Activity: Introduction to Waves

In today's activity, you will examine the properties of waves in three different ways: analytically using the wave equation, experimentally using a signal generator and an oscilloscope, and virtually using an Excel spreadsheet.

Equipment needed: Each group needs a signal generator with power cord and connecting cable, and an oscilloscope.

Before You Start:
Answer the following questions to the best of your ability before doing the experiment.

Today we will be examining the voltage wave described by

V (x,t) = (2.0 V) sin[(0.245 rad/m)x - (230 rad/s)t + f]

Identify each of the following parameters of this wave: amplitude V₀, wave number k, wavelength l, angular speed w, frequency f, and period T. Be sure to show all of your calculations
Sketch this wave for the position x=0 from time t=-5.0 ms to t=5.0 ms, with a phase constant f of 0.
Sketch this wave for the postion x=0 from time t=-5.0 ms to t=5.0 ms, with a phase constant f of -0.1 rad. What effect does the phase constant have on the wave?
Sketch this wave for the time t=0 from position x=-50 to x=50 m, with a phase constant f of 0.
In which direction is the wave moving? How do you know this?

After you have thought about your answers, compare notes with your group members. Does everyone have the same predictions, or are there differing opinions?

Looking at the Wave as a Function of Time

Hook up the equipment as illustrated to the right, using CH 1 of the oscilloscope. Turn on both the signal generator and the oscilloscope. Make sure the sine wave button (as opposed to the square wave or triangular wave button) on the signal generator is depressed.
Set the frequency on the signal generator to the value of f you calculated in the Before You Start section. (Choose the appropriate Range button, then turn the Frequency dial to get the appropriate factor.) There may be a slight delay before the change is registered in the digital display. Turn the amplitude knob on the signal generator all the way to its minimum value.
Adjust the time/div knob (horizontal scale) on the oscilloscope until the time scale factor M displayed at the bottom of the oscilloscope screen is 10 ms/div. Adjust the volts/div knob (vertical scale) for CH 1 on the oscilloscope until the CH1 voltage scale factor at the bottom of the oscilloscope screen is 1.0 V/div.
If you do not see a sine wave after making these changes to the oscilloscope, raise your hand and ask for assistance.

1. Adjust the amplitude knob on the signal generator to increase the size of the wave, and use the vertical position knob on the oscilloscipe to move the wave up or down. Your goal is to have a sine wave centered vertically, with an amplitude of 2.0 V. (Each centimeter-high square on the oscilloscope screen is called a division. You can find the voltage by multiplying the number of divisions by the CH1 conversion factor displayed at the bottom of the oscilloscope screen, e.g., 1.0 V/div.) Write down the voltage scale factor and the number of vertical divisions of the wave's amplitude.

2. Sketch the resulting graph. Be sure label your axes and include scale markings on your sketch. Does the graph match your f = 0 prediction? Describe any differences.

3. Find the period of the wave by counting the horizontal divisions required for the wave to complete one cycle. Write down this number of divisions, along with the time scale factor M, then calculate the period. Does your value agree with your prediction from the Before You Start?

4. One possible difference between your prediction graph and the oscilloscope display may be that the wave displayed on the oscilloscope does not pass through the origin in the center of the screen. Adjust the triggering knob until the display matches your predicted graph. Sketch the result.

5. What is the effect of the triggering? What variable in the wave equation does the triggering affect?

Looking at the Wave as a Function of Distance
Open up the Excel spreadsheet act02xl.xls. This spreadsheet already contains a plot of the wave at two different times. The blue heavy line is the wave at t=0. The pink lighter line is the wave at the time indicated in cell D4. The data in column A is the different values of x. Column B contains the values for y at t=0, and column C contains y at the later time.

6. Determine the wavelength of the wave. The graph is not detailed enough for an accurate determination, so you should use the data in Column B to find the distance to complete one cycle.

7. Slightly increase the time in D4 and notice what happens to the pink wave. Does it move toward positive x or toward negative x? Does this match your prediction?

8. Make incremental increases in t (Cell D4) until the pink wave moves one complete cycle and lines up with the blue wave. Record this last value of t that causes the waves to align. How does it compare with the period you predicted before?

9. Change t to zero in D4. Vary f and see what happens to the pink wave. Does the graph when f = -0.1 rad match your prediction?

10. Describe the effect of f on the wave. How does the effect depend on the sign of f?

11. What value of f produces the same wave as f = 0? (Blue wave) Does this make sense?

12. The equation we have been using is V (x,t) = (2.0 V) sin[(0.245 rad/m)x - (23.0 rad/s)t + f]. How would this change if the wave were to move to the left (toward negative x)?

13. Change the equation in column C according to your answer to 12, and try out different values of t to verify it works.

1.	Adjust the amplitude knob on the signal generator to increase the size of the wave, and use the vertical position knob on the oscilloscipe to move the wave up or down. Your goal is to have a sine wave centered vertically, with an amplitude of 2.0 V. (Each centimeter-high square on the oscilloscope screen is called a division. You can find the voltage by multiplying the number of divisions by the CH1 conversion factor displayed at the bottom of the oscilloscope screen, e.g., 1.0 V/div.) Write down the voltage scale factor and the number of vertical divisions of the wave's amplitude.
2.	Sketch the resulting graph. Be sure label your axes and include scale markings on your sketch. Does the graph match your f = 0 prediction? Describe any differences.
3.	Find the period of the wave by counting the horizontal divisions required for the wave to complete one cycle. Write down this number of divisions, along with the time scale factor M, then calculate the period. Does your value agree with your prediction from the Before You Start?
4.	One possible difference between your prediction graph and the oscilloscope display may be that the wave displayed on the oscilloscope does not pass through the origin in the center of the screen. Adjust the triggering knob until the display matches your predicted graph. Sketch the result.
5.	What is the effect of the triggering? What variable in the wave equation does the triggering affect?

6.	Determine the wavelength of the wave. The graph is not detailed enough for an accurate determination, so you should use the data in Column B to find the distance to complete one cycle.
7.	Slightly increase the time in D4 and notice what happens to the pink wave. Does it move toward positive x or toward negative x? Does this match your prediction?
8.	Make incremental increases in t (Cell D4) until the pink wave moves one complete cycle and lines up with the blue wave. Record this last value of t that causes the waves to align. How does it compare with the period you predicted before?
9.	Change t to zero in D4. Vary f and see what happens to the pink wave. Does the graph when f = -0.1 rad match your prediction?
10.	Describe the effect of f on the wave. How does the effect depend on the sign of f?
11.	What value of f produces the same wave as f = 0? (Blue wave) Does this make sense?
12.	The equation we have been using is V (x,t) = (2.0 V) sin[(0.245 rad/m)x - (23.0 rad/s)t + f]. How would this change if the wave were to move to the left (toward negative x)?
13.	Change the equation in column C according to your answer to 12, and try out different values of t to verify it works.