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Chapter 15: Problem 40

15.40\(\cdot\) A 1.50 -m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 \(\mathrm{m} / \mathrm{s}\) . What are the wavelength and frequency of (a) the fundamental: (b) the second overtone; (c) the fourth harmonic?

Short Answer

Expert verified
(a) Fundamental: wavelength 3.00 m, frequency 16.0 Hz. (b) Second overtone: wavelength 1.00 m, frequency 48.0 Hz. (c) Fourth harmonic: wavelength 0.75 m, frequency 64.0 Hz.

Step by step solution

01

Understanding Wave Properties

The speed of a wave on a string is given by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( v \) is the speed of the wave, \( T \) is the tension, and \( \mu \) is the linear mass density. For a string fixed at both ends, the wavelength \( \lambda \) and frequency \( f \) of transverse waves are related by the wave speed: \( v = f \cdot \lambda \). We are given the wave speed is 48.0 m/s and the length of the rope is 1.50 m.
02

Fundamental Frequency (First Harmonic)

For the fundamental frequency, the wavelength \( \lambda_1 \) is twice the length of the rope: \( \lambda_1 = 2 \times 1.50 \text{ m} = 3.00 \text{ m} \). Using the wave speed and wavelength, the frequency \( f_1 \) can be calculated as \( f_1 = \frac{v}{\lambda_1} = \frac{48.0 \text{ m/s}}{3.00 \text{ m}} = 16.0 \text{ Hz} \).
03

Second Overtone Frequency (Third Harmonic)

The second overtone corresponds to the third harmonic, where three half-wavelengths fit into the length of the rope: \( 3 \times \frac{\lambda_3}{2} = 1.50 \text{ m} \). Solving for \( \lambda_3 \), we get \( \lambda_3 = 1.00 \text{ m} \). The frequency \( f_3 \) is then \( f_3 = \frac{v}{\lambda_3} = \frac{48.0 \text{ m/s}}{1.00 \text{ m}} = 48.0 \text{ Hz} \).
04

Fourth Harmonic Frequency

The fourth harmonic, where four half-wavelengths fit into the length of the rope, gives: \( 4 \times \frac{\lambda_4}{2} = 1.50 \text{ m} \). Solving for \( \lambda_4 \), we find \( \lambda_4 = 0.75 \text{ m} \). The frequency \( f_4 \) is calculated as \( f_4 = \frac{v}{\lambda_4} = \frac{48.0 \text{ m/s}}{0.75 \text{ m}} = 64.0 \text{ Hz} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Transverse Waves
Transverse waves are a type of wave where the motion of the medium is perpendicular to the direction of the wave travel. They occur in a variety of physical contexts, including the waves observed on a stretched string or rope, as in the given exercise. When you pluck a guitar string, the wave travels across the string, but the string itself vibrates up and down.
Transverse waves are characterized by properties such as amplitude, wavelength, frequency, and speed:
  • **Amplitude**: The maximum distance from the rest position that particles of the medium move when a wave passes.
  • **Wavelength** (\( \lambda \)): The distance between two consecutive similar points of the wave, like crest to crest or trough to trough.
  • **Frequency** (\( f \)): The number of complete wave cycles passing a point per unit time, usually per second.
  • **Wave Speed** (\( v \)): The speed at which the wave propagates through the medium, given by \( v = f \cdot \lambda \).
In the exercise, knowing that transverse waves travel at 48 m/s on the 1.50 m rope helps to find various harmonic frequencies and wavelengths.
Harmonic Frequencies
Harmonic frequencies are specific frequencies at which a medium naturally resonates. They result from standing waves forming along the medium, like the rope in the exercise. These frequencies are integral multiples of the fundamental frequency.
  • **Fundamental Frequency (First Harmonic)**: This is the lowest frequency at which the mode of vibration occurs. For the rope, the first harmonic wavelength is twice the rope length, resulting in a frequency of 16 Hz.
  • **Second Overtone (Third Harmonic)**: This involves three half-wavelengths fitting into the rope length. The third harmonic hence corresponds to triple the fundamental frequency, calculated to be 48 Hz.
  • **Fourth Harmonic**: In this case, four half-wavelengths are accommodated in the rope's length. The frequency is four times that of the fundamental frequency, resulting in 64 Hz.
Each harmonic is an integer multiple of the base, allowing the waves to reinforce at certain points creating a standing wave pattern, which is integral to musical acoustics.
Wavelength Calculation
Wavelength calculation for waves on a rope or string is integral to determining harmonic frequencies. It requires understanding the relationship between wavelength, frequency, and wave speed. The relationship is given by \( v = f \cdot \lambda \), where \( v \) is the speed, \( f \) is the frequency, and \( \lambda \) is the wavelength.To calculate the wavelength for a specific harmonic frequency:
  • Identify the harmonic number. The first harmonic has one antinode and a single loop that is half a wavelength. The third harmonic (second overtone) includes three loops (one-and-a-half wavelengths), and the fourth harmonic contains two complete wavelengths on the rope.
  • Use the media's properties: Given the rope's speed of 48 m/s and its length of 1.50 m, the equations illustrate how many antinodes fit within the medium length, guiding the \( \lambda \) calculation.
  • For practical problems, matching these calculated wavelengths to real-world phenomena explains the sound produced by string instruments or the patterns seen in vibrational modes.
Efficiently applying these calculations can help make precise predictions about wave behavior and assist in designing systems that use wave characteristics effectively.

