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Question #42886

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The Question

A rigid pipe of length $L$ , closed at both ends, contains an ideal gas of adiabatic index $\gamma$ . The pipe is accelerated uniformly with a constant acceleration $a$ along its longitudinal axis. The gas is maintained at a constant uniform temperature $T$ such that the local speed of sound in the unaccelerated gas is $c = \sqrt{\gamma R T / M}$ . Assuming the gas is in equilibrium in the accelerating frame before any acoustic waves are excited, what is the fundamental frequency $f_1$ of the longitudinal acoustic standing waves in the pipe? (Assume the amplitude of oscillations is small and neglect any dissipative effects).

Options

A.
$\frac{c}{2L}$
B.
$\frac{c}{2\pi} \sqrt{ \frac{\pi^2}{L^2} + \frac{\gamma^2 a^2}{4 c^4} }$
C.
$\frac{c}{2\pi} \sqrt{ \frac{\pi^2}{L^2} - \frac{\gamma^2 a^2}{4 c^4} }$
D.
$\frac{c}{2L} \left( 1 + \frac{\gamma a L}{2 c^2} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Harmonic Content in Pipes|Frame of Reference Inertial vs Non-Inertial|Constraint Relations

Why This Question Is Tricky

Common Mistakes

Students incorrectly assume that because the local speed of sound $c = \sqrt{\gamma RT/M}$ remains constant in an isothermal gas, the standard wave equation and harmonic frequencies $f = c/2L$ apply unchanged.

They fail to account for the non-inertial frame's pseudo-force, which creates an equilibrium pressure gradient.

When deriving the wave equation, this gradient couples with the acoustic displacement, introducing a first-order spatial derivative term (a 'damping' term in space).

This modifies the Sturm-Liouville eigenvalue problem, shifting the fundamental frequency.

The trap tests the transition from elementary algebraic formulas to rigorous continuum mechanics in non-inertial frames.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Harmonic Content in Pipes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free