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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 speed of the wave is 0.2 m/s.

Step by step solution

01

Identify the form of the wave equation

The general form of a wave equation is \( y(x, t) = A \cdot h(kx + \omega t) \), where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.
02

Recognize given parameters

From the equation \( y(x, t) = (4.00 \text{ mm}) h\left[\left(30 \text{ m}^{-1}\right)x + \left(6.0 \text{ s}^{-1}\right) t\right]\), we identify that \( k = 30 \text{ m}^{-1} \) and \( \omega = 6.0 \text{ s}^{-1} \).
03

Calculate wave speed

The speed \( v \) of the wave is given by \( v = \frac{\omega}{k} \). Substitute \( \omega = 6.0 \text{ s}^{-1} \) and \( k = 30 \text{ m}^{-1} \) to find the speed: \[ v = \frac{6.0 \text{ s}^{-1}}{30 \text{ m}^{-1}} = 0.2 \text{ m/s} \].

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Key Concepts

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

Wave Speed
Wave speed is a measure of how fast the wave travels through a medium. Imagine watching waves move across a pond. Just like those ripples, wave speed tells us how quickly the wave crests go from one spot to another.
  • In mathematics, the wave speed \( v \) can be calculated using the formula \( v = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wave number.
  • The wave speed depends on the properties of the wave and the medium through which it is moving.

In the original equation given, \( \omega = 6.0 \text{ s}^{-1} \) and \( k = 30 \text{ m}^{-1} \), we use these values to find that the speed of the wave is \( 0.2 \text{ m/s} \).
Remember, understanding wave speed is key to analyzing and predicting how waves behave in various environments.
Amplitude
Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. Think of it as the height of a wave crest above its average position.
  • It is represented by the symbol \( A \) and tells us how far the wave moves from its resting position.
  • In the context of sound, larger amplitudes mean louder sounds. For light, larger amplitudes mean brighter light.

In our wave equation \( y(x, t) = (4.00 \text{ mm}) h(...) \), the amplitude is \( 4.00 \text{ mm} \).
This indicates how high above and below the central value the waves move due to oscillations. Understanding amplitude is crucial for interpreting the energy and intensity of waves in different scenarios.
Wave Number
Wave number \( k \) is like a measure of how many wave crests are packed into a certain distance. It's a little like frequency, but instead of time, it involves space.
  • The wave number is defined as \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength.
  • A higher wave number means more wave crests in a given length, indicating shorter wavelengths.

For the wave in our equation, \( k = 30 \text{ m}^{-1} \).
This means in just one meter, there are 30 crests of the wave, portraying a short wavelength. Understanding wave number helps in analyzing the spatial distribution of waves.
Angular Frequency
Angular frequency \( \omega \) is a bit like the heartbeat of the wave. It tells us how fast the wave oscillates or how quickly it cycles back to its starting point.
  • Angular frequency is connected to the frequency of the wave by the formula \( \omega = 2\pi f \), where \( f \) is the usual frequency measured in cycles per second or Hertz.
  • It is measured in radians per second, making it a handy tool for relating rotational and wave motion.

In the given equation, \( \omega = 6.0 \text{ s}^{-1} \).
This tells us that the wave completes 6 cycles every second, making it quite fast. Grasping the concept of angular frequency is essential for exploring the nature and behavior of periodic waves.

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Most popular questions from this chapter

One of the harmonic frequencies for a particular string under tension is \(325 \mathrm{~Hz}\). The next higher harmonic frequency is \(390 \mathrm{~Hz}\). What harmonic frequency is next higher after the harmonic frequency \(195 \mathrm{~Hz}\) ?

A sinusoidal transverse wave of amplitude \(y_{m}\) and wavelength \(\lambda\) travels on a stretched cord. (a) Find the ratio of the maximum particle speed (the speed with which a single particle in the cord moves transverse to the wave) to the wave speed. (b) Does this ratio depend on the material of which the cord is made?

A sinusoidal wave of angular frequency \(1200 \mathrm{rad} / \mathrm{s}\) and amplitude \(3.00 \mathrm{~mm}\) is sent along a cord with linear density \(2.00 \mathrm{~g} / \mathrm{m}\) and tension \(1200 \mathrm{~N}\). (a) What is the average rate at which energy is transported by the wave to the opposite end of the cord? (b) If, simultaneously, an identical wave travels along an adjacent, identical cord, what is the total average rate at which energy is transported to the opposite ends of the two cords by the waves? If, instead, those two waves are sent along the same cord simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) \(0,(\mathrm{~d}) 0.4 \pi \mathrm{rad}\), and \((\mathrm{e}) \pi \mathrm{rad} ?\)

A sinusoidal transverse wave traveling in the negative direction of an \(x\) axis has an amplitude of \(1.00 \mathrm{~cm}\), a frequency of \(550 \mathrm{~Hz}\), and a speed of \(330 \mathrm{~m} / \mathrm{s}\). If the wave equation is of the form \(y(x, t)=y_{m} \sin (k x \pm \omega t)\), what are (a) \(y_{m},(\) b) \(\omega,(\mathrm{c}) k\), and \((\mathrm{d})\) the correct choice of sign in front of \(\omega ?\)

The type of rubber band used inside some baseballs and golf balls obeys Hooke's law over a wide range of elongation of the band. A segment of this material has an unstretched length \(\ell\) and a mass \(m .\) When a force \(F\) is applied, the band stretches an additional length \(\Delta \ell .\) (a) What is the speed (in terms of \(m, \Delta \ell\), and the spring constant \(k\) ) of transverse waves on this stretched rubber band? (b) Using your answer to (a), show that the time required for a transverse pulse to travel the length of the rubber band is proportional to \(1 / \sqrt{\Delta \ell}\) if \(\Delta \ell \& \ell\) and is constant if \(\Delta \ell\) ? \(\ell\).
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