135. Superposition xjf Coincident Trains of
Waves. In the case of Coincident Trains we have the
same difference of phase at every point of the trains and at
every moment. Whatever may be the distance between two
adjoining crests — one taken from each train^— that distance is

-the same everywhere.- If we compound two such trains in
the usual way we get merely a third harmonic train of the
same wave-length as the constituents, and "of an amplitude
which depends on the difference of phase of the constituents.
The student should find no difficulty in. proving this graphic-
ally.
^136. Equation of the Resultant Train. We

-can prov& analytically that two harmonic trains of tliOzSame



^^^ INTERFERENCE [CH. VII

wave-length and in the same straight line must always give a
third harmonic train.
We saw that we can represent one of the trains by the
equation

y = a sin — (vt - x),

where y is the displacement at time t and at a distance x from
the origin, X is the wave-length, v the velocity of the waves,
and a the amplitude.
The second constituent differs from this only in amplitude
and phase, and so can be written

2ir
y' = h sin -y- («< - 03 + a).
These when compounded give the wave

7 = y + y' =as,m.-Y{vt~x) + 6sin %-{vt-x + a)

which can be written in the form

J" = c sin -r- {vi — x + p)

.if c cos P = a +.h cos o

and - c sin yS = 6 siii a

h sin a

or tan a = =

a + b cos a

and c" = a" + 6^ + lab cos a.
