence depends upon the nature of the resonator as regards iass and spring, and not merely upon the ratio of these quantities. Taking first the case of a single resonator, regarded as bounded at the irface of a small sphere, we have to establish the connexion between the lotion of this surface and the aerial pressure operative upon it as the result '. vibration. We suppose that the vibrations have such a high degree of arsistence that we may calculate the pressure as if they were permanent, hus if ty be the velocity-potential, we have as before with sufficient approxi-tation _ 1 - ikr 1 d^ __ 1 "VT / 66 — ~"~ ? 7 — """" I 5 T / r a dr r- > that, if p be the radial displacement of the spherical surface, dpjdt = — a/r2, ad .......................(47) .gain, if or be the density of the fluid and Sp the variable part of the ressure, ............... (48) •hich gives the pressure in terms of the displacement p at the surface of a phere of small radius r. Under the circumstances contemplated we may se (48) although the vibration slowly dies down according to the law of eint, rhere n is not wholly real. If M denotes the " mass " and n the coefficient of restitution applicable ) p, the equation of motion is r if we introduce eint and write M' for M + ^ircrr3, Lpproximately, n nd if we write n = p + iq, (49) (50) '), q = p . 2irtrkr* / M' ................... (51) f T be the time in which vibrations die down in the ratio of e : 1, T= l/q. If there be a second precisely similar vibrator at a distance R from the rst, we have for the potential — R dt' R. VI. 19es further examination'.