Problem 28 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

(a) 2.30 mm, (b) 277.3 Hz, (c) 0.9003 m, (d) 249.78 m/s, (e) negative x-direction, (f) 156.1 N, (g) 14.0 W.

Step 1: Identify the Amplitude

The amplitude of a wave is the coefficient in front of the cosine function in the wave equation. In the given wave function, the amplitude is 2.30 mm. Therefore, the amplitude is \( A = 2.30 \, \text{mm} \).

Step 2: Calculate the Frequency

The angular frequency \( \omega \) is given by the term \( 1742 \, \text{rad/s} \) in the wave function. The frequency \( f \) can be found using the formula \( f = \frac{\omega}{2\pi} \). Therefore, \( f = \frac{1742}{2\pi} \approx 277.3 \, \text{Hz} \).

Step 3: Determine the Wavelength

The wave number \( k \) is given by \( 6.98 \, \text{rad/m} \). The wavelength \( \lambda \) is related to the wave number by \( \lambda = \frac{2\pi}{k} \). Thus, \( \lambda = \frac{2\pi}{6.98} \approx 0.9003 \, \text{m} \).

Step 4: Calculate the Wave Speed

The wave speed \( v \) can be calculated using the formula \( v = f\lambda \). Given \( f \approx 277.3 \, \text{Hz} \) and \( \lambda \approx 0.9003 \, \text{m} \), the wave speed is \( v = 277.3 \times 0.9003 \approx 249.78 \, \text{m/s} \).

Step 5: Identify the Direction of the Wave

The phase factor in the wave function includes \( +kx+\omega t \); the presence of \(+\omega t\) indicates the wave is traveling in the negative x-direction.

Step 6: Calculate the Tension in the Rope

The wave speed on a rope is given by \( v = \sqrt{\frac{T}{\mu}} \), where \( \mu \) is the linear mass density \( \frac{m}{L} \). With \( m = 0.00338 \, \text{kg} \) and \( L = 1.35 \, \text{m} \), \( \mu = \frac{0.00338}{1.35} \approx 0.002504 \, \text{kg/m} \). Then, solving \( v = 249.78 = \sqrt{\frac{T}{0.002504}} \) gives \( T \approx \mu v^2 \approx 156.1 \, \text{N} \).

Step 7: Determine the Average Power Transmitted by the Wave

The average power \( P \) transmitted by a wave traveling along a string is given by \( P = \frac{1}{2} \mu \omega^2 A^2 v \). Using \( \mu \approx 0.002504 \, \text{kg/m} \), \( \omega = 1742 \, \text{rad/s} \), \( A = 0.0023 \, \text{m} \), and \( v \approx 249.78 \, \text{m/s} \), we compute \( P = \frac{1}{2} (0.002504)(1742^2)(0.0023)^2(249.78) \approx 14.0 \, \text{W} \).

Key Concepts

Wave Function

A wave function is a mathematical representation of a wave. It describes how the wave travels through space and time. In this case, the wave function provided is \( y(x, t) = 2.30 \, \text{mm} \, \cos [(6.98 \, \text{rad/m})x + 1742 \, \text{rad/s} \)\(t)] \).
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

Amplitude is a fundamental characteristic of waves. It represents the maximum extent of a wave's displacement from its equilibrium position.
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

Wave speed is a critical concept in wave motion, defining how fast a wave propagates through a medium.
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

• \( 249.78 \, \text{m/s} \).

The speed of a wave depends on the medium through which it travels.
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

Frequency refers to how often the wave oscillates or completes a cycle in one second.
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:

• \( f = \frac{\omega}{2\pi} \).

For this example, the frequency is approximately \( 277.3 \, \text{Hz} \).
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.

A higher frequency indicates more cycles per unit time, which means more energy in the wave.

Wavelength

Wavelength is the distance between successive crests (or any successive identical points) of a wave.
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:

• \( \lambda = \frac{2\pi}{k} \).

For our wave, this computes to about \( 0.9003 \, \text{m} \). Wavelength directly affects the wave's speed along with frequency.
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.