Section 7.3 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. Sinusoidal waves are periodic in both space and time, so the displacement of a particle in a medium is symbolized by a function like \(D(x,t) \) or \(y(x,t) \text{.}\)
\begin{equation*}
y(x,t) = y_{\mathrm{max}} \sin\left(\frac{2\pi}{\lambda}x \pm \frac{2\pi}{T}t + \phi_i\right)
\end{equation*}
A sinusoidal wave of this form is also known as a traveling wave. Why do you think this might be a good name?
Definition 7.3.2. Initial Phase.
The phase of a wave, typically written as \(\phi\text{,}\) refers to where in a cycle (from \(0\) to \(2\pi\)) a sinusoidal wave is at any given point in time and space. The initial phase, \(\phi_i\text{,}\) specifies the phase at the origin at \(t = 0\text{.}\)
Principle 7.3.3. Wave Speed.
For sinusoidal waves, the wave speed can be written
\begin{equation*}
v = \frac{\lambda}{T} = \mathrm{\frac{distance}{time}} = \lambda f
\end{equation*}
For a traveling wave, each wave crest travels forward a distance of one wavelength \(\lambda\) during a time interval of one period \(T\text{.}\)
Additional Detail 7.3.4. Wave Speed and Elasticity.
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{v}{f} = \frac{\text{medium}}{\text{source}}
\end{equation*}
Exercises Activities
1. Graphing Phase.
Using a graphing program (like Desmos or Mathematica), try graphing the following functions (use \(\lambda = 6 \mathrm{~m}\)).
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda})} \)
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda} + \frac{\pi}{3})}\)
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda} + \frac{\pi}{6})}\)
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda} - \frac{\pi}{3})}\)
2. 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?
