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Chapter 15: Problem 28

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is \(y(x, t) = 2.30 \, \mathrm{mm} \, \mathrm{cos} [(6.98 \, \mathrm{rad/m})x \space + 1742 \, \mathrm{rad/s})t]\). Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

Short Answer

Expert verified
(a) 2.30 mm, (b) 277.3 Hz, (c) 0.9003 m, (d) 249.78 m/s, (e) negative x-direction, (f) 156.1 N, (g) 14.0 W.

Step by step solution

01

Identify the Amplitude

The amplitude of a wave is the coefficient in front of the cosine function in the wave equation. In the given wave function, the amplitude is 2.30 mm. Therefore, the amplitude is \( A = 2.30 \, \text{mm} \).
02

Calculate the Frequency

The angular frequency \( \omega \) is given by the term \( 1742 \, \text{rad/s} \) in the wave function. The frequency \( f \) can be found using the formula \( f = \frac{\omega}{2\pi} \). Therefore, \( f = \frac{1742}{2\pi} \approx 277.3 \, \text{Hz} \).
03

Determine the Wavelength

The wave number \( k \) is given by \( 6.98 \, \text{rad/m} \). The wavelength \( \lambda \) is related to the wave number by \( \lambda = \frac{2\pi}{k} \). Thus, \( \lambda = \frac{2\pi}{6.98} \approx 0.9003 \, \text{m} \).
04

Calculate the Wave Speed

The wave speed \( v \) can be calculated using the formula \( v = f\lambda \). Given \( f \approx 277.3 \, \text{Hz} \) and \( \lambda \approx 0.9003 \, \text{m} \), the wave speed is \( v = 277.3 \times 0.9003 \approx 249.78 \, \text{m/s} \).
05

Identify the Direction of the Wave

The phase factor in the wave function includes \( +kx+\omega t \); the presence of \(+\omega t\) indicates the wave is traveling in the negative x-direction.
06

Calculate the Tension in the Rope

The wave speed on a rope is given by \( v = \sqrt{\frac{T}{\mu}} \), where \( \mu \) is the linear mass density \( \frac{m}{L} \). With \( m = 0.00338 \, \text{kg} \) and \( L = 1.35 \, \text{m} \), \( \mu = \frac{0.00338}{1.35} \approx 0.002504 \, \text{kg/m} \). Then, solving \( v = 249.78 = \sqrt{\frac{T}{0.002504}} \) gives \( T \approx \mu v^2 \approx 156.1 \, \text{N} \).
07

Determine the Average Power Transmitted by the Wave

The average power \( P \) transmitted by a wave traveling along a string is given by \( P = \frac{1}{2} \mu \omega^2 A^2 v \). Using \( \mu \approx 0.002504 \, \text{kg/m} \), \( \omega = 1742 \, \text{rad/s} \), \( A = 0.0023 \, \text{m} \), and \( v \approx 249.78 \, \text{m/s} \), we compute \( P = \frac{1}{2} (0.002504)(1742^2)(0.0023)^2(249.78) \approx 14.0 \, \text{W} \).

