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Learning Introductory Physics with Activities

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

Section 7.2 Representing Waves

A first step in building a wave model is to consider different representations (words, graphs, equations, etc.) to help describe the behaviors of waves. While the figure below shows actual waves in time and space, simplified graphical representations will be more useful.
Figure 7.2.1. An oscillating source creates a wave with the same frequency.

Exercises History Graphs

When modeling the motion of point particles, you primarily used a Motion Graph to show, for example, the position of an object across many different instants in time. This sort of graph can still be used to show the motion of one position in a medium over time. This is known as a history graph.
Figure 7.2.2. A graphical representation of a sinusoidal wave in time (holding space fixed).
When you have a history graph, a quantity of note is the period \(T\text{:}\) the time it takes to complete one full oscillation. As in simple harmonic motion, the period is related to the frequency by
\begin{equation*} T = \frac{1}{f} = \frac{2\pi}{\omega} \end{equation*}
Another quantity of note is the amplitude: the maximum displacement away from equilibrium of a particle in the medium.

Exercises Snapshot Graphs

Alternatively, you can graph the displacement of the medium over space at a single instant in time, creating a snapshot graph.
Figure 7.2.4. A graphical representation of a sinusoidal wave in space (holding time fixed).
Now, the quantity of note is the distance spanned from one crest of the wave to the next, or the wavelength, symbolized by \(\lambda \text{.}\)
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.

Exercises Activities

1. Exploring Wavenumber.

A friend tells you about a wave with wavenumber \(k = \frac{4\pi}{7} \mathrm{~\frac{rad}{m}}\text{.}\) What is the wavelength of this wave? What would the wavenumber be for a different wave with double the wavelength?

References References

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