MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Wave Motion Page 2

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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 n1, and n2 can be represented by the equations,

 y1 = A sin 2pn1t and 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) t ] Cos [ 2 p (n1 - n2) t ] 2 2

\ y = R sin 2p nt

 where, R = 2 A cos 2 p (n1 - n2) t and n = n1 + n2 2 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 ( n1 - n2 ) t = ± 1 2

 i.e. when 2p ( n1 - n2 ) t = kp (k = 0, 1, 2…) 2

 \ (n1 - n2 ) t = k

 or when t = k i.e. when t = 0, 1 , 2 , .... n1 - n2 n1 - n2 n1 - n2`

Case II

R = Rmin = 0

 i.e. R is min. when cos 2p ( n1 - n2 ) t = 0 2

 i.e. when, 2p ( n1 - n2 ) t = (k + 1 ) p, k = 0, 1, 2.... 2 2

 Thus, (n1 - n2) t = k + 1 2

 or when t = 1 , 3 , 5 , ... 2(n1 - n2) 2(n1 - n2) 2(n1 - n2)

 Time interval between two maxima = Time interval between two minima = 1 (n1 - n2)

 \ Period of beats = 1 (n1 - n2)

and frequency of beats = |(
n1 - n2)|

 Uses of Beats Determination of an unknown frequency. Tuning of musical instruments. Production of sound notes of very low frequency. 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 Radar - Determination of the speed of a moving object like a vehicle. In astronomy - To determine the speed of distant astronomical subjects like stars or galaxies.

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