Eigenfunctions

  The previous exercises introduced some of WAVE's features. They suggest similarities and differences among the various equations; we will examine many of these similarities and differences in the following exercises. We begin by considering the eigenfunctions or modes for the four linear PDEs for a uniform medium of length L with fixed boundaries.

Various equations and eigenfunctions can be selected by first choosing an EquationType under [Parameters] followed by [Init]|[Modes] and then selecting the eigenfunction number. Modes 1 through 5 can be selected by checking the appropriate box. Modes that are not listed can be selected using the two input fields at the bottom of the screen. Select the Fixed boundary condition before you begin the following exercises; eigenfunctions are different for different boundary conditions.

  
Table: Eigenfunctions for various PDEs with fixed boundaries on .

• Fixed-Boundary Eigenfunctions
     

  a.

  Select [Parameters]|[Electromagnetic] and set the initial condition to mode 4. Create a detector by clicking on the blue button (labeled ``d") in the lower left corner of the wave graph and then clicking inside the wave graph. Create two additional detectors. Run the program. Notice that all the detectors have the same frequency but different phases. Measure the frequency of this mode by holding down the mouse inside the detector graph and reading the time at two different maxima. Measure the wavelength by holding the mouse down in the wave graph and reading the separation between two maxima. Calculate the phase velocity, . Place the system in another mode and repeat. Are the phase velocities the same?

   

  b.

  Repeat [a.] for the Schrödinger equation. Since the quantum detectors measure propability current, they cannot be used to measure phase oscillations. You can, however, follow a phase oscillation in the bottom graph by looking at the real (or the imaginary) part of the wavefunction. Examine modes 1, 4, 16 and determine the frequency for each mode. Are the phase velocities the same?

   

  c.

  Repeat [a.] for the Klein-Gordon equation but set the ends of the medium to be at . Try modes 1, 4, 16. Are the phase velocities the same?  

  d.

  The functional relationship between wave number, and the angular frequency, for a harmonic wave is called a dispersion function. Determine the dispersion relationship, , for the equations in Table that exhibit oscillatory behavior. A wave equation is said to be nondispersive if and k are proportional. Which of the equations in Table are nondispersive? Which are dispersive?  

  e.

  Repeat [a.] for the diffusion equation. Even though these solutions do not oscillate, there is a time associated with each mode, called the characteristic time, that determines how long it takes for the function to decay to of its initial value at a point. Determine the characteristic time for modes 1, 4, 16.

• Periodic-Boundary Eigenfunctions
      Change the boundary to Periodic and examine the modes as in the previous exercise. It is important to note that these modes are not the same as for Fixed boundaries.

  a.

  How do the eigenfunctions change when you switch to periodic boundary conditions? Try both positive and negative modes. Write the analytical form of the eigenfunctions for periodic boundaries.

  b.

  Using Periodic boundary conditions with an [Electromagnetic] wave, place the system simultaneously in modes 4 and -4 and run the simulation. Is it possible to write the eigenfunctions for Periodic boundaries in terms of the functions listed in ? Is the reverse possible?

   

• Beats
    The phenomena of beats is usually covered in introductory physics texts for the classical wave equation. You should now be familiar enough with the program to construct simulations that demonstrate the phenomenon of beats for other equations. Do you notice differences in the beat patterns when the simulation is run for the Schrödinger equation? The Klein-Gordon eqution? Hint: Set the medium to for interesting effects with the Klein-Gordon equation.