Representation 17.2.2. History Graph.
A history graph for wave motion shows the displacement of a fixed part of a medium over time.
Section 17.2 Representing Waves
Subsubsection 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.
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.
Subsubsection Snapshot Graphs
Alternatively, you can graph the displacement of the medium over space at a single instant in time, creating a snapshot graph.
Representation 17.2.4. Snapshot Graph.
A snapshot graph for wave motion shows the displacement of all parts of a medium at a fixed time.
Now, the quantity of note is the distance spanned from one crest of the wave to the next, or the wavelength, symbolized by \(\lambda \text{.}\)
Definition 17.2.5. Wavelength.
The wavelength \(\lambda\) is the amount of distance for a wave to progress through one full cycle.
Definition 17.2.6. Wavenumber.
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 described by angular frequency, frequency, and period. There is an analogous spatial relationship between wavelength and a quantity called the wavenumber \(k\text{:}\)
\begin{equation*}
k = \frac{2 \pi}{\lambda}
\end{equation*}
The wavenumber represents the spatial frequency of a wave over a unit distance, and is measured in radians per meter.
Subsubsection Wave Speed Relation
A traveling wave propagates away from its source at a particular speed called the wave speed. Since both the period \(T\) and the wavelength \(\lambda\) correspond to a single cycle of a wave, they may be combined as follows:
\begin{equation*}
v = \mathrm{\frac{distance}{time}} = \frac{\lambda}{T} = \lambda f
\end{equation*}
However, it is worth noting that wave speed is not determined by the frequency of the wavelength. Instead, for mechanical waves, wave speed is a property of the medium’s elasticity! Combining this with knowledge of the frequency of the oscillating source allows you to determine the wavelength:
\begin{equation*}
\lambda = \frac{v}{f} = \frac{\text{medium}}{\text{source}}
\end{equation*}
Principle 17.2.7. Wave Speed.
Wave speed, wavelength, and frequency are related as follows:
\begin{equation*}
\lambda = \frac{v}{f}
\end{equation*}
\begin{equation*}
v = \lambda f
\end{equation*}
In some cases, it is useful to represent \(v\) in terms of \(k\) and \(\omega\)
\begin{equation*}
v = \frac{\omega}{k}
\end{equation*}
Subsubsection Activities
Activity 17.2.1. Exploring Wavenumber.
A friend tells you about a wave with wavenumber \(k = \frac{4\pi}{7} \mathrm{~\frac{rad}{m}}\text{.}\)
(a)
What is the wavelength of this wave?
(b)
What would the wavenumber be for a different wave with double the wavelength?
(c)
Make a sketch of both waves on the same axes. Which kind of graph do you need to choose to show the difference between the two waves?
References References
[1]
History Graph Figure created by Rebecka Tumblin.
