and
y — A cos nt cos x — \A"- cos2 nt cos 2# + f A3 cos3 wi cos 3a?. . . .(40)
When cos nt — 0, y = 0 throughout ; when cos nt = I,
y — A cos so — \A- cos 2* + %A3 cos 3#,
so that at this moment of maximum displacement the form is the same as for the progressive wave (36).
We have still to determine n so as to satisfy (28), (29), with evanescent IB, /3'. The first is satisfied by a = 0, since a' = 0. The second becomes
d-<y! , . ,da? ,
f a A '
In the small terms we may take a — — g ladt = — £--. sinnt, so that
J *t
d*a.' a~A3
, — h go? + --. — (sin nt + 5 sin Snt) = 0. dp y 4n ^ '
To satisfy this we assume
a' ==• H sin nt + K sin 3nt,
Then H(g-n^+^=0, . K(q - 9<) + ^±- = 0,
^"•^ 4jil *• ZLoi
jf/t/ ^t?/6
from the first of which
%2==^ + ^! = ^_^2)...............
or, if we restore homogeneity by introduction of k,
,...............(42)
With this value of n the stationary vibration
y = A cos nt cos kx — ^kA* cos2 nt cos 2/b? + f k*As cos3 w< cos Skx,.. .(43) satisfies all the conditions. It may be remarked that according to (42)- the frequency of vibration is diminished by increase of amplitude.
The special cases above considered of purely progressive or purely stationary waves possess an exceptional simplicity. In general, with omission of (3, 0', equations (28), (29), become
2 da d a2 + a/2 , . .