1916] ENERGY ACQUIRED BY SMALL RESONATORS FROM INCIDENT WAVES 417
Thus, if in free .vibration p is proportional to eint, where n is complex, the equation for n is
•n? (- M' + i. 4Wcr4) -I- ^ = 0......................(4)
The free vibrations are assumed to have considerable persistence, and the coefficient of decay is &~qt, where
q =
(5)
We now suppose that the resonator is exposed to primary waves whose velocity-potential is there
<jE> = ae^ .................................. (6)
The effect is to introduce on the right hand of (3) the term 4tirr2cra.ipeipt; and since the resonance is supposed to be accurately adjusted, p'2 = //,/JI/'. Under the same conditions icPp/dtf in the third term on the left 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
' -~ dt
— 4)7rrV apsinpt, ............ (7)
which is of the usual form for vibrations of systems of one degree of freedom. For the permanent forced vibration M'd*p/dtf + /up = 0 absolutely, and
dp___asinpt
dt~ kr* ............................... w
The energy located in the resonator is then
^ (9)
2/cV .................................... (J)
and it may become very great when M is large and r small.
But when M is large, it may take a considerable time to establish the permanent regime after the resonator starts from rest. The approximate solution of (7), applicable in that case, is
'-2p£<1
q being regarded as small in comparison with p ; and the energy located in the resonator at time t
We may now inquire what time is required for the accumulation of energy equal (say) to one quarter of the limiting value. This occurs when e~^ = £,
or by (5) when
Iog2 Iog2.if'
j, -- — , _
q p . kr . 2ircrr3
B. vi. 27e are no complications to confuse the interpretation.