Question 18

Two coherent sources of sound, $$S_1$$ and $$S_2$$, produce sound waves of the same wavelength $$\lambda = 1\,\text{m}$$ are in phase. $$S_1$$ and $$S_2$$ are placed $$1.5\,\text{m}$$ apart (see fig). A listener, located at L, directly in front of $$S_2$$, finds that the intensity is at a minimum when he is $$2\,\text{m}$$ away from $$S_2$$. The listener moves away from $$S_1$$, keeping the distance from $$S_2$$ fixed. The adjacent maximum of intensity is observed when the listener is at a distance $$d$$ from $$S_1$$. Then $$d$$ is:

$$\text{Initial path length from } S_1 \text{ at the minimum point:}$$

$$S_1L_{\text{initial}} = \sqrt{1.5^2 + 2^2} = 2.5\ \text{m}$$

$$\Delta x_{\text{initial}} = S_1L_{\text{initial}} - S_2L = 2.5 - 2 = 0.5\ \text{m} = \frac{\lambda}{2}$$

$$\text{Since } \Delta x \text{ increases as the listener moves along the arc away from } S_1\text{:}$$

$$\text{Adjacent maximum condition:} \quad \Delta x_{\text{final}} = \lambda = 1\ \text{m}$$

$$d - S_2L = 1 \implies d - 2 = 1 \implies d = 3\ \text{m}$$

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