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Key Concepts

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

Wave Function
A wave function is a mathematical representation of a wave. It describes how the wave travels through space and time. In this case, the wave function provided is \( y(x, t) = 2.30 \, \text{mm} \, \cos [(6.98 \, \text{rad/m})x + 1742 \, \text{rad/s} \)\(t)] \).
The wave function typically includes parameters like amplitude, angular wave number \( k \), and angular frequency \( \omega \).
Understanding each component in the wave function helps in analyzing the wave's various properties, such as its direction and speed. For instance, the cosine function indicates this is a periodic wave, which means it repeats itself at regular intervals.
  • The coefficient \( 2.30 \, \text{mm} \) is the amplitude, representing the maximum displacement from rest.
  • The term \( 6.98 \, \text{rad/m} \) is the wave number, which relates to the wavelength.
  • The \( 1742 \, \text{rad/s} \) term is the angular frequency, relating to the frequency.
Amplitude
Amplitude is a fundamental characteristic of waves. It represents the maximum extent of a wave's displacement from its equilibrium position.
In simple terms, it is how far the wave "reaches" or "grows" from its baseline up to the top-most point known as the wave crest.
In the wave equation \( y(x, t) = 2.30 \, \text{mm} \, \cos [(6.98 \, \text{rad/m})x + 1742 \, \text{rad/s})t] \), the amplitude is given by the coefficient in front, which is \( 2.30 \, \text{mm} \).
  • This parameter indicates the energy of the wave; a larger amplitude means the wave carries more energy.
  • Amplitude does not affect the speed of the wave; it solely impacts the energy level and intensity.
Wave Speed
Wave speed is a critical concept in wave motion, defining how fast a wave propagates through a medium.
You determine wave speed by multiplying the frequency of the wave \( f \) by its wavelength \( \lambda \).
Mathematically expressed as \( v = f\lambda \). For the given wave, with frequency \( 277.3 \, \text{Hz} \) and wavelength \( 0.9003 \, \text{m} \), the wave speed is
  • \( 249.78 \, \text{m/s} \).
The speed of a wave depends on the medium through which it travels.
For instance, waves move faster in less dense materials and slower in denser ones. Understanding wave speed is vital for numerous practical applications, from engineering to communication.
Frequency
Frequency refers to how often the wave oscillates or completes a cycle in one second.
It is measured in Hertz (Hz), where one Hertz equals one cycle per second.
From the angular frequency \( \omega \) provided in the wave equation, \( \omega = 1742 \, \text{rad/s} \), you can determine the frequency using the formula:
  • \( f = \frac{\omega}{2\pi} \).
For this example, the frequency is approximately \( 277.3 \, \text{Hz} \).
Frequency is inversely related to the period of the wave, which is the time taken to complete one cycle.
  • Higher frequencies imply shorter periods and vice versa.
A higher frequency indicates more cycles per unit time, which means more energy in the wave.
Wavelength
Wavelength is the distance between successive crests (or any successive identical points) of a wave.
It is usually denoted by \( \lambda \) and is measured in meters (m).
Using the wave equation, the wave number \( k = 6.98 \, \text{rad/m} \) helps find the wavelength with the equation:
  • \( \lambda = \frac{2\pi}{k} \).
For our wave, this computes to about \( 0.9003 \, \text{m} \). Wavelength directly affects the wave's speed along with frequency.
It is an essential element in determining how waves interact with each other and their surrounding environment.
  • In optics, for instance, wavelengths determine the color of light.
  • In sound, they influence pitch.

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Most popular questions from this chapter

The speed of sound in air at 20\(^\circ\)C is 344 m/s. (a) What is the wavelength of a sound wave with a frequency of 784 Hz, corresponding to the note G\(_5\) on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave one octave higher (twice the frequency) than the note in part (a)?

Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and wavelength 0.320 m. The waves travel in the \(-x\)-direction, and at \(t = 0\) the \(x = 0\) end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write a wave function describing the wave. (c) Find the transverse displacement of a particle at \(x = 0.360\) m at time \(t = 0.150\) s. (d) How much time must elapse from the instant in part (c) until the particle at \(x = 0.360\) m next has maximum upward displacement?

One string of a certain musical instrument is 75.0 cm long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 m/s. (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength 0.765 m? (Assume that the breaking stress of the wire is very large and isn't exceeded.) (b) What frequency sound does this string produce in its fundamental mode of vibration?

A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation \(y(x, t) = (5.60 \, \mathrm{cm}) \mathrm{sin} [(0.0340 \mathrm{rad/cm})x] \mathrm{sin} [(150.0 \, \mathrm{rad/s})t]\), where the origin is at the left end of the string, the x-axis is along the string, and the \(y\)-axis is perpendicular to the string. (a) Draw a sketch that shows the standing-wave pattern. (b) Find the amplitude of the two traveling waves that make up this standing wave. (c) What is the length of the string? (d) Find the wavelength, frequency, period, and speed of the traveling waves. (e) Find the maximum transverse speed of a point on the string. (f) What would be the equation \(y(x,t)\) for this string if it were vibrating in its eighth harmonic?

A thin, 75.0-cm wire has a mass of 16.5 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 cm makes 625 vibrations per second? (b) How fast would this wave travel?
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