Discussion
List-I shows four configurations made of straight and semi-circular narrow tubes containing air. A sound wave of wavelength \(\lambda = 0.29\) m enters these structures at the point \(S\) and a sound detector is placed at \(D\). Between the points \(S\) and \(D\), the sound travels only through the tubes. List-II contains the possible smallest values of \(l\) (refer to the figures) for which the detector \(D\) records maximum amplitude. Ignore effects of sharp corners. [Given \(\cos(15^{\circ}) = 0.97\)]
Choose the option that best describes the match between the entries in List-I to those in List-II.
| List-I | List-II |
|---|---|
 | (1) 1.32 m (2) 1.19 m (3) 0.51 m (4) 0.29 m (5) 0.13 m |
Answer: D
The correct match is P → 3, Q → 4, R → 5, S → 2.
The detector hears maximum amplitude when the two routes differ in length by a whole number of wavelengths. The smallest \(l\) uses the difference equal to one \(\lambda = 0.29\) m.
(P) The semicircular route has length \(\frac{\pi l}{2}\) against \(l\): \((\frac{\pi}{2} - 1)l = 0.571\,l = 0.29\), so \(l = 0.51\) m (3).
(Q) The rectangular route is \(0.5l + l + 0.5l = 2l\) against \(l\): \(l = 0.29\) m (4).
(R) The route is a side \(l\) plus a semicircle on the diagonal \(\sqrt{2}\,l\), that is \(l + \frac{\pi\sqrt{2}}{2}l = 3.22\,l\). The difference is \(2.22\,l = 0.29\), so \(l = 0.13\) m (5).
(S) In the triangle the angles are 45°, 105° and 30°. By the sine rule the two sides are \(\frac{l\sin 30^{\circ}}{\sin 105^{\circ}}\) and \(\frac{l\sin 45^{\circ}}{\sin 105^{\circ}}\), with \(\sin 105^{\circ} = \cos 15^{\circ} = 0.97\). The route is \(1.244\,l\), the difference is \(0.244\,l = 0.29\), and \(l = 1.19\) m (2).
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