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Principle 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.

Beats

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.

Conditions 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 waves.

Analytical Treatment - Theory of Beats

The two sound waves of equal amplitude (A) and slightly different frequencies n_{1}, and n_{2} can be represented by the equations,

y_{1} = A sin 2pn_{1}t

and

y_{2} = A sin 2pn_{2}t

By the principle of superposition of waves, the resultant displacement is given by y = y_{1} + y_{2} = A sin 2pn_{1}t + A sin 2pn_{2}t

\ y = 2A sin [ 2 p

(n_{1} + n_{2})

t ]

Cos [ 2 p

(n_{1} - n_{2})

2

\ y = R sin 2p nt

where, R = 2 A cos 2 p

t and n =

n_{1} + n_{2}

R represents the amplitude of the resultant wave motion. Now, Intensity a (amplitude|^{2}.

Case I

R = Rmax = 2A

i.e. R is max when Cos 2p

(

n_{1} - n_{2}

)

t = ± 1

i.e. when 2p

t = kp

(k = 0, 1, 2…)

\ (n_{1} - n_{2} ) t = k

or when t =

k

i.e. when t = 0,

1

,

, ....

n_{1} - n_{2}`

Case II

R = Rmin = 0

i.e. R is min. when cos 2p

t = 0

i.e. when, 2p

t = (k +

p, k = 0, 1, 2....

Thus, (n_{1} - n_{2}) t = k +

3

5

, ...

2(n_{1} - n_{2})

\ Period of beats =

and frequency of beats = |(n_{1} - n_{2})|

Uses of Beats

Doppler Effect

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 Doppler Effect.

Applications