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\right)
\end{equation*}
For sinusoidal waves, the wave speed can be written
\begin{equation*}
v = \frac{\lambda}{T} = \mathrm{\frac{distance}{time}} = \lambda f
\end{equation*}
Each wave crest travels forward a distance of one wavelength \(\lambda\) during a time interval of one period \(T\text{.}\)
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.
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 \(\lambda = 6 \mathrm{~m}\)).
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda})} \)
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda} + \frac{\pi}{3})}\)
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda} + \frac{\pi}{6})}\)
- \(\displaystyle \sin{(2\pi\frac{x}{\lambda} - \frac{\pi}{3})}\)
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?
