Resonance

• E&M Sources
  Set the equation type to electromagnetic and select [Init]|[Zero]

  to clear the wavefunction. Click on the red source button in the function display and create a source by clicking in the middle of the display. Exit the inspector panel and run the simulation with the default parameters. Notice that a right-traveling Gaussian pulse is produced at a time of sec and that this pulse passes through the source after reflecting from the fixed boundary. Double-click on the source to bring up the inspector panel again. You can now adjust the source parameters using the sliders and buttons in the panel.

  Clicking the Adv Options button displays a second panel with additional source properties. Forcing the wavefunction to follow the source has a dramatic effect on the simulation, since it effectively adds a boundary condition to the medium. Run the program to see the effect. The Periodic switch allows the user to select a repetition rate for Gaussian and most other pulses. Periodic has no effect on sinusoidal or user-defined functions. Coherence Time , on the other hand, affects only sinusoidal sources.

• Power
  Consider the standard introductory physics demonstration consisting of a string fixed at one end with a tuning fork at the other. The forced end produces a wave which is reflected from the fixed end, then reflected again from the forced end. This gives rise to resonance effects; the phase relationship between the traveling waves and the source determines the amplitude of the resonance. Start this exercise by selecting [Parameters]|[Electromagnetic] and [Init]|[Zero] and choosing Fixed boundary conditions. Create a source and set the following properties:
  • Function Type: Sine
  • No Drag: TRUE
  • Amp:
  • Freq:
  • Pos:
  • (Advanced Options) Direction: Right
  • (Advanced Options) Track Source: TRUE
  Select space integrated, E(t)=, and source power, , analysis plots.

  a.

  Run the program and explain the behavior of the two analysis graphs. What is the phase relationship between the reflected, i.e, left-traveling, wave and the source? Change the source frequency to and repeat after hitting [F7] to reset. What happens now?

  b.

  In order to simulate the ``real'' system you will need to add some damping. Create a segment and edit its properties as follows:

  • Position:
  • Width:
  • Index:
  • Gain: -0.1
  • Color: Whatever looks good!
  • Able to Drag: FALSE

  Run the program until a steady-state is reached and record the energy for frequencies from to in increments of . Plot these energies and predict the steady-state energy of the system as a function of frequency. Note that the source end will not be a node even though it acts like a fixed boundary for the traveling wave. Can you predict how the position of the node closest to the source will vary with frequency?