Sound waves are passing through two routes—on | vedclass

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(A) The path length of the straight route is $2r$ (the diameter of the semicircle).The path length of the semicircular route is $\pi r$.The path diffe

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$\frac{v}{r(\pi-2)}$

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(A) The path length of the straight route is $2r$ (the diameter of the semicircle).The path length of the semicircular route is $\pi r$.The path difference $\Delta x$ between the two routes is $\Delta x = \pi r - 2r = r(\pi - 2)$.For maximum amplitude (constructive interference), the path difference must be an integral multiple of the wavelength $\lambda$, so $\Delta x = n\lambda$, where $n = 1, 2, 3, \dots$.Substituting the path difference, we get $r(\pi - 2) = n\lambda$.Since the velocity of sound is $v = f\lambda$, we have $\lambda = \frac{v}{f}$.Substituting this into the equation: $r(\pi - 2) = n \frac{v}{f}$.Rearranging for frequency $f$: $f = n \left[ \frac{v}{r(\pi - 2)} \right]$.Thus, the frequencies of the resultant waves of maximum amplitude are integral multiples of $\frac{v}{r(\pi - 2)}$.

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