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Chapter 16: Problem 7

A transverse sinusoidal wave is moving along a string in the positive direction of an \(x\) axis with a speed of \(80 \mathrm{~m} / \mathrm{s}\). At \(t=0\), the string particle at \(x=0\) has a transverse displacement of \(4.0 \mathrm{~cm}\) from its equilibrium position and is not moving. The maximum transverse speed of the string particle at \(x=0\) is \(16 \mathrm{~m} / \mathrm{s}\). (a) What is the frequency of the wave? (b) What is the wavelength of the wave? If \(y(x, t)=y_{m} \sin (k x \pm \omega t+\phi)\) is the form of the wave equa- of sign in front of \(\omega ?\)

Short Answer

Expert verified
Frequency is 63.66 Hz, wavelength is 1.256 m, and the sine term is \( \sin(kx - \omega t + \phi) \).

Step by step solution

01

Understand the given information

The wave is sinusoidal and moves with a speed of 80 m/s. At t=0, the displacement of a particle at x=0 is 4 cm, and it's not moving. The maximum transverse speed is 16 m/s.
02

Determine angular frequency

The maximum transverse speed of the particle at x=0 is given by \( v_{max} = \omega y_m \). Given \( v_{max} = 16 \, \text{m/s} \) and \( y_m = 0.04 \, \text{m} \), we find \[ \omega = \frac{16}{0.04} = 400 \, \text{rad/s}. \]
03

Calculate frequency

Frequency \( f \) can be found using the relation \( \omega = 2\pi f \). Thus, \[ f = \frac{\omega}{2\pi} = \frac{400}{2\pi} \approx 63.66 \, \text{Hz}. \]
04

Determine wavelength

Use the wave speed relation \( v = f \lambda \). Substitute \( v = 80 \, \text{m/s} \) and \( f = 63.66 \, \text{Hz} \) to find \[ \lambda = \frac{v}{f} = \frac{80}{63.66} \approx 1.256 \, \text{m}. \]
05

Determine the sign in the wave equation

The wave is moving in the positive x-direction, so we use \( \sin(kx - \omega t + \phi) \). The sign in front of \( \omega \) should be negative.

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Key Concepts

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

Transverse Wave
Transverse waves are a type of wave where particle displacement is perpendicular to the direction of wave propagation. Think of snapping a long jump rope鈥攗p and down motion travels along the rope. This is similar to the wave on a string described in your problem.
  • **Directionality**: In transverse waves, like the one in this problem, energy travels horizontally while the medium's particles move vertically.
  • **Medium**: The wave travels through a medium, in this case, a string. The medium must be able to support shear stress for a transverse wave to form.
  • **Real-Life Examples**: Light waves and surface ripples on water likewise exhibit transverse characteristics.
The string's particle at an initial position has a transverse displacement from its equilibrium, a tell-tale sign of a transverse wave in action. This type of wave can be beautifully described using sine and cosine functions, leading perfectly into wave equations and their analyses.
Frequency Calculation
Frequency is a fundamental property of a wave that tells us how many wave cycles pass a given point per second, measured in Hertz (Hz). You can calculate it if you know the wave's angular frequency.
  • **Relation to Angular Frequency**: Angular frequency (\( \omega \)) and frequency (\( f \)) are linked by the formula \( \omega = 2\pi f \).
  • **Given Values**: In this scenario, the maximum transverse speed was provided, leading to \( \omega = 400 \, \text{rad/s} \).
  • **Calculate Frequency**: With the angular frequency, use \( f = \frac{\omega}{2\pi} \) to find \( f \approx 63.66 \, \text{Hz} \). This tells us how lively the wave activities are!
Identifying frequency is crucial as it allows you to determine other significant aspects of wave motion, like its wavelength. It鈥檚 like discovering the beat to which the wave "dances" as it travels.
Wavelength Determination
A wave's wavelength is the distance over which the wave's shape repeats. It's a key characteristic that works together with speed and frequency to describe wave behavior. Here鈥檚 how you determine it:
  • **Using Wave Speed**: The formula \( v = f \lambda \) connects a wave's speed (\( v \)), frequency (\( f \)), and wavelength (\( \lambda \)).
  • **Given Values**: In your problem, the wave speed is \( 80 \, \text{m/s} \) and frequency is \( 63.66 \, \text{Hz} \).
  • **Calculate Wavelength**: Substitute these values into \( \lambda = \frac{v}{f} \) to find \( \lambda \approx 1.256 \text{ m} \).
This expresses the length of one complete wave cycle and reveals how compact or stretched the wave appears in its environment. Wavelength is instrumental in understanding not just the visible "spacing out" of waves but also their interaction with barriers and openings鈥攅ssential for sound and light studies.
Wave Equation Analysis
The wave equation is a mathematical model that describes wave motion. It allows you to predict how a wave varies over time and space. The generic form is \( y(x, t) = y_m \sin(kx \pm \omega t + \phi) \).
  • **Components of the Equation**: The term \( y_m \) represents the maximum displacement, or amplitude. \( \omega t \) is the time-dependent part, while \( kx \) depends on space. \( \phi \) is the phase shift.
  • **Wave Directionality**: The sign in front of \( \omega t \) determines the direction of wave travel. For a wave traveling in the positive x-direction, use a negative sign \( kx - \omega t \).
  • **Practical Understanding**: Analyzing the equation helps understand how the wave changes with time or distance. It informs you about speed, frequency, and initial conditions described in real-world setups.
Understanding a wave鈥檚 equation including its terms and correct phase choice is critical, as it embodies all dynamic information about the wave on a string, letting you evaluate and anticipate its behavior in practically any context!

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Most popular questions from this chapter

The speed of a transverse wave on a string is \(170 \mathrm{~m} / \mathrm{s}\) when the string tension is \(120 \mathrm{~N}\). To what value must the tension be changed to raise the wave speed to \(180 \mathrm{~m} / \mathrm{s} ?\)

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 \(85.0 \mathrm{~W} ?\)

A generator at one end of a very long string creates a wave given by $$y=(6.0 \mathrm{~cm}) \cos \frac{\pi}{2}\left[\left(2.00 \mathrm{~m}^{-1}\right) x+\left(8.00 \mathrm{~s}^{-1}\right) t\right] $$ and a generator at the other end creates the wave $$ y=(6.0 \mathrm{~cm}) \cos \frac{\pi}{2}\left[\left(2.00 \mathrm{~m}^{-1}\right) x-\left(8.00 \mathrm{~s}^{-1}\right) t\right] $$ Calculate the (a) frequency, (b) wavelength, and (c) speed of each wave. For \(x \geq 0,\) what is the location of the node having the (d) smallest, (e) second smallest, and (f) third smallest value of \(x\) ? For \(x \geq 0,\) what is the location of the antinode having the \((g)\) smallest, (h) second smallest, and (i) third smallest value of \(x\) ?

(a) Write an equation describing a sinusoidal transverse wave traveling on a cord in the positive direction of a \(y\) axis with an angular wave number of \(60 \mathrm{~cm}^{-1}\), a period of \(0.20 \mathrm{~s}\), and an amplitude of \(3.0 \mathrm{~mm}\). Take the transverse direction to be the \(z\) direction. (b) What is the maximum transverse speed of a point on the cord?

When played in a certain manner, the lowest resonant frequency of a certain violin string is concert \(\mathrm{A}(440 \mathrm{~Hz}) .\) What is the frequency of the (a) second and (b) third harmonic of the string?
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