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| Analytical Treatment of Interference of Waves |
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| Let two similar waves of amplitudes A1 and A2 having same frequency and wavelength travel past a point in a medium, producing individual displacements y1 and y2. |
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| Then, |
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| Where f is the phase difference between the waves at the point under consideration. Then, according to the principle of superposition, |
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| y = y1 + y2 |
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| Expanding the 2nd term in the above expression using the formula |
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| sin (A + B) = sin A cos B + cos A sin B, we get, |
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| The above equation represents a simple harmonic wave of wavelength l, velocity v and amplitude R. It differs in phase with respect to the first wave by an angle q, which can be found by referring to figure. |
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| In this figure, |
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| In the same figure, applying Pythagoras theorem, |
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| Case (i) When f = 0 |
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| When the two waves are in phase, the resultant amplitude is given by |
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| The maximum value of cos q is 1. |
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| Thus, R will be maximum when f = 0 |
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| Two waves are said to interfere constructively or reinforce each other, if the resultant amplitude is maximum. The condition for constructive interference is that the phase difference |
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| A phase difference of 2p corresponds to a path difference l. |
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| Let the phase difference f correspond to a path difference d. |
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| Thus, two waves will undergo constructive interference at all points at which the path difference is an even multiple of half a wavelength. |
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| Two waves are said to interfere destructively, if the resultant amplitude is minimum. This happens if |
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| Thus, the condition for destructive interference is that the path difference should be an odd multiple of half a wavelength. |
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