TI-83 Beat Investigation

Objective:

Use the TI-83 calculator to simulate constructive and destructive wave interference.

Background:

The mathematic relationship known as the sine function (sin) can be used to draw a waveform. This waveform could represent sound waves, light waves or water waves. The amplitude (height) of a waveform (displayed on the y-axis) is given by the formula Y = SIN (X) where X is the x-axis value and sets the wave frequency. A larger coefficient on X means a faster frequency. We will typically use units of time (seconds) or length (meters) for the x-axis.

When two waves interact (interference), their amplitudes are added together instant by instant. If the polarity of the waves are the same (both positive or both negative) the wave amplitudes are added in a constructive manner. If the wave amplitudes are of opposite polarity the addition is destructive, meaning the amplitude of the resulting wave less than the original waves.

This real-life phenomenom can be simulated by adding two sine waves and having the calculator display the graph. We will do this and simulate two waves interacting when they have slightly different frequencies. This will show us the beat frequency of the two waves.

Procedure:

On the TI-83:

  1. Turn off all plots (STAT PLOT) and clear equations in the equation editor (Y=).
  2. Set the mode to degrees (MODE).
  3. Set the graph window (WINDOW) to Xmin=0, Xmax= 360, Xscl=90, Ymin=-2, Ymax=2, Yscl=1.
  4. In the equation editor (Y=): set Y1 = sin(X).
  5. Press GRAPH and observe the waveform.
  6. In the equation editor (Y=): set Y2 = sin(1.1X).
  7. Press GRAPH and observe the waveform.
  8. Display more of the waveform by setting the graph window (WINDOW) to Xmax= 540. Note that you have not changed the frequency of the wave, just more of the wave train is shown.
  9. In the equation editor (Y=): set Y3 = sin(X) + sin(1.1X).
  10. Press GRAPH and observe the waveform.
  11. Use TRACE and the arrow keys to verify the constructive and destructive interference of waveforms Y1 and Y2 as graphed by Y3.
  12. Clear equations Y1 and Y2 in the equation editor (Y=).
  13. Press GRAPH and observe the waveform.
  14. Display more of the waveform by setting the graph window (WINDOW) to Xmax= 1000.
  15. Press GRAPH and observe the waveform.
  16. Display more of the waveform by setting the graph window (WINDOW) to Xmax= 5000.
  17. Press GRAPH and observe the waveform.
  18. Display more of the waveform by setting the graph window (WINDOW) to Xmax= 8000.
  19. Press GRAPH and observe the waveform.
  20. Use TRACE and the arrow keys to measure the beat frequency .
  21. In the equation editor (Y=): set Y3 = sin(X) + sin(0.9X).
  22. Press GRAPH and observe the waveform.
  23. Use TRACE and the arrow keys to measure the beat frequency .

Observations: