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of Superposition of Waves
When two or more waves travelling
through a medium arrive at a point simultaneously, each wave produces its
own displacement at that point independently of the other waves. Hence, the
resultant displacement at that point is the vector sum of the displacements
due to each of the waves.
When two sound notes of the same amplitude but and slightly
different frequencies are sounded together, the resultant intensity of the
sound alternately increases and decreases. This rise and fall in the
intensity of the sound is called beats.
The rise in intensity is called the "waxing" of sound and the
fall in intensity is called the "waning" of sound.
for the Formation of Beats
i) Two waves should travel through the same medium.
ii) The amplitudes of the two waves should be nearly equal.
iii) There must be a slight difference between the frequencies of the two
Treatment - Theory of Beats
sound waves of equal amplitude (A) and slightly different frequencies n1, and n2 can be
represented by the equations,
y1 = A sin 2pn1t
y2 = A sin 2pn2t
By the principle of superposition of waves, the resultant
displacement is given by
y = y1 + y2 = A sin 2pn1t + A sin 2pn2t
\ y = 2A sin [ 2 p
(n1 + n2)
Cos [ 2 p
(n1 - n2)
\ y = R sin 2p nt
where, R = 2 A cos 2 p
t and n =
n1 + n2
R represents the amplitude of the resultant wave motion.
Now, Intensity a (amplitude|2.
R = Rmax
i.e. R is max when Cos 2p
n1 - n2
t = ± 1
i.e. when 2p
t = kp
(k = 0, 1, 2…)
\ (n1 - n2 ) t = k
or when t =
i.e. when t = 0,
n1 - n2`
R = Rmin
i.e. R is min. when cos 2p
t = 0
i.e. when, 2p
t = (k +
p, k = 0, 1, 2....
Thus, (n1 - n2) t = k +
2(n1 - n2)
\ Period of beats =
and frequency of beats = |(n1 - n2)|
The apparent change in the frequency of a sound due to the
relative motion between the source of the sound and the observer is called