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Waves reaching a point from two different paths from two different sources interfere. If the path lengths differ by integral multiples of a wavelength the interference will be constructive and if the path lengths differ by odd multiples of half wavelengths interference will be destructive. In the given situation waves travel from A to B along path 1 and path 2. If the frequency of the waves is $$n$$ and the velocity of the waves is $$v$$, find the least difference between length of paths 2 and 1 that produces constructive interference at B. If the velocity along path 1 is $$v_1$$ and along path 2 is $$v_2$$, find the least difference between length of paths 2 and 1 that produces destructive interference at B.
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Correct Answer: 0
PLACEHOLDER ANSWER - the question has two answers. With one speed, the wavelength is $$\lambda = \frac{v}{n}$$, and the least non-zero path difference for constructive interference is one wavelength, $$L_2 - L_1 = \frac{v}{n}$$. With different speeds, what matters is the number of wavelengths along each path, $$\frac{nL_1}{v_1}$$ and $$\frac{nL_2}{v_2}$$, and destructive interference needs these to differ by $$\frac{1}{2}$$ at least. So $$\frac{L_2}{v_2} - \frac{L_1}{v_1} = \frac{1}{2n}$$, i.e. $$L_2 - L_1 = L_1\left(\frac{v_2}{v_1} - 1\right) + \frac{v_2}{2n}$$, which reduces to $$\frac{v}{2n}$$ when $$v_1 = v_2 = v$$.
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