Section 5.11: Exploring the Dispersion of Classical Waves

This simulation has not been converted from Java to JavaScript.

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.

  1. 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)?
  2. 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?
  3. Continue changing the wavelength and measuring the period and the speed of the wave. How does the wave speed change with the wavelength?
  4. 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.