1. Exploring Wavenumber.
A friend tells you about a wave with wavenumber \(k = \pi/2 \frac{\text{rad}}{\text{m}}\text{.}\) What is the wavelength of this wave? What would the wavenumber be for a different wave with double the wavelength?
Section 7.2 Sinusoidal Waves
Waves can take any shape or size, and do not necessarily have a regular, smooth, repeating pattern. However, if a wave source oscillates with simple harmonic motion, then the wave that is generated will be a sinusoidal wave. A first step in building a wave model is to build different representations (words, graphs, equations, etc.) to help describe the behaviors of waves.
An important aspect of sinusoidal waves is that they are periodic in both space and time. The displacement \(D(x,t) \) of a particle in the medium depends on both the particle’s position \(x \) and the time \(t\) when you observe it. In Figure 7.2.2, we show a graphical representation of a sinusoidal wave in space and time. The period of the wave, is related to the frequency of the wave
\begin{equation*}
T = \frac{1}{f}
\end{equation*}
and has the same meaning as in simple harmonic motion. Period is the time it takes to complete one oscillation and frequency is the number of cycles per second. The amplitude of the wave is the maximum displacement away from equilibrium of a particle in the medium.
In the second panel, the displacement of the medium over space is shown at one instant in time. The distance spanned from one crest of the wave to the next is called the wavelength \(\lambda \text{.}\)
Recall from simple harmonic motion the relationship between angular frequency, frequency, and period,
\begin{equation*}
\omega = 2 \pi f = \frac{2 \pi}{T}
\end{equation*}
Time is the physical quantity that connects angular frequency, frequency and period. There is an analogous spatial relationship between wavelength and a quantity called the wavenumber \(k \)
\begin{equation*}
k = \frac{2 \pi}{\lambda}
\end{equation*}
The wavenumber represents the spatial frequency of a wave over a unit distance. Another fundamental relationship of sinusoidal waves is the wave speed represented by the symbol \(\nu \text{,}\)
\begin{equation*}
\nu = \frac{\lambda}{T} = \frac{\text{distance}}{\text{time}} = \lambda f
\end{equation*}
For sinusoidal waves, each wave crest travels forward a distance of one wavelength \(\lambda\) during a time interval of one period \(T\text{.}\)
It is important to note that, for mechanical waves, wave speed is a property of the medium’s elasticity while the frequency of the wave is the frequency of the oscillating source of the wave. To keep this important aspect of waves in mind, you can use the relationship
\begin{equation*}
\lambda_{\text{mechanical}} = \frac{\nu}{f} = \frac{\text{medium}}{\text{source}}
\end{equation*}
Now that you have a grasp on the verbal and graphical representations of waves, you can examine the mathematics of sinusoidal waves. The displacement of a sinusoidal wave can be represented mathematically using the displacement function
\begin{equation*}
D(x,t) = A \sin(kx \pm \omega t + \phi_o)
\end{equation*}
which describes the displacement at time \(t \) of a particle at position \(x \) in the medium.
Exercises Activities
2. Graphing Phase.
You have probably noticed the letter \(\phi\) (phi) typically represents a phase inside of a sine or cosine function. Using a graphing program (like Desmos or Mathematica), try graphing the following functions (use the value for \(k\) from the previous activity).
- \(\displaystyle \sin{(kx)} \)
- \(\displaystyle \sin{(kx + \frac{\pi}{3})}\)
- \(\displaystyle \sin{(kx + \frac{\pi}{6})}\)
- \(\displaystyle \sin{(kx - \frac{\pi}{3})}\)
3. Exploring Phase.
Based on your graphs, what is the mathematical role of the phase in an oscillation or wave equation? What do you think the physical role of the phase is?
References References
[1]
Gif image in Figure 7.2.1 courtesy of Veritasium.
https://www.youtube.com/watch?v=Iuv6hY6zsd0
[2]
Figure 7.2.2 created by Rebecka Tumblin.
[3]
Animations courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State. licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. For more information see
https://www.acs.psu.edu/drussell/demos.html
