Visualization and Analysis Tools

Exercises in this subsection are designed to introduce important properties such as wavelength, frequency, eigenfunctions (i.e., modes) and dispersion. Readers who are familiar with these topics are urged to scan these exercises since the visualization and analysis tools that will be used for more advanced problems are also introduced.

• Classical Wave
    Load the WAVE program from DOS. The default values for the analysis graphs (energy density and contour) should be visible. Select a right-traveling Gaussian wave from the [Init] main menu. Run the program until the contour plot first appears and pause the program.

  a.

  What happens to the wavefunction at the right boundary? What happened to the energy distribution?

  b.

  Why does the density plot zig-zag and why do the colors alternate between red and blue? Expand the contour plot to full screen using the red expand button in the upper right-hand corner. Click on the blue button on the bottom right corner of the contour plot and change the attribute to 3D.

  c.

  Using the main menu, change the boundary condition to Periodic and rerun the experiment. (Click the [F7-Reset]

  hot key to reset the initial condition. Don't go to the main menu!) Explain the differences.

Note: You may need to adjust the program execution speed for subsequent exercises. The default setting may be too fast for the classical wave equation and too slow for the diffusion equation. Refer to the section on running the program for hints on how to adjust various parameters for optimum performance.

• Diffusion
    Change the equation type to diffusion under the [Parameters]

  menu, change the boundary condition back to Fixed , and place a Gaussian near the right boundary as the initial condition. Select the contour plot and Space Integrated I(t)=y*y as the analysis plots. Run the program until the contour plot first appears and pause the simulation.

  a.

  What happens to the function at the right boundary? Hint: you may need to rescale the Space Integrated I(t)=y*y graph.

  b.

  Using the main menu, change the boundary condition to Periodic and repeat the experiment. Explain the differences between [a.] and [b.].

• Schrödinger
    Change the equation type to Schrödinger under [Preferences] . Select a Gaussian initial condition. Reselect the contour plot in one of the analysis plots and change the boundary condition back to Fixed . Run the program until the contour plot first appears and pause the program.

  a.

  What happens to the wavefunction at the right boundary?

  b.

  Change the boundary condition using the main menu to Periodic and repeat. Are there any differences?

  c.

  Select the Gaussian initial condition but change the default conditions by setting the momentum--- which happens to be equal to the wavenumber, k, since --- to zero. Run the program. What similarities do you see with the solution to the classical wave equation? To the diffusion equation?

• Klein-Gordon
    The behavior of a wavefunction can change markedly at different scales for wavefunctions with frequency-dependent phase velocity. In this exercise, the behavior of short wavelengths will be compared to that of longer wavelengths in a Klein-Gordon equation. Change the equation type to Klein-Gordon under [Parameters] . Resize the medium by setting the left boundary to -50 and the right boundary to 50 under [Parameters]|[Space Parameters] . Reselect the contour plot in one of the analysis plots and change the boundary condition back to Fixed .

  a.

  Select a bidirectional Gaussian initial condition with a width of . Run the program until the contour plot first appears and pause the program. Notice the phenomenon of pulse propagation, as one might expect after having selected a bidirectional Gaussian. Although the pulses do not remain well-formed, disturbances propagate in both positive and negative directions, as before. Estimate the velocity of the disturbance from the slope of the outer contour isoclines.

  b.

  Now select a Gaussian initial pulse with a width of . Run the program and note the differences from [a.]. Clearly, the width of the wavefunction has a dramatic effect on the time evolution of the system modeled by the Klein-Gordon equation.

You should reset the Space Parameters to and after running the Klein-Gordon exercise. Rather than resetting the size of the medium, the boundary conditions, and the number of grid points to default values, we have chosen to change as few variables as possible when switching types. Unfortunately, the analysis graphs are very dependent on the equation type and must be reselected by the user whenever the equation type is changed.