Section 5.11: Exploring the Dispersion of Classical Waves
The most common example of dispersion is the splitting of white light into a rainbow of color as it passes through a prism. Different wavelengths of light travel though glass at different speeds and so different colors of light are refracted different amounts resulting in a rainbow. This animation allows you to explore dispersion of waves on a string as modeled by a chain of coupled oscillators. You can set the chain in motion by typing in different functions or dragging a red ball before you start the simulation and watch the oscillation of the chain over time (time is given in seconds and position in cm). Note that if you change a parameter, the text box stays yellow until you push "enter" and then you must push the "Init" button to initialize the animation.
- Start with the initial equation of sin(2*pi*x) and #Part = 16 and determine the period of the oscillation. What is the speed of the wave (v = ω/k = 2π/kT)?
- Half the wavelength (Input the equation: sin(4*pi*x) and initialize the animation). What is the period of the oscillation? Speed of the wave?
- Continue changing the wavelength and measuring the period and the speed of the wave. How does the wave speed change with the wavelength?
- Enter e^(-25*(x-0.5)*(x-0.5)) for the initial condition of a Gaussian wave packet. Describe how its shape changes over time.
Note that this varying wave speed is a variation in the phase velocity of the wave. This is dispersion of a wave: a difference in phase velocity depending on the wave number. This has implications for the time evolution of a wave packet constructed out of a series of these waves. Here the dispersion leads to the change of the wave packet shape over time.
This Exploration uses an Open Source Physics Quantum Mechanics Applet.