Superposition and Interference

    Restart the program or select Electromagnetic and Fixed boundaries if the default conditions have been changed. Select an energy density data analysis graph, , and use the [F3 Offset] hot key to separate the left, right, and total wave components.

• Energy
  Use [Init]|[Gaussian] to select an initial right-traveling wave centered at with an amplitude of and a width of .

  a.

  Integrate the square of the initial wavefunction, . What is the relationship of this integral to the Energy reading displayed in the blue stripe in the wave graph?

  b.

  Run the simulation until the peak of the Gaussian coincides with the right boundary. What is the value of the integral now?

  c.

  Use [Init] to create a second left traveling Gaussian centered at with an amplitude of and a width of . Check the add to existing wave button and run the simulation until the two pulses overlap. Does the energy reading change when the two waves overlap? Does the integral change?

  Be sure to consider the magnetic field for a traveling wave when you answer the previous questions. If the electric field is positive, a right-traveling wave will have a magnetic field pointing out of the screen while a left traveling wave will have a magnetic field pointing into the screen. Note also that the phase shift at a boundary is different for magnetic and electric fields.
• Standing Wave Energy
  Use [Init]|[Modes] and select mode 4. Select two energy distribution data analysis graphs, . Click the blue attributes button on the left analysis graph and select Electric ; click the blue attributes button on the right analysis graphs and select Magnetic . Run the program and explain the behavior of these two graphs. What is the relationship between the nodes of the standing wave and the zeros in the analysis graphs?
• Quantum interference
  Select Schrödinger and Fixed and an energy distribution data analysis graph, . Select an initial Gaussian wavefunction centered at with an amplitude of , a width of and a momentum of +50. Add to this a second Gaussian centered at with an amplitude of , a width of and a momentum of -50. Collide these two Gaussian pulses but pause the simulation when the pulses overlap. What do you observe in the analysis graph? Use a mouse-down to measure the separation of the zeros in the analysis graph. How is this separation related to the momentum?
• Multiple Sources
  Set both boundary conditions to absorbing. Create three overlapping sinusoidal sources at with the following frequencies and amplitudes respectively: and . Set the NoDrag button to true so that the sources will not inadvertently be moved. The output of the sources should approximate a square wave, since we have selected the first three nonzero Fourier coefficients of this function. Verify this.

• Coherence Time
    No natural sinusoidal source is a perfect monochromatic wave; it must have a starting and a stopping time. If we assume that a one Hertz source is at maximum amplitude at some time, t=0, then it might be safe to assume it will be at maximum at t=2 or t=3 sec. But is it safe to assume a maximum at 100 sec? The length of time that we can make a prediction about the phase of a sinusoidal source is called the coherence time. The coherence time has been implemented for sinusoidal sources in the Adv Options submenu of the Source Inspector .

  a.

  Create a medium with absorbing boundaries containing a detector at and a sinusoidal source with the following properties:

  • Position: -0.4
  • Frequency: 10 Hertz
  • Amp: 0.5
  • No Drag: TRUE
  • Noise: TRUE
  • Amp Noise: 0.0
  • Coherence Time: 1.0
  • Direction: Right

  Select a detector analysis plot and an FFT of the detector readings, FFT-Y(t). Use the blue attributes button to set the number of points in the FFT to 1024. Set the display speed to 1 for maximum time resolution. Run the program. Notice the abrupt phase change approximately every second. The FFT is no longer a single peak but a distribution of frequencies. Using paper and pencil, calculate the Fourier transform of a sine wave of finite duration.

  b.

  Create 8 additional sources identical to and overlapping the first source. These sources are initially in phase and their amplitudes add to . They slowly drift out of phase with each other and after about 1 second their phases are random. Estimate the r.m.s. amplitude and standard deviation of N sources with random phases and compare to the readings on the detector plot.