Chapter 15: Problem 28
A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is \(y(x, t) = 2.30 \, \mathrm{mm} \, \mathrm{cos} [(6.98 \, \mathrm{rad/m})x \space + 1742 \, \mathrm{rad/s})t]\). Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.
Short Answer
Step by step solution
Identify the Amplitude
Calculate the Frequency
Determine the Wavelength
Calculate the Wave Speed
Identify the Direction of the Wave
Calculate the Tension in the Rope
Determine the Average Power Transmitted by the Wave
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Function
The wave function typically includes parameters like amplitude, angular wave number \( k \), and angular frequency \( \omega \).
Understanding each component in the wave function helps in analyzing the wave's various properties, such as its direction and speed. For instance, the cosine function indicates this is a periodic wave, which means it repeats itself at regular intervals.
- The coefficient \( 2.30 \, \text{mm} \) is the amplitude, representing the maximum displacement from rest.
- The term \( 6.98 \, \text{rad/m} \) is the wave number, which relates to the wavelength.
- The \( 1742 \, \text{rad/s} \) term is the angular frequency, relating to the frequency.
Amplitude
In simple terms, it is how far the wave "reaches" or "grows" from its baseline up to the top-most point known as the wave crest.
In the wave equation \( y(x, t) = 2.30 \, \text{mm} \, \cos [(6.98 \, \text{rad/m})x + 1742 \, \text{rad/s})t] \), the amplitude is given by the coefficient in front, which is \( 2.30 \, \text{mm} \).
- This parameter indicates the energy of the wave; a larger amplitude means the wave carries more energy.
- Amplitude does not affect the speed of the wave; it solely impacts the energy level and intensity.
Wave Speed
You determine wave speed by multiplying the frequency of the wave \( f \) by its wavelength \( \lambda \).
Mathematically expressed as \( v = f\lambda \). For the given wave, with frequency \( 277.3 \, \text{Hz} \) and wavelength \( 0.9003 \, \text{m} \), the wave speed is
- \( 249.78 \, \text{m/s} \).
For instance, waves move faster in less dense materials and slower in denser ones. Understanding wave speed is vital for numerous practical applications, from engineering to communication.
Frequency
It is measured in Hertz (Hz), where one Hertz equals one cycle per second.
From the angular frequency \( \omega \) provided in the wave equation, \( \omega = 1742 \, \text{rad/s} \), you can determine the frequency using the formula:
- \( f = \frac{\omega}{2\pi} \).
Frequency is inversely related to the period of the wave, which is the time taken to complete one cycle.
- Higher frequencies imply shorter periods and vice versa.
Wavelength
It is usually denoted by \( \lambda \) and is measured in meters (m).
Using the wave equation, the wave number \( k = 6.98 \, \text{rad/m} \) helps find the wavelength with the equation:
- \( \lambda = \frac{2\pi}{k} \).
It is an essential element in determining how waves interact with each other and their surrounding environment.
- In optics, for instance, wavelengths determine the color of light.
- In sound, they influence pitch.