418 ENERGY ACQUIRED BY SMALL RESONATORS FROM INCIDENT WAVES [409
The energy propagated in time t across the area S of primary wave-front is (Theory of Sound, § 245)
*t, ................................. (13)
where a is the velocity of propagation, so that p — ak. If we equate (13) fco one quarter of (9) and identify t with the value given by (12), neglecting the distinction between M and M', we get
n 4A* ^ }
_ _ 21og2./c2"8-7rlog2
The resonator is thus able to capture an amount of energy equal to that passing in the same time through an area of primary wave-front comparable with V/TT, an area "which may exceed any number of times the cross-section of the resonator itself.
* log 2 = 0-693..ft of (3) may be replaced by ~pdp/dt, whether we are dealing with the permanent forced vibration or with free vibrations of nearly the same period which gradually die away. Thus our equation becomes on rejection of the imaginary part