Light waves from two coherent sources superimpose at a point. The waves, at this point, can be expressed as `y_(1) = a sin [10^(15) pi t]` and `y_(2)a sin [10^(15) pi t + phi]`. Find the resultant amplitude if phase difference `phi` is (a) zero (b) `pi//3` (c) `pi` Also find the frequency (Hz) of resultant wave in each case.
Question
Light waves from two coherent sources superimpose at a point. The waves, at this point, can be expressed as y_(1) = a sin [10^(15) pi t] and y_(2)a sin [10^(15) pi t + phi]. Find the resultant amplitude if phase difference phi is (a) zero (b) pi//3 (c) pi Also find the frequency (Hz) of resultant wave in each case.
Solution
The resultant wave is given by the principle of superposition, which states that the displacement of the resultant wave is the vector sum of the displacements of the individual waves.
(a) If the phase difference phi is zero, the two waves are in phase and the resultant amplitude is the sum of the individual amplitudes. Therefore, the resultant amplitude is 2a. The frequency of the resultant wave is the same as the frequency of the individual waves, which is 10^(15) Hz.
(b) If the phase difference phi is pi/3, the resultant amplitude is given by the formula R = sqrt[(a^2) + (a^2) + 2*a*a*cos(phi)]. Substituting phi = pi/3 and a = a into the formula, we get R = a*sqrt[3]. The frequency of the resultant wave is the same as the frequency of the individual waves, which is 10^(15) Hz.
(c) If the phase difference phi is pi, the two waves are out of phase and the resultant amplitude is the difference of the individual amplitudes. Therefore, the resultant amplitude is 0. The frequency of the resultant wave is the same as the frequency of the individual waves, which is 10^(15) Hz.
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