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Chapter 16: Problem 27

A wave traveling in the \(+x\) direction has an amplitude of \(0.35 \mathrm{m}\) a speed of \(5.2 \mathrm{m} / \mathrm{s},\) and a frequency of \(14 \mathrm{Hz} .\) Write the equation of the wave in the form given by either Equation 16.3 or 16.4

Short Answer

Expert verified
The wave equation is \( y(x, t) = 0.35 \sin(16.93 x - 28\pi t) \).

Step by step solution

01

Identify known values

We have been given the amplitude \(A = 0.35 \, \text{m}\), speed \(v = 5.2 \, \text{m/s}\), and frequency \(f = 14 \, \text{Hz}\) of the wave.
02

Use wave equation

A wave traveling in the positive \(x\) direction can be described by the equation: \[ y(x, t) = A \sin(kx - \omega t)\] where \(y(x, t)\) represents the wave function, \(A\) is the amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, \(x\) is the position, and \(t\) is the time.
03

Calculate the angular frequency \(\omega\)

Angular frequency \(\omega\) is given by \(\omega = 2\pi f\). Thus, \(\omega = 2\pi \times 14 \, \text{Hz} = 28\pi \, \text{rad/s}\).
04

Calculate the wave number \(k\)

The wave number \(k\) can be calculated from \(k = \frac{2\pi}{\lambda}\). The wavelength \(\lambda\) is given by \(\lambda = \frac{v}{f}\). Thus, \(\lambda = \frac{5.2 \, \text{m/s}}{14 \, \text{Hz}} \approx 0.371 \, \text{m}\). Therefore, \(k = \frac{2\pi}{0.371} \approx 16.93 \, \text{m}^{-1}\).
05

Write the wave equation

Substitute the values into the wave equation: \[y(x, t) = 0.35 \sin(16.93 x - 28\pi t)\] This equation describes the wave traveling in the positive \(x\) direction.

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Key Concepts

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

Amplitude
Amplitude is a fundamental feature of a wave, representing the wave's maximum displacement from its rest position. In simpler terms, it's the height of the wave's peaks. This concept is essential because it tells us about the wave's energy. The larger the amplitude, the more energy the wave carries.

In our exercise, the amplitude is given as \(0.35\, \text{m}\). This means that at its highest or lowest point, the wave reaches 0.35 meters above or below its equilibrium position. Understanding amplitude helps in visualizing how "tall" or "strong" a wave is and plays a key role in applications like sound waves, where higher amplitudes result in louder sounds.

  • Energy and Amplitude: More energy typically means a larger amplitude.
  • Effects: In sound, larger amplitude translates to louder volume.
Wave Speed
Wave speed describes how quickly a wave propagates through a medium. It is a crucial property as it influences how fast the wave reaches a certain point. In the given problem, the wave speed is 5.2 \(\text{m/s}\). This means the wave advances by 5.2 meters every second.

Wave speed depends on the medium through which the wave is traveling. For example, sound waves travel faster in water than in air, while light waves move slower in glass than in a vacuum. Understanding wave speed is key in various fields such as telecommunications and acoustics.

  • Formula: Wave speed \(v\) is related to frequency \(f\) and wavelength \(\lambda\) by \(v = f\lambda\).
  • Media Dependence: Different materials affect the speed of wave travel.
Angular Frequency
Angular frequency indicates how rapidly a wave oscillates. It is a measure of how many cycles a wave completes in a given period and is expressed in radians per second \(\text{rad/s}\). It is a key concept when dealing with sinusoidal waves like sound or light.

For a wave with a frequency of 14 Hz, the angular frequency \(\omega\) is calculated using the formula \(\omega = 2\pi f\). In our example, \(\omega = 2\pi \times 14 = 28\pi\, \text{rad/s}\). This tells us how fast the wave's phase changes over time.

  • Relation to Frequency: Angular frequency is derived directly from frequency using \(\omega = 2\pi f\).
  • Importance: It helps in analyzing circuits in physics and understanding the behavior of waves in various media.

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Most popular questions from this chapter

A convertible moves toward you and then passes you; all the while, its loudspeakers are producing a sound. The speed of the car is a constant \(9.00 \mathrm{m} / \mathrm{s},\) and the speed of sound is \(343 \mathrm{m} / \mathrm{s} .\) What is the ratio of the frequency you hear while the car is approaching to the frequency you hear while the car is moving away?

A man stands at the midpoint between two speakers that are broadcasting an amplified static hiss uniformly in all directions. The speakers are \(30.0 \mathrm{m}\) apart and the total power of the sound coming from each speaker is \(0.500 \mathrm{W}\). Find the total sound intensity that the man hears (a) when he is at his initial position halfway between the speakers, and (b) after he has walked \(4.0 \mathrm{m}\) directly toward one of the speakers.

Two blocks are connected by a wire that has a mass per unit length of \(8.50 \times 10^{-4} \mathrm{kg} / \mathrm{m} .\) One block has a mass of \(19.0 \mathrm{kg},\) and the other has a mass of \(42.0 \mathrm{kg} .\) These blocks are being pulled across a horizontal frictionless floor by a horizontal force \(\overrightarrow{\mathbf{P}}\) that is applied to the less massive block. A transverse wave travels on the wire between the blocks with a speed of \(352 \mathrm{m} / \mathrm{s}\) (relative to the wire). The mass of the wire is negligible compared to the mass of the blocks. Find the magnitude of \(\overrightarrow{\mathbf{P}}\)

A wave has the following properties: amplitude \(=0.37 \mathrm{m},\) period \(=0.77 \mathrm{s},\) wave speed \(=12 \mathrm{m} / \mathrm{s} .\) The wave is traveling in the \(-x\) direction. What is the mathematical expression (similar to Equation 16.3 or 16.4 ) for the wave?

A loudspeaker in a parked car is producing sound whose frequency is \(20510 \mathrm{Hz} .\) A healthy young person with normal hearing is standing nearby on the sidewalk but cannot hear the sound because the frequency is too high. When the car is moving, however, this person can hear the sound. (a) Is the car moving toward or away from the person? Why? (b) If the speed of sound is \(343 \mathrm{m} / \mathrm{s},\) what is the minimum speed of the moving car?
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