Figure shows a closed pipe (closed at both ends) of length 10 m. MN is a membrane (thin sheet) capable of doing simple harmonic motion of amplitude 1 mm along left and right from shown situation. On two sides of membrane are two gases $G_{1}$ and $G_{2}$ in which sound can travel at 300 m/s and 100 m/s respectively. Mark the INCORRECT statement(s). [For options C, D take maximum amplitude of standing wave patterns to be 1 mm]

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If resonance is obtained in right side $(G_{2}),$ side $(G_{1})$ might not resonate.

For angular frequency $75 π r a d / s,$ a particle right of AA’ would be oscillating with maximum displacement 0.5 mm.

For angular frequency $75 π r a d / s,$ a particle $\frac{2}{3} m$ right of AA’ would be oscillating with maximum displacement $\frac{\sqrt{3}}{2} m m$ .

Resonance in both gases will be possible for some specific frequencies only.

answer is D.

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Detailed Solution

$\begin{array}{l} V_{1} = 300 m / s \\ V_{2} = 100 m / s \\ f_{1} = f_{2} \end{array}$

$\begin{array}{l} λ_{1} : λ_{2} = V_{1} : V_{2} = 3 : 1 \\ ω = 75 π r a d / \sec \\ V = 300 m / s \\ λ = 8 m \\ A = A_{x} \sin k x \\ = 1 \sin \frac{2 π}{8} \times \frac{2}{3} = 0.5 m m \end{array}$