550 NOTE ON THE THEOKY OF THE DOUBLE RESONATOR [432
a long cylindrical neck of radius R and length L we should have c = TrR2/L. An application of Lagrange's method gives as the differential equations of motion,
1 -"-3 , n<i ^ a -"-a _ /A
-7-7- T Cl 57 — U.
By addition and integration
i + ^+i^O, .......... ... ................. (4)
G! C2 C3
since in the case of free vibrations all the quantities X may be supposed proportional to ept, so that d/dt may be replaced by p.
From (3) and (4) by elimination of Xs,
Cs . . « Y -0
+ "S+ «/ J -A2 — ^J
whence as the equation for js2
In the use of double resonance to secure an exalted effect, as in the experiments of Boys and of Callendar, we may suppress the direct communi-. cation between the second resonator 8' and the external air. Then cs = 0, and (6) -becomes
To interpret the c's suppose first the passage between S and S' abolished, so that c2 = 0. The first resonator then acts as a simple resonator, and if pl be the corresponding p, we have pf/a? — — C:/S, as usual. Again, if S be infinite, we have for the second resonator acting alone, p^fa?= —c2/$/; and (7) may be written
In (8) if 8' 1 8 be very small, p2 approximates to ^i2 or to pz*, and this is the case of greatest importance in experiment.0*00366, the pressure being atmospheric. The effect of temperature upon sound is thus about 2000 times greater than upon light. If we suppose the system of temperature differences to be altered in this proportion, the course of rays of light and of sound will be the same.