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Chapter 16: Problem 28

Use the wave equation to find the speed of a wave given by $$ y(x, t)=(3.00 \mathrm{~mm}) \sin \left[\left(4.00 \mathrm{~m}^{-1}\right) x-\left(7.00 \mathrm{~s}^{-1}\right) t\right] . $$

Short Answer

Expert verified
The speed of the wave is 1.75 m/s.

Step by step solution

01

Identify Wave Equation Parameters

The given wave equation is \( y(x, t) = (3.00 \, \text{mm}) \sin \left( \left(4.00 \, \text{m}^{-1}\right) x - \left(7.00 \, \text{s}^{-1}\right) t \right) \). From this equation, we identify the wave number \( k = 4.00 \, \text{m}^{-1} \) and the angular frequency \( \omega = 7.00 \, \text{s}^{-1} \).
02

Apply Wave Speed Formula

The speed \( v \) of a wave is given by the formula \( v = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wave number. You'll use the values obtained in Step 1 to calculate the wave speed.
03

Perform the Calculation

Substitute the values \( \omega = 7.00 \, \text{s}^{-1} \) and \( k = 4.00 \, \text{m}^{-1} \) into the formula: \( v = \frac{\omega}{k} = \frac{7.00}{4.00} \, \text{m/s} = 1.75 \, \text{m/s} \).

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Key Concepts

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

Wave Equation
The wave equation is an essential formula used to describe the motion of waves. It takes the form \( y(x, t) = A \sin(kx - \omega t) \), where:
  • \( y(x, t) \) is the wave function that describes the displacement at position \( x \) and time \( t \).
  • \( A \) is the amplitude, representing the wave's maximum displacement from its rest position.
  • \( k \) is the wave number, indicating how many wavelengths fit into a unit distance.
  • \( \omega \) is the angular frequency, describing how fast the wave oscillates.
Understanding this equation helps differentiate between various parameters that affect a wave's shape and propagation. It provides insights into the behavior of different types of waves, such as sound waves and light waves. The given equation \( y(x, t) = (3.00 \, \text{mm}) \sin \left(\left(4.00 \, \text{m}^{-1}\right) x - \left(7.00 \, \text{s}^{-1}\right) t\right) \) helps us determine key characteristics like amplitude, wave number, and angular frequency.
Angular Frequency
Angular frequency, denoted as \( \omega \), is a crucial concept when examining wave motion. It is defined as the rate at which the wave oscillates per unit time and is measured in radians per second (\( \text{s}^{-1} \)).
  • Angular frequency relates to the wave's speed and its periodic behavior.
  • It determines how many oscillations occur in a given time.
  • It is linked to the ordinary frequency \( f \) by the relation \( \omega = 2\pi f \).
In our given wave equation, the angular frequency is \( 7.00 \, \text{s}^{-1} \), indicating how quickly the wave cycles through its pattern. A higher angular frequency means that more wave cycles occur in the same duration, resulting in a faster oscillating wave.
Wave Number
The wave number \( k \) represents the number of wave cycles per unit distance and is measured in inverse meters (\( \text{m}^{-1} \)). It provides a spatial understanding of the wave's nature.
  • Wave number is linked to the wavelength \( \lambda \) via the formula \( k = \frac{2\pi}{\lambda} \).
  • A higher wave number implies a smaller wavelength, meaning the wave oscillates more frequently over a given distance.
  • Wave number can help determine how waves propagate through space.
In our problem, the wave number is \( 4.00 \, \text{m}^{-1} \), which helps us comprehend how the wave's spatial properties relate to its speed. With both the wave number and angular frequency, we utilized the formula \( v = \frac{\omega}{k} \) to calculate the wave speed, reflecting the wave's propagation characteristics.

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Most popular questions from this chapter

At time \(t=0\) and at position \(x=0 \mathrm{~m}\) along a string, a traveling sinusoidal wave with an angular frequency of \(440 \mathrm{rad} / \mathrm{s}\) has displacement \(y=+4.5 \mathrm{~mm}\) and transverse velocity \(u=-0.75 \mathrm{~m} / \mathrm{s}\). If the wave has the general form \(y(x, t)=y_{m} \sin (k x-\omega t+\phi)\), what is phase constant \(\phi ?\)

A human wave. During sporting events within large, densely packed stadiums, spectators will send a wave (or pulse) around the stadium (Fig. \(16-29)\). As the wave reaches a group of spectators, they stand with a cheer and then sit. At any instant, the width \(w\) of the wave is the distance from the leading edge (people are just about to stand) to the trailing edge (people have just sat down). Suppose a human wave travels a distance of 853 seats around a stadium in \(39 \mathrm{~s}\), with spectators requiring about \(1.8 \mathrm{~s}\) to respond to the wave's passage by standing and then sitting. What are (a) the wave speed \(v\) (in seats per second) and (b) width \(w\) (in number of seats)?

A string under tension \(\tau_{i}\) oscillates in the third harmonic at frequency \(f_{3}\), and the waves on the string have wavelength \(\lambda_{3}\). If the tension is increased to \(\tau_{f}=4 \tau_{i}\) and the string is again made to oscillate in the third harmonic, what then are (a) the frequency of oscillation in terms of \(f_{3}\) and (b) the wavelength of the waves in terms of \(\lambda_{3}\) ?

What phase difference between two identical traveling waves, moving in the same direction along a stretched string, results in the combined wave having an amplitude \(1.50\) times that of the common amplitude of the two combining waves? Express your answer in (a) degrees, (b) radians, and (c) wavelengths.

A string along which waves can travel is \(2.70 \mathrm{~m}\) long and has a mass of \(260 \mathrm{~g}\). The tension in the string is \(36.0 \mathrm{~N}\). What must be the frequency of traveling waves of amplitude \(7.70 \mathrm{~mm}\) for the average power to be \(85.0 \mathrm{~W} ?\)
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