Problem 25 - Physics

Question

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?

Step-by-Step Solution

Answer

\(\psi(x, t) = 0.37 \sin(0.68x + 8.16t)\) is the wave equation.

Step 1: Identify the formula structure

The general form for a wave traveling in the negative x-direction can be expressed as:\[ \psi(x, t) = A \sin(kx + \omega t + \phi) \]. Here, \(A\) is the amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, and \(\phi\) is the phase constant.

Step 2: Find angular frequency

The angular frequency \(\omega\) is related to the period \(T\) by the formula: \(\omega = \frac{2\pi}{T}\). Given that the period \(T = 0.77\, \mathrm{s}\), we calculate \(\omega\) as follows:\[ \omega = \frac{2\pi}{0.77} \approx 8.16\, \mathrm{rad/s} \].

Step 3: Find wave number

The wave number \(k\) is related to the wavelength \(\lambda\) via the formula: \(k = \frac{2\pi}{\lambda}\). Since wave speed \(v\) is given by \(v = \lambda f\) and frequency \(f = \frac{1}{T}\), first calculate the wavelength:\[ \lambda = \frac{v}{f} = \frac{12\, \mathrm{m/s}}{\frac{1}{0.77\, \mathrm{s}}} = 12 \times 0.77 = 9.24\, \mathrm{m} \].\Then, calculate \(k\):\[ k = \frac{2\pi}{9.24} \approx 0.68\, \mathrm{rad/m} \].

Step 4: Write the wave equation

Now, with values calculated, plug them into the wave equation formula. The wave equation is now:\[ \psi(x, t) = 0.37 \sin(0.68x + 8.16t) \]. Since the wave travels in the negative x-direction, maintain the plus sign with \(x\) term.

Key Concepts

Wave Properties

Waves are traveling disturbances that carry energy from one place to another without the accompanying transport of matter. Several key properties define a wave: amplitude, wavelength, frequency, period, and wave speed. Understanding these properties helps us describe and analyze wave behavior.

- **Amplitude** refers to the maximum displacement of the wave from its rest position. It determines the wave's intensity.
- **Wavelength** (\( \lambda \)) is the distance between successive points of similar phase on the wave, such as crest to crest or trough to trough.
- **Frequency** (\( f \)) is the number of cycles the wave completes in one second and is measured in hertz (Hz).
- **Period** (\( T \)) is the time it takes for one complete cycle of a wave to pass a given point.
- **Wave speed** (\( v \)) is the rate at which the wave propagates through a medium and is calculated by the product of frequency and wavelength: \( v = f \lambda \).

These basic properties remain consistent for all forms of waves, whether they're sound, light, or water waves.

Wave Number

The wave number (\( k \)) is a crucial concept when discussing wave mathematics. It is defined as the number of wavelengths per unit distance and is given by the formula:\[ k = \frac{2\pi}{\lambda} \]where \( \lambda \) is the wavelength.

The wave number indicates how many wavelengths fit into a given length and is measured in radians per meter (rad/m). By providing insights into the spatial frequency of the wave, the wave number helps describe the wave's structure and is pivotal for solving wave equations. In the context of the given exercise, the wave number gives us a direct understanding of how tight or spread apart the wave oscillations are over a particular distance.

Angular Frequency

Angular frequency (\( \omega \)) is a measure of how quickly the phase of the wave changes with time. It is related to the period (\( T \)) of the wave by the equation:\[ \omega = \frac{2\pi}{T} \]where \( T \) is the period.

Typically measured in radians per second (rad/s), angular frequency tells us how many radians the wave travels through per unit of time. This concept is essential when considering cyclical processes, as it provides an understanding of how fast the wave's oscillations occur over time.

In our exercise, angular frequency is calculated based on the given period, allowing us to express the wave's speed in terms of how rapidly a point on the wave oscillates up and down.

Amplitude Calculation

Amplitude calculation involves determining the maximum extent of oscillation of the wave from its equilibrium position. This parameter is crucial because it dictates the energy carried by the wave. In mathematical terms, the amplitude \( A \) appears as a coefficient in the wave equation:\[ \psi(x, t) = A \sin(kx + \omega t + \phi) \]where \( A \) represents the amplitude.

For our specific exercise, the amplitude is given as 0.37 meters. This value directly influences the intensity of the wave's oscillations, and it is integral to constructing the complete wave equation. Accurately calculating and substituting amplitude into the wave equation helps describe the wave's behavior visually and physically.