---
title: "A horizontal rope of length \\(L\\) is held under uniform tension \\(F_T\\). The linear mass density of the rope increases linearly from \\(\\mu_0\\) at \\(x = 0\\) to \\(3\\mu_0\\) at \\(x = L\\). A continuous wave generator at \\(x = 0\\) drives the rope, sending waves toward \\(x = L\\). To estimate the transit time \\(\\Delta t\\) of a wave pulse along the rope, a student uses the wave speed calculated at the rope’s average linear mass density \\(\\mu_{\\text{avg}}\\). Which choice correctly states the expression for the estimated transit time \\(\\Delta t\\) and describes the behavior of the wave frequency \\(f\\) as the wave travels toward \\(x = L\\)?"
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url: "https://nerd-notes.com/ubq/116744/"
date_modified: "2026-08-04T05:25:50+00:00"
---

# A horizontal rope of length \(L\) is held under uniform tension \(F_T\). The linear mass density of the rope increases linearly from \(\mu_0\) at \(x = 0\) to \(3\mu_0\) at \(x = L\). A continuous wave generator at \(x = 0\) drives the rope, sending waves toward \(x = L\). To estimate the transit time \(\Delta t\) of a wave pulse along the rope, a student uses the wave speed calculated at the rope’s average linear mass density \(\mu_{\text{avg}}\). Which choice correctly states the expression for the estimated transit time \(\Delta t\) and describes the behavior of the wave frequency \(f\) as the wave travels toward \(x = L\)?

A horizontal rope of length \(L\) is held under uniform tension \(F_T\). The linear mass density of the rope increases linearly from \(\mu_0\) at \(x = 0\) to \(3\mu_0\) at \(x = L\). A continuous wave generator at \(x = 0\) drives the rope, sending waves toward \(x = L\). To estimate the transit time \(\Delta t\) of a wave pulse along the rope, a student uses the wave speed calculated at the rope's average linear mass density \(\mu_{\text{avg}}\). Which choice correctly states the expression for the estimated transit time \(\Delta t\) and describes the behavior of the wave frequency \(f\) as the wave travels toward \(x = L\)?

![A horizontal rope of length L extends from a wave generator on the left at x = 0 to a fixed support on the right at x = L. The left end of the rope is attached to a vertical mechanical oscillator labeled Source. The rope is drawn with a continuously increasing thickness from left to right, representing increasing linear mass density. At x = 0, the rope has linear mass density label \mu_0. At x = L, the rope has linear mass density label 3\mu_0. A horizontal force arrow pointing to the right at x = L is labeled F_T, indicating uniform tension throughout the rope. A sinusoidal wave pulse is drawn near x = 0 traveling to the right, with a small rightward arrow above it labeled v. A horizontal axis below the rope is labeled x, with tick marks at 0 and L. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785821150-blcteZ.jpg)

- **A.** Estimated transit time: \(L \sqrt{\dfrac{\mu_0}{2F_T}}\); Frequency: Decreases
- **B.** Estimated transit time: \(L \sqrt{\dfrac{2\mu_0}{F_T}}\); Frequency: Remains constant
- **C.** Estimated transit time: \(L \sqrt{\dfrac{2\mu_0}{F_T}}\); Frequency: Decreases
- **D.** Estimated transit time: \(L \sqrt{\dfrac{4\mu_0}{F_T}}\); Frequency: Remains constant

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/116744/*