Wave motion visualization

How to run a CDF Player simulation
download WaveMotion.cdf
download WaveMotion.nb (Mathematica notebook)

Overview

This is a demonstration, created with the software Wolfram Mathematica, about wave motion in a 1-dimensional media (i.e. a string).
The demonstration can be interactively used with the free CDF Player (see here how to install it).

Wave motion

It’s not easy to understand the mathematical inner workings of moving waves.
That’s because a wave is a perturbation (in a media) that changes with space and time.
So, even in the simplest case (i.e. waves propagating in a 1-dimensional media like a string) there are at least three variables involved: amount of the perturbation, position and time.

This means that the mathematical description of a wave (called wave equation or wave function) must have the general form:

$y = \psi \left( {x,t} \right)$

where $y$ is the perturbation (displacement with respect of the equilibrium position), $x$ is the position and $t$ is the time.
In above function there will also be the main wave constant parameters i.e. the wavelength $\lambda$ and the period $T$ .
The most commonly used wave function is that of a harmonic wave that can be written in the form:

$y=A \cos\left( {2\pi \left( {\frac{x}{\lambda } - \frac{t}{T}} \right)}+ {\varphi _0} \right)$

where ${\varphi _0}$ is the initial phase and $A$ is the wave amplitude.
All the other interesting wave parameters can be derived from $\lambda$ and $T$ :
$f = \frac{1}{T}$ (frequency)
$k = \frac{{2\pi }}{\lambda }$ (wave number)
$\omega = \frac{{2\pi }}{T}$ (angular frequency)
$v = \frac{\lambda }{T}$ (wave phase speed)

The aim of this simulation is to better understand the interplay between the wave function variables, the wave parameters and the moving wave corresponding properties.

The interactive demonstration shows 4 different views;

• The moving wave
• The time view at a fixed position ( $x_0$ )
• The space view at fixed times (when the flash comes)
• The 3D view, in which there is also the moving point representing the state of the perturbation at $x_0$ as time goes by and its space-time trajectory.
In the demonstration panel you can change the wave parameters (the wavelength $\lambda$ and the period $T$ ), or the point $x_0$ in which the time evolution is examined and see how the wave dynamic evolution changes accordingly.
The wave speed cannot be directly changed as it depends on $\lambda$ and $T$ through the formula $v = \frac{\lambda }{T}$ .

Using the simulation

Click on the play button () in the time animation section to start the time evolution. See what happens by changing the wave parameters $\lambda$ and $T$ .
The first graph represent a dynamic 2D view of wave moving in space.
The second graph is the time view and represents the wave perturbation at a fixed position ( $x_0$ ). It’s possible to change this position by moving the $x_0$ slider.
The third graph is intended to highlight the wave profile at some fixed times and will represent the space view wave snapshots taken at some chosen times (every 7 time units). The snapshot is taken when there’s a flash in the first graph and the wave profile in the third graph stay still till the next update.
The fourth graph is the overall mathematical representation of the time view and the space view, combined in a single 3D graph. The red path represents the time evolution of the perturbation at the position $x_0$ .
It’s also possible to click and drag with the mouse on the 3D graph to change the point of view.

In the first row of the controls section it’s possible to choose between two different kinds of wave function, select the animation rate (speed of the animation) and whether to show the 3D view.
For the ones more curious about the mathematical form of the wave functions used in the function type selector here are the details:
• Asymmetric wave:

$y = \psi \left( {x,t} \right) = {\sin ^2}\left[ {{{\left( {\sqrt \pi \left( { - \frac{x}{\lambda } + \frac{t}{T}} \right)\bmod \sqrt \pi } \right)}^2}} \right]$

The square on the $\sin$ argument has the effect to make the wave profile asymmetric with respect to the half wavelength (or half period) point. The modulus operator (by $\sqrt \pi$ ) is needed to readjust things and make the wavelength/wave period parameters constant.
The reason for creating an asymmetric wave is the interesting fact that the wave profile in the space view and in the time view are similar but somewhat mirrored…
• Harmonic wave:

$\[y = \psi \left( {x,t} \right) = - \frac{1}{2}\cos \left[ {2\pi \left( { - \frac{x}{\lambda } + \frac{t}{T}} \right)} \right] + \frac{1}{2}$

Here the $\frac{1}{2}$ factor is used to make the wave oscillate in the range $\left[ {0,1} \right]$ .
The minus sign before the $\cos$ function is equivalent to set an initial phase ${\varphi _0} = \pm \pi$ , that was just appropriate for the graph visualization.

Note: selecting the check-box to show the 3D view can make the simulation much slower and jerky in the browser.
Anyway the simulation will run smoother and faster (also with the 3D graph enabled) if you download the WaveMotion.cdf file on your PC and then and run it directly in the CDF Player program (free – see the page “How to run CDF demonstrations”).