---
title: "A cylindrical pipe of fixed length \\(L\\) is closed at one end and open at the other. An audio speaker positioned at the open end emits sound at a constant frequency \\(f\\), which establishes a standing wave corresponding to the third harmonic (first overtone) in the gas inside the pipe at initial absolute temperature \\(T_1\\). The gas is then uniformly heated to a new absolute temperature \\(T_2\\) without changing its composition or the pipe length. If the same speaker emitting the same frequency \\(f\\) now excites the fundamental standing wave mode of the pipe, by what factor must the absolute temperature have changed (\\(T_2/T_1\\)), and by what factor did the wavelength of the sound wave inside the pipe change (\\(\\lambda_2/\\lambda_1\\))?"
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url: "https://nerd-notes.com/ubq/123206/"
date_modified: "2026-09-28T11:57:54+00:00"
---

# A cylindrical pipe of fixed length \(L\) is closed at one end and open at the other. An audio speaker positioned at the open end emits sound at a constant frequency \(f\), which establishes a standing wave corresponding to the third harmonic (first overtone) in the gas inside the pipe at initial absolute temperature \(T_1\). The gas is then uniformly heated to a new absolute temperature \(T_2\) without changing its composition or the pipe length. If the same speaker emitting the same frequency \(f\) now excites the fundamental standing wave mode of the pipe, by what factor must the absolute temperature have changed (\(T_2/T_1\)), and by what factor did the wavelength of the sound wave inside the pipe change (\(\lambda_2/\lambda_1\))?

A cylindrical pipe of fixed length \(L\) is closed at one end and open at the other. An audio speaker positioned at the open end emits sound at a constant frequency \(f\), which establishes a standing wave corresponding to the third harmonic (first overtone) in the gas inside the pipe at initial absolute temperature \(T_1\). The gas is then uniformly heated to a new absolute temperature \(T_2\) without changing its composition or the pipe length. If the same speaker emitting the same frequency \(f\) now excites the fundamental standing wave mode of the pipe, by what factor must the absolute temperature have changed (\(T_2/T_1\)), and by what factor did the wavelength of the sound wave inside the pipe change (\(\lambda_2/\lambda_1\))?

![A horizontal cylindrical tube of length \(L\) is oriented with its open end on the left and a flat closed boundary on the right. Just to the left of the open end, a small speaker cone faces into the tube opening. Inside the tube, two dashed sinusoidal curves form a standing wave envelope with zero displacement at the right wall, a displacement antinode at the left opening, and two interior zero-crossing nodes. Beneath the tube, a horizontal dimension arrow extends from the open left end to the closed right boundary and is labeled \(L\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790596674-UW4xHZ.jpg)

- **A.** \(T_2/T_1 = \dfrac{1}{9}\) and \(\lambda_2/\lambda_1 = \dfrac{1}{3}\)
- **B.** \(T_2/T_1 = 3\) and \(\lambda_2/\lambda_1 = 3\)
- **C.** \(T_2/T_1 = 9\) and \(\lambda_2/\lambda_1 = 1\)
- **D.** \(T_2/T_1 = 9\) and \(\lambda_2/\lambda_1 = 3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123206/*