174
ON SOUND.
Making this = 0 when x — 0 and when x — l,
' (at) — (at) = 0
(ft (at — — (at + T) = 0,
which are the terminal equations for this case.
The first equation, which is general for all values of
t, gives = <£>'. The second gives
' is a periodical
function, going through all its changes while the quan¬
tity affected by it is changed by 21, or while t increases
<2i
by — : giving the same number of complete vibrations
per second, and therefore the same fundamental note, as
a pipe closed at both ends, Article 78. The velocity of
the particle, or
a. <£' (at — x) + a yjr' (at + x),
may be represented (for the same reasons as in the
beginning of Article 74) by
^ ( n . nir (at — x))
2 I<7*. sin-^-j-
, * {-n nnr(at — x))
+ 2\Dn. cos —-4
, v f n • nir (at + a?)]
+ 2 j Cn . Bin--4
^ nir (at + x)
+ S \Dn. cos-j--