The air column in a pipe closed at one end is made to vibrate in its second overtone by a tuning fork of frequency 440 Hz. The speed of sound in air is 330 m s⁻¹. End corrections may be neglected. Let P₀ denote the mean pressure at any point in the pipe and ΔP₀ the maximum amplitude of pressure variation. (a) Find the length L of the air column. (b) What is the amplitude of pressure variation at the middle of the column? (c) What are the maximum and minimum pressure at the open end of the pipe? (d) What are the maximum and minimum pressure at the closed end of the pipe?
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Question

The air column in a pipe closed at one end is made to vibrate in its second overtone by a tuning fork of frequency 440 Hz. The speed of sound in air is 330 m s⁻¹. End corrections may be neglected. Let P₀ denote the mean pressure at any point in the pipe and ΔP₀ the maximum amplitude of pressure variation. (a) Find the length L of the air column. (b) What is the amplitude of pressure variation at the middle of the column? (c) What are the maximum and minimum pressure at the open end of the pipe? (d) What are the maximum and minimum pressure at the closed end of the pipe?

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G1 Physics · Accepted Answer

Step 1: Use v = √(B/ρ) in a fluid, v = √(Y/ρ) for longitudinal waves in a solid, or v = √(γRT/M) for an ideal gas. Step 2: Form any required ratio before substituting values. Step 3: Evaluate with the data given. Final: C: (a) 15/16 m; (b) ΔP/2; (c) equal to mean pressure; (d) P₀ + ΔP₀ and P₀ − ΔP₀.