ELASTIC VIBRATIONS
381
(4)
or
(5)
и-Л , UJO . , -»JT • ,
(- к 1- n 2 x = E sm pt.
dt 2 dt
5. Integration of the Differential Equation. We can effect the
complete integration of the differential equation which governs the
motion, § 4, (5), if we find one single special solution ; cf. Chap. XIV,
§ 11. Now, it was long since known or surmised that the system on
which a periodic impressed force acts ultimately gives up its own
note and takes on the period of the impressed force. But the phase
of the one oscillation is different from that of the other; the tides
lag behind the moon.
We are thus moved to try an experiment and see if we cannot de
termine a particular periodic solution of (5), § 4, built on the simplest
lines imaginable. So we set
(6) x = A sin (pt — a),
when A and a are undetermined constants, and try to determine
these so that (6) will be a solution.
Substituting the function (6) in equation (5), we find :
A(?i 2 — p 2 ) sin (pt — a) + Акр cos (pt — a) = E sin pt.
This equation is equivalent to the following:
\A(n 2 — p 2 ) cos a + Акр sin a — E\sinpt
— \A (n 2 — p 2 ) sin а — Акр cos a\cos pt = 0.
The latter equation will be true for all values of t if
A(n 2 — p 2 ) cos a + Акр sin a = E
A(n 2 —p 2 ) sin a — Акр cos a = 0.
From the last equation follows that
(8)
We will agree to understand by a that root of this equation for
which
0 < a < 7Г.
The first equation (7) gives