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Most popular questions from this chapter

Combining Standing Waves. A guitar string of length \(L\) is plucked in such a way that the total wave produced is the sum of the fundamental and the second harmonic. That is, the standing wave is given by $$y(x, t)=y_{1}(x, t)+y_{2}(x, t)$$ where $$\begin{aligned} y_{1}(x, t) &=C \sin \omega_{1} t \sin k_{1} x \\ y_{2}(x, t) &=C \sin \omega_{2} t \sin k_{2} x \end{aligned}$$ with \(\omega_{1}=v k_{1}\) and \(\omega_{2}=v k_{2}\) . (a) At what values of \(x\) are the nodes of \(y_{1} ?\left(\) b) At what values of \(x\) are the nodes of \(y_{2} ?(\mathrm{c})\) Graph \right the total wave at \(t=0, t=\frac{1}{8} f_{1}, t=\frac{1}{4} f_{1}, t=\frac{3}{8} f_{1},\) and \(t=\frac{1}{2} f_{1}\) . (d) Does the sum of the two standing waves \(y_{1}\) and \(y_{2}\) produce a standing wave? Explain.

A thin, 75.0 -cm wire has a mass of 16.5 \(\mathrm{g} .\) One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire. (a) To what tension (in newtons) must you adjust the screw so that a transverse wave of wavelength 3.33 \(\mathrm{cm}\) makes 875 vibrations per second? (b) How fast would this wave travel?

A certain transverse wave is described by $$y(x, t)=(6.50 \mathrm{mm}) \cos 2 \pi\left(\frac{x}{28.0 \mathrm{cm}}-\frac{t}{0.0360 \mathrm{s}}\right)$$ Determine the wave's (a) amplitude; (b) wavelength; (c) frequency; (d) speed of propagation; (e) direction of propagation.

Three pieces of string, each of length \(L,\) are joined together end to end, to make a combined string of length 3\(L .\) The first piece of string has mass per unit length \(\mu_{1},\) the second piecehas mass per unit length \(\mu_{2}=4 \mu_{1},\) and the third piece has mass per unit length \(\mu_{3}=\mu_{1} / 4 .(a)\) If the combined string is under tension \(F,\) how much time does it take a transverse wave to travel the entire length 3\(L ?\) Give your answer in terms of \(L, F,\) and \(\mu_{1}\) . (b) Does your answer to part (a) depend on the order in which the three pieces are joined together? Explain.

Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are 7.5 \(\mathrm{m}\) from it, you measure its intensity to be 0.11 \(\mathrm{W} / \mathrm{m}^{2}\) . An intensity of 1.0 \(\mathrm{W} / \mathrm{m}^{2}\) is often used as the "threshold of pain." How much closer to the source can you move before the sound intensity reaches this threshold?
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