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At the request of Gibson (GIBSON@VELA.ACS.OAKLAND.EDU)
I am posting the following lab instruction sheet
(refered to in a previous message) that I use with my
high school physics classes to have them incorporate
graphing calculators in the study of waves.

Ed Schweber
e-mail: EdSchweb@ix.netcom.com


Physics Lab: Using a Graphing Calculator to Study Waves


Today we will use the graphing calculators to study the basic
properties of waves.

Author's note: The following settings apply to the TI-81
               calculator, since that's what I have at
               home. My school had TI-85 calculators.
               The settings on these were slightly different,
               but easy to figure out  if you are familiar
               with the TI-81

Mode Settings: Norm; Float 2; Rad; Function; Connected; Sequence;
               Grid Off; Rect.

Range Settings: Xmin = 0
                Xmax = 2
                Xscl = 0
                Ymin = -3
                Ymax = +3
                Yscl = 0
                Xres = 1

Part I: Graphing Basic Waves
        Author's note: My students typically know right triangle
                       but don't know what a sine curve is. This
                       first part of the lab enables them to see
                       the connection between the sine curves
                       (the waves) and the mathematical formulas
                       they will be graphing.:

   (1) Enter Y1 = sin[2(pi)x] and graph it
        (a) Trace the graph to determine the wavelength and the
             amplitude of the wave.

   (2) Enter Y1 = 2sin[2(pi)x] and graph it
        (a) How does this wave differ from the previous wave?
            Be specific! Be numerical!

   (3) Enter Y1 = sin[4(pi)x] and graph it
        (a) What is the wavelength and the amplitude?

   (4) Write the equation of a wave that would have the same
       wavelength as in the previous part; but which would have
       an amplitude of 3. Test your prediction by entering your
       equation on the calculator and graphing it Does it work?

   (5) Write the equation of a wave that has an amplitude of 2
       and a wavelength of 2. Test your prediction. Does it work?

   (6) Write the equation of a wave that has an amplitude of 1
       and a wavelength of 0.25. Test your prediction. Does it work?

   (7) What is the general rule for determining the amplitude from
       the equation for a wave?

   (8) What is the general rule for determing the wavelength from the
       equation for a wave?

Part II: Traveling Waves:
            (Author's note: Here the students see how to use equations
                            to move their sine curves horizontally -
                            that is how to represent traveling waves)

(1) Graph the following. Be careful with the parentheses as you type
    the equations in and watch the screen carefully as the calculator
    performs the graphing

                  Y1 = sin[2(pi)x]
                  Y2 = sin[2(pi)(x - 0.25)]
                  Y3 = sin[2(pi)(x - 0.50)]
                  Y4 = sin[(2pi)(x - 0.75))

    As the calculator graphs each equation sequentially; would you say
    the waves are moving to the right or moving to the left. By how
    much does each wavecrest move from one wave to the next?

(2) Graph the following. Notice that all you are doing is changing the
    minus signs to plus signs. Watch the screen carefully as the
    calculator performs the graphing

                  Y1 = sin[2(pi)x]
                  Y2 = sin[2(pi)(x + 0.25)]
                  Y3 = sin[2(pi)(x + 0.50)]
                  Y4 = sin[(2pi)(x + 0.75))

       As the calculator graphs each equation sequentially; would you
       say the waves are moving to the right or moving to the left?
       By how much does each wavecrest move from one wave to the next?

Part III: Combining Moving Waves
              Author's note: Now that the students have seen which
                             equatiions to use to draw waves and
                             how to modify the equations to give their
                             waves a horizontal displacement, they will
                             study standing waves by graphing a sequence
                             of two waves traveling in opposite
directions
                             and their supperposition.

   (1) Enter Y1 = sin[2(pi)x]; Y2 = (1.1)sin[2(pi)x]; Y3 = Y1 + Y2.
       The first two waves are being drawn with slightly different
       amplitudes so you can distinguish between them on the screen.
       Ideally, they should have the same amplitudes.
       (a) Note the location and height of all crests and troughs
       (b) Use a full sheet of paper to draw a carefull sketch of all
           three lines
   (2) Enter Y1 = sin[2(pi)(x - 0.25)]; Y2 = sin[2(pi)(x + 0.25));
       Y3 = Y1 + Y2. Be careful with the parentheses when entering the
       equations.
       (a) Note the location and the height of all crests and troughs.
       (b) Use a full sheet of paper to draw a sketch of all three
lines.
   (3) Repeat the previous part using the equations:
        Y1 = sin[2(pi)(x -0.5)]; Y2 = sin[2(pi)(x + 0.5)]; Y3 = Y1 + Y2

  (4) Repeat the previous part using the equations:
        Y1 = sin[2(pi)(x -0.75)]; Y2 = sin[2(pi)(x + 0.75)]; Y3 = Y1 +
Y2.
   (4) Repeat the previous part using the equations:
        Y1 = sin[2(pi)(x - 1.0)]; Y2 = sin[2(pi)(x + 1.0]; Y3 = Y1 + Y2.
   (5) Explain how this part of the experiment illustrates what happens
       with standing waves.

Part IV: Combining waves with distinctly different wavelengths and
         amplitudes
         Author's note: This part helps the students see how a complex
                        wave can be generated by he supperposition of
                        a fundamental frequency and some of its
                        overtones.
(1) On the (Y=  screen) enter
                Y1 = 2 sin[2(pi)x]
                Y2 = 0.5sin[10(pi)x]
                Y3 = Y1 + Y1
   (2) First just graph Y1 and Y2. If you dont know how to turn
       functions on and off for graphing, please ask for assistance.
       How many wavelengths of Y2 fit into one wavelength of Y1?
   (3) Then turn off Y1 and Y2 and turn on Y3. Discuss how the graph
       of Y3 shows both the properties of Y1 and of Y2.
   (4) Repeat all the previous steps of this part leaving everything
       the same with the exception of changing the Y2 equation to:
                   Y2 = 0.5 sin[20(pi)x]
   (5) Discuss how this part of the experiment shows how a string can
       simultaneously be vibrating in its fundamental frequency and
       in some of its overtones and how the different possible
       mixtures of overtones can contribute to the timbre or the
       quality of the sound being produced

PART V: Combining waves with the same amplitude and two frequencies
        that are close to each other.
        Author's note: This part helps students see how beats form.

   (1) On the y= screen set:
         Y1 = sin[15(pi)x]
         Y2 = sin[20(pi)x]
         Y3 = Y1 + Y2
   (2) First just graph Y1 and Y2. Comment on how the graphs.
   (3) Then just graph Y3. Sketch this graph on your lab report and
       describe what it looks like in words



  (4) Change Y1 and Y2 to:
        Y1 = sin[17(pi)x]
        Y2 = sin[20(pi)x] and leave
        Y3 = Y1 + Y2
  (5) Just graph Y3. Sketch this graph on your lab report and describe
      how it differs from the graph of Y3 in # 3.
   (7) Set Y1 = sin[19(pi)x] and leave Y2 and Y3 as
           Y2 = sin[20(pi)x] and
           Y3 = Y1 and Y2
   (8) Just graph Y3 and sketch the graph on your lab report.
       Comment on how it differs from the other two graphs of Y3 in
       this part.
   (9) Comment on how this part of the exercise illustrates how beats
       are formed and use your results to discuss how the nature of
       the beats formed depends on the frequency differences between
       the two individual of waves.