Problem 28 - Fundamentals of Physics

Question

Use the wave equation to find the speed of a wave given by $$ y(x, t)=(3.00 \mathrm{~mm}) \sin \left[\left(4.00 \mathrm{~m}^{-1}\right) x-\left(7.00 \mathrm{~s}^{-1}\right) t\right] . $$

Step-by-Step Solution

Answer

The speed of the wave is 1.75 m/s.

Step 1: Identify Wave Equation Parameters

The given wave equation is \( y(x, t) = (3.00 \, \text{mm}) \sin \left( \left(4.00 \, \text{m}^{-1}\right) x - \left(7.00 \, \text{s}^{-1}\right) t \right) \). From this equation, we identify the wave number \( k = 4.00 \, \text{m}^{-1} \) and the angular frequency \( \omega = 7.00 \, \text{s}^{-1} \).

Step 2: Apply Wave Speed Formula

The speed \( v \) of a wave is given by the formula \( v = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wave number. You'll use the values obtained in Step 1 to calculate the wave speed.

Step 3: Perform the Calculation

Substitute the values \( \omega = 7.00 \, \text{s}^{-1} \) and \( k = 4.00 \, \text{m}^{-1} \) into the formula: \( v = \frac{\omega}{k} = \frac{7.00}{4.00} \, \text{m/s} = 1.75 \, \text{m/s} \).

Key Concepts

Wave Equation

The wave equation is an essential formula used to describe the motion of waves. It takes the form \( y(x, t) = A \sin(kx - \omega t) \), where:

• \( y(x, t) \) is the wave function that describes the displacement at position \( x \) and time \( t \).
• \( A \) is the amplitude, representing the wave's maximum displacement from its rest position.
• \( k \) is the wave number, indicating how many wavelengths fit into a unit distance.
• \( \omega \) is the angular frequency, describing how fast the wave oscillates.

Understanding this equation helps differentiate between various parameters that affect a wave's shape and propagation. It provides insights into the behavior of different types of waves, such as sound waves and light waves. The given equation \( y(x, t) = (3.00 \, \text{mm}) \sin \left(\left(4.00 \, \text{m}^{-1}\right) x - \left(7.00 \, \text{s}^{-1}\right) t\right) \) helps us determine key characteristics like amplitude, wave number, and angular frequency.

Angular Frequency

Angular frequency, denoted as \( \omega \), is a crucial concept when examining wave motion. It is defined as the rate at which the wave oscillates per unit time and is measured in radians per second (\( \text{s}^{-1} \)).

• Angular frequency relates to the wave's speed and its periodic behavior.
• It determines how many oscillations occur in a given time.
• It is linked to the ordinary frequency \( f \) by the relation \( \omega = 2\pi f \).

In our given wave equation, the angular frequency is \( 7.00 \, \text{s}^{-1} \), indicating how quickly the wave cycles through its pattern. A higher angular frequency means that more wave cycles occur in the same duration, resulting in a faster oscillating wave.

Wave Number

The wave number \( k \) represents the number of wave cycles per unit distance and is measured in inverse meters (\( \text{m}^{-1} \)). It provides a spatial understanding of the wave's nature.

• Wave number is linked to the wavelength \( \lambda \) via the formula \( k = \frac{2\pi}{\lambda} \).
• A higher wave number implies a smaller wavelength, meaning the wave oscillates more frequently over a given distance.
• Wave number can help determine how waves propagate through space.

In our problem, the wave number is \( 4.00 \, \text{m}^{-1} \), which helps us comprehend how the wave's spatial properties relate to its speed. With both the wave number and angular frequency, we utilized the formula \( v = \frac{\omega}{k} \) to calculate the wave speed, reflecting the wave's propagation characteristics.