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Chapter 16: Problem 30

Use the wave equation to find the speed of a wave given in terms of the general function \(h(x, t)\) $$ y(x, t)=(4.00 \mathrm{~mm}) h\left[\left(30 \mathrm{~m}^{-1}\right) x+\left(6.0 \mathrm{~s}^{-1}\right) t\right] $$

Short Answer

Expert verified
The wave speed is 0.20 m/s.

Step by step solution

01

Identify the wave equation

The general form of a traveling wave along a string is given by \( y(x, t) = h(kx + \omega t) \), where \( k \) is the wave number and \( \omega \) is the angular frequency. In our function, it is given as \( y(x, t) = (4.00 \mathrm{~mm}) h\left[\left(30 \mathrm{~m}^{-1}\right) x + \left(6.0 \mathrm{~s}^{-1}\right) t\right] \). Hence, \( k = 30 \; \mathrm{m}^{-1} \) and \( \omega = 6.0 \; \mathrm{s}^{-1} \).
02

Identify the formula to calculate wave speed

The wave speed \( v \) is calculated using the relationship between \( k \), the wave number, and \( \omega \), the angular frequency. The formula is given by \( v = \frac{\omega}{k} \).
03

Compute the wave speed

Substitute the values of \( \omega = 6.0 \; \mathrm{s}^{-1} \) and \( k = 30 \; \mathrm{m}^{-1} \) into the speed formula: \[ v = \frac{6.0 \; \mathrm{s}^{-1}}{30 \; \mathrm{m}^{-1}} = 0.20 \; \mathrm{m/s} \].

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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 a fundamental mathematical representation used to describe how waves travel through different mediums. It's key in physics and engineering for understanding wave phenomena such as sound, light, and water waves. A wave can be expressed mathematically using the function \( y(x, t) = A h(kx + \omega t) \).
  • \( y(x, t) \) is the wave function.
  • \( A \) represents the amplitude of the wave, which is the maximum displacement from its equilibrium position.
  • \( h \) is a general wave function.
  • \( k \) is the wave number, indicating how many waves fit into a unit distance.
  • \( \omega \) is the angular frequency, representing how many oscillations occur in a unit time.
  • \( t \) is the time variable.
This equation provides the backbone for analyzing numerous wave-related problems, offering insights into wave speed, propagation, and behavior over time.
Wave Number
The wave number, denoted as \( k \), is a crucial concept in wave mechanics. It essentially describes how many wavelengths fit into a given unit of space. Mathematically, it is defined as the reciprocal of the wavelength \( \lambda \), such that \( k = \frac{2\pi}{\lambda} \).
  • \( k \) is measured in meters to the power of negative one (\( \mathrm{m}^{-1} \)), indicating the number of cycles per meter.
  • A higher wave number means more waves are present over a shorter length.
  • In the original exercise, the wave number is given as \( 30 \, \mathrm{m}^{-1} \).
Understanding wave numbers helps in identifying the frequency and energy of the wave, offering insights into its high or low frequency nature depending on the number of oscillations.
Angular Frequency
Angular frequency \( \omega \) plays a vital role in describing the oscillatory motion of waves. It signifies how many cycles or oscillations a wave completes in a unit of time, usually measured in radians per second (\( \mathrm{s}^{-1} \)). It links directly to the frequency \( f \) of the wave through the relationship \( \omega = 2\pi f \).
  • \( \omega \) illuminates the energetic and temporal characteristics of a wave.
  • A higher angular frequency indicates quicker oscillations, signifying a more energetic wave.
  • In the step-by-step solution, the given angular frequency is \( 6.0 \, \mathrm{s}^{-1} \).
Angular frequency is central to analyzing wave phenomena, as it helps determine the wave speed and its interaction with mediums, influencing how waves spread and dissipate.
Traveling Wave
A traveling wave is a type of wave that moves or propagates through a medium with time. It's characterized by a consistent transfer of energy from one point to another without the permanent displacement of the medium. Mathematically, it's expressed in a form \( y(x, t) = A h(kx + \omega t) \).
Key features include:
  • Direction: Traveling waves can move in positive or negative directions depending on the sign in the wave function.
  • Propagation: They demonstrate wave-like behaviors such as interference and diffraction.
  • Speed: The speed of a traveling wave, \( v \), is determined by the formula \( v = \frac{\omega}{k} \), where in the exercise it is calculated as \( 0.20 \, \mathrm{m/s} \).
Understanding traveling waves is crucial for analyzing how waves are utilized in real-world applications like radio transmissions, oceanography, and acoustics.

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Most popular questions from this chapter

A string oscillates according to the equation \(y^{\prime}=(0.50 \mathrm{~cm}) \sin \left[\left(\frac{\pi}{3} \mathrm{~cm}^{-1}\right) x\right] \cos \left[\left(40 \pi \mathrm{s}^{-1}\right) t\right]\) What are the (a) amplitude and (b) speed of the two waves (identical except for direction of travel) whose superposition gives this oscillation? (c) What is the distance between nodes? (d) What is the transverse speed of a particle of the string at the position \(x=1.5 \mathrm{~cm}\) when \(t=\frac{9}{8} \mathrm{~s} ?\)

The heaviest and lightest strings on a certain violin have linear densities of 3.0 and \(0.29 \mathrm{~g} / \mathrm{m} .\) What is the ratio of the diameter of the heaviest string to that of the lightest string, assuming that the strings are of the same material?

At time \(t=0\) and at position \(x=0 \mathrm{~m}\) along a string, a traveling sinusoidal wave with an angular frequency of 440 rad/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 \(\phi ?\)

The function \(y(x, t)=(15.0 \mathrm{~cm}) \cos (\pi x-15 \mathrm{zt}),\) with \(x\) in meters and \(t\) in seconds, describes a wave on a taut string. What is the transverse speed for a point on the string at an instant when that point has the displacement \(y=+12.0 \mathrm{~cm} ?\)

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 \(\lambda_{3} ?\)
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