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Chapter 16: Problem 32

Use the wave equation to find the speed of a wave given by $$ y(x, t)=(3.00 \mathrm{~mm}) \sin \left[\left(3.00 \mathrm{~m}^{-1}\right) x-\left(8.00 \mathrm{~s}^{-1}\right) t\right] . $$

Short Answer

Expert verified
The speed of the wave is 2.67 m/s.

Step by step solution

01

Understand the wave equation

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

Identify the parameters

The given wave equation is \( y(x, t) = (3.00 \text{ mm}) \sin [(3.00 \text{ m}^{-1}) x - (8.00 \text{ s}^{-1}) t] \). Here, the wave number \( k = 3.00 \text{ m}^{-1} \) and the angular frequency \( \omega = 8.00 \text{ s}^{-1} \).
03

Use the wave speed formula

The speed \( v \) of a wave is given by the formula \( v = \frac{\omega}{k} \). This formula relates the wave speed to the angular frequency \( \omega \) and the wave number \( k \).
04

Calculate the wave speed

Substitute \( \omega = 8.00 \text{ s}^{-1} \) and \( k = 3.00 \text{ m}^{-1} \) into the wave speed formula: \( v = \frac{8.00 \text{ s}^{-1}}{3.00 \text{ m}^{-1}} = \frac{8.00}{3.00} \text{ m/s} = 2.67 \text{ m/s} \).

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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 crucial concept when working with waves, especially in various fields like physics and engineering. The wave speed refers to how fast the wave propagates through a medium. Think of it as the speed at which the wave's shape moves in space. In the case of electromagnetic waves, this could mean how quickly light travels through air or a vacuum.

To calculate the wave speed, we use the formula:
  • \( v = \frac{\omega}{k} \)
Here, \( v \) is the wave speed, \( \omega \) is the angular frequency, and \( k \) is the wave number. This equation highlights the relationship between frequency-related characteristics and spatial characteristics of a wave.

For example, in the given wave equation, the values were \( \omega = 8.00 \text{ s}^{-1} \) and \( k = 3.00 \text{ m}^{-1} \), leading to a speed \( v \) of 2.67 \( \text{ m/s} \). This means the wave moves at 2.67 meters per second.
Wave Number
The wave number is an essential component of the wave equation and relates to how "packed" the wave's cycles are over a particular distance. It is comparable to spatial frequency, measuring how many wave cycles exist in a certain length.

Mathematically, the wave number \( k \) is defined as:
  • \( k = \frac{2 \pi}{\lambda} \)
where \( \lambda \) is the wavelength of the wave. The wave number has units of \( \text{m}^{-1} \), emphasizing its role in translating spatial information into numerical values.

In our given equation, \( k = 3.00 \text{ m}^{-1} \), indicating that every meter contains about 3 complete wave cycles. Knowing the wave number helps in understanding the wave's spatial structure and is vital when calculating the wave speed using the relationship between wave number and angular frequency.
Angular Frequency
Angular frequency is a fundamental notion in understanding time-based wave behavior. It explains how quickly the wave oscillations repeat themselves over time and is measured in radians per second.

The angular frequency \( \omega \) can be expressed as:
  • \( \omega = 2 \pi f \)
where \( f \) is the frequency measured in Hertz (Hz), which tells us how many cycles occur in one second. Thus, \( \omega \) accounts for the circular nature of wave oscillations, converting the linear frequency into a rotational frequency.

In the context of our exercise, the given wave's angular frequency is \( 8.00 \text{ s}^{-1} \). This value demonstrates that the wave completes \( 8.00 \) radians of phase, or just over one full cycle, per second. Understanding angular frequency provides insight into the temporal dynamics and speed of the wave's oscillations.

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Most popular questions from this chapter

A sand scorpion can detect the motion of a nearby beetle (its prey) by the waves the motion sends along the sand surface (Fig. 16-35). The waves are of two types: transverse waves traveling at \(v_{t}=50 \mathrm{~m} / \mathrm{s}\) and longitudinal waves traveling at \(v_{l}=150 \mathrm{~m} / \mathrm{s}\). If a sudden motion sends out such waves, a scorpion can tell the distance of the beetle from the difference \(\Delta t\) in the arrival times of the waves at its leg nearest the beetle. What is that time difference if the distance to the beetle is \(37.5 \mathrm{~cm}\) ?

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 \(\tau_{f}=8 \tau_{i}\) 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 \(\lambda_{3}\) ?

Two sinusoidal waves of the same frequency travel in the same direction along a string. If \(y_{m 1}=2.0 \mathrm{~cm}, y_{m 2}=\) \(4.0 \mathrm{~cm}, \phi_{1}=0\), and \(\phi_{2}=\pi / 2 \mathrm{rad}\), what is the amplitude of the resultant wave?

A string oscillates according to the equation $$ y^{\prime}=(0.80 \mathrm{~cm}) \sin \left[\left(\frac{\pi}{3} \mathrm{~cm}^{-1}\right) x\right] \cos \left[\left(40 \pi \mathrm{s}^{-1}\right) t\right] $$ 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 \(x=2.1 \mathrm{~cm}\) when \(t=0.50 \mathrm{~s} ?\)

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 \(4.00 \mathrm{~g} / \mathrm{m}\) 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 , (d) \(0.4 \pi \mathrm{rad}\), and (e) \(\pi \mathrm{rad}\) ?
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