Problem 86 - Fundamentals of Physics

Question

(a) Write an cquation describing a sinusoidal transverse wave traveling on a cord in the positive direction of a \(y\) axis with an angular wave number of \(60 \mathrm{~cm}^{-1}\), a period of \(0.20 \mathrm{~s}\), and an amplitude of \(3.0 \mathrm{~mm}\). Take the transverse direction to be the \(z\) direction. (b) What is the maximum transverse speed of a point on the cord?

Step-by-Step Solution

Answer

The wave equation is \( z(y, t) = 0.003 \sin(600y - 10\pi t) \) and \( v_{\text{max}} = 0.03\pi \, \text{m/s} \).

Step 1: Determine the Wave Equation Parameters

A sinusoidal wave traveling in the direction of the y-axis typically has an equation of the form \( z(y, t) = A \sin(ky - \omega t) \) for a wave moving in the positive direction.- Given amplitude \( A = 3.0 \, \text{mm} = 0.003 \, \text{m} \).- Given angular wave number \( k = 60 \, \text{cm}^{-1} = 600 \, \text{m}^{-1} \).- The period \( T = 0.20 \, \text{s} \), thus the angular frequency \( \omega = \frac{2\pi}{T} = \frac{2\pi}{0.20} = 10\pi \, \text{s}^{-1} \).

Step 2: Write the Wave Equation

Substitute the known parameters into the general sinusoidal wave equation. The wave equation becomes:\[ z(y, t) = 0.003 \sin(600y - 10\pi t) \] where \( z(y, t) \) is the transverse displacement of a point on the cord at position \( y \) and time \( t \).

Step 3: Calculate the Maximum Transverse Speed

The transverse speed of a point on the cord is given by the partial derivative of \( z(y, t) \) with respect to time \( t \). Thus, the transverse speed is:\[ v_y = \frac{\partial z}{\partial t} = 0.003 \times (-10\pi) \cos(600y - 10\pi t) \] The maximum value of \( \cos(600y - 10\pi t) \) is 1, so the maximum transverse speed is:\[ v_{\text{max}} = 0.003 \times 10\pi = 0.03\pi \, \text{m/s} \]

Key Concepts

Transverse Wave

A transverse wave is a type of wave where the particle displacement is perpendicular to the direction of wave propagation. Imagine a wave traveling along a stretched rope. As the wave moves horizontally along the rope, each segment of the rope moves up and down. This up and down motion of the rope is characteristic of a transverse wave.
The equation describing a sinusoidal transverse wave in one dimension is given as \( z(y, t) = A \sin(ky - \omega t) \). Here, \( z(y, t) \) represents the displacement of the particles in the transverse direction (perpendicular to the direction of wave travel).

• Transverse Direction: In this context, the transverse direction is the \( z \)-axis, meaning the wave travels along the \( y \)-axis, while the vibrations occur in the \( z \)-axis.
• Wave Equation Components: It includes quantities like amplitude, wave number, and angular frequency, crucial to defining the physical attributes of the wave.

Angular Wave Number

The angular wave number \( k \) is a critical parameter in the description of a wave. It is related to the wavelength of the wave, which is the distance over which the wave's shape repeats.
The equation for the angular wave number is \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength. In a given problem, the angular wave number simplifies into a measure of how many radians the wave advances per unit distance.

• Value: In the exercise, \( k = 600 \, \text{m}^{-1} \), which means each meter along the wave's path corresponds to a phase change of 600 radians.
• Role in Wave Equation: The wave number provides information about the spatial frequency of the wave, complementing the understanding provided by the angular frequency related to time.

Maximum Transverse Speed

The maximum transverse speed of a point on a wave is the highest speed attained by the particles during their oscillation. Understanding this concept will help us discern the wave's energy and dynamics.
It is calculated by differentiating the wave equation \( z(y, t) \) with respect to time \( t \), providing the velocity function \( v_y = \frac{\partial z}{\partial t} \).

• Formula: For the given wave, the transverse speed is \( v_y = -0.03\pi \cos(600y - 10\pi t) \). The maximum speed occurs when the cosine term equals 1, leading to \( v_{\text{max}} = 0.03\pi \, \text{m/s} \).
• Importance: Understanding the maximum transverse speed is essential for analyzing wave interactions and potential energy transfer along the wave, as it indicates how fast the wave's amplitude changes.

Amplitude

Amplitude \( A \) is a key parameter that determines the extent of displacement of particles from their equilibrium position due to a wave. It represents the maximum displacement experienced by the particles as the wave passes through.
In the context of a sinusoidal wave, the amplitude gives a measure of the wave's energy. A higher amplitude results in more energy being carried by the wave.

• Value: In the exercise, the amplitude \( A = 0.003 \text{ m} \), indicating a maximum displacement of 3 mm.
• Wave Equation Impact: The amplitude directly affects the magnitude of the wave in the wave equation \( z(y, t) = A \sin(ky - \omega t) \).
• Physical Implication: It is a straightforward visual way to assess how strong or vibrant the wave is; larger amplitudes often correlate with more significant impacts on the medium through which the wave travels.

Period

The period \( T \) of a wave indicates the time taken for one complete cycle of the waveform. It is a fundamental property that helps define the temporal characteristics of a wave.
The Period is inversely related to the wave's frequency, which is the number of cycles per unit time. The relationship is given by the formula \( T = \frac{1}{f} \).

• Value: In the solution, the period given is \( 0.20 \, \text{s} \), meaning each complete oscillation or cycle of the wave takes 0.20 seconds.
• Role in Wave Behavior: The period helps describe how fast the oscillations are occurring and is crucial for applications where timing is vital, such as resonance phenomena.

Angular Frequency

Angular frequency \( \omega \) is related to how fast the wave oscillates in time. It provides a measure of the wave's oscillatory speed, distinct from its propagation speed.
It is calculated using the period, \( T \), with the formula \( \omega = \frac{2\pi}{T} \). This relationship derives from the standard periodic function expression \( y = \sin(\omega t) \).

• Value: For the wave described in the exercise, we have \( \omega = 10\pi \, \text{s}^{-1} \), given a period of 0.20 seconds.
• Significance: Angular frequency offers insight into the temporal frequency of the wave, enabling predictions of the oscillation rate experienced by a medium point, making it crucial for analyzing wave behaviors such as phase shifts and natural frequencies in systems.