---
title: "A string of length \\(L\\) fixed at both ends is driven by a mechanical vibrator operating at a constant frequency \\(f\\). Under an initial string tension \\(T_1\\), a standing wave with four antinodes is established, as shown in the diagram. The tension is then gradually altered to a new value \\(T_2\\) without changing the frequency \\(f\\) or length \\(L\\), causing the string to shift into a standing wave pattern with two antinodes. The wave speed on a string of linear mass density \\(\\mu\\) under tension \\(T\\) is given by \\(v = \\sqrt{\\dfrac{T}{\\mu}}\\). What is the ratio \\(\\dfrac{T_2}{T_1}\\) of the new tension to the initial tension?"
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url: "https://nerd-notes.com/ubq/117439/"
date_modified: "2026-08-04T06:52:10+00:00"
---

# A string of length \(L\) fixed at both ends is driven by a mechanical vibrator operating at a constant frequency \(f\). Under an initial string tension \(T_1\), a standing wave with four antinodes is established, as shown in the diagram. The tension is then gradually altered to a new value \(T_2\) without changing the frequency \(f\) or length \(L\), causing the string to shift into a standing wave pattern with two antinodes. The wave speed on a string of linear mass density \(\mu\) under tension \(T\) is given by \(v = \sqrt{\dfrac{T}{\mu}}\). What is the ratio \(\dfrac{T_2}{T_1}\) of the new tension to the initial tension?

A string of length \(L\) fixed at both ends is driven by a mechanical vibrator operating at a constant frequency \(f\). Under an initial string tension \(T_1\), a standing wave with four antinodes is established, as shown in the diagram. The tension is then gradually altered to a new value \(T_2\) without changing the frequency \(f\) or length \(L\), causing the string to shift into a standing wave pattern with two antinodes. The wave speed on a string of linear mass density \(\mu\) under tension \(T\) is given by \(v = \sqrt{\dfrac{T}{\mu}}\). What is the ratio \(\dfrac{T_2}{T_1}\) of the new tension to the initial tension?

![A horizontal string of length L is fixed between two vertical rigid supports located at the left and right ends. The string forms a standing wave pattern consisting of four identical, evenly spaced loops between the supports, with fixed node points at both ends and three intermediate nodes along the string. A double-headed vertical arrow at the center of each loop indicates the peak-to-peak amplitude of vibration. A horizontal dimension line below the string spans the entire distance between the supports and is labeled with the length L. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785826330-N5PQcY.jpg)

- **A.** \(\dfrac{1}{2}\)
- **B.** \(4\)
- **C.** \(8\)
- **D.** \(16\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117439/*