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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 17.3 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 (more specifically, a traveling 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 \(y(x,t) \) or \(D(x,t) \text{.}\)
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Definition 17.3.1. Sinusoidal Wave.

A sinusoidal wave with amplitude \(y_{\mathrm{max}}\text{,}\) wavelength \(\lambda\text{,}\) period \(T\text{,}\) and initial phase \(\phi_i\) can be written
\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*}
Alternatively, the wave may be written in terms of the wavenumber \(k = \frac{2\pi}{\lambda}\) and the angular frequency \(\omega = \frac{2\pi}{T}\text{:}\)
\begin{equation*} y(x,t) = y_{\mathrm{max}} \sin\left(kx \pm \omega t + \phi_i\right) \end{equation*}
The sign of the \(\pm\) determines the direction of travel: choosing \(-\) corresponds to a wave moving in the positive direction, while choosing \(+\) corresponds to a wave moving in the negative direction.
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Definition 17.3.2. Initial Phase.

The phase of a wave, typically written as \(\phi\text{,}\) refers to where in a cycle (from \(0\) to \(2\pi\)) a wave is at any given point in time and space. The initial phase, \(\phi_i\text{,}\) specifies the phase at the origin at \(t = 0\text{.}\)
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Subsubsection Activities

Activity 17.3.1. Graphing Phase.

Use a graphing program (like Desmos or Mathematica), to graph the following functions (use \(\lambda = 6 \mathrm{~m}\)).
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  • \(\displaystyle \sin{(2\pi\frac{x}{\lambda})} \)
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  • \(\displaystyle \sin{(2\pi\frac{x}{\lambda} + \frac{\pi}{3})}\)
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  • \(\displaystyle \sin{(2\pi\frac{x}{\lambda} + \frac{\pi}{6})}\)
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  • \(\displaystyle \sin{(2\pi\frac{x}{\lambda} - \frac{\pi}{3})}\)
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Activity 17.3.2. 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?
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