Grade Level/Subject / High school physics, all levels
Unit / Waves
Enduring Understanding / All waves, regardless of origin or type, transmit energy from one place to another and share a common set of characteristics that explain their behavior. Wave behavior can be used to understand and explain many everyday phenomena.
SOL Objectives / PH.8 The student will investigate and understand wave phenomena.
Key concepts include
a) wave characteristics;
b) fundamental wave processes; and
c) light and sound in terms of wave models.
Title / Inquiry with Beats
Lesson Objective / To understand beat frequency, both phenomenologically and mathematically, through analogy, experimentation, data analysis, and simulation.
Inquiry Level / 2
Materials Required / Two cars, two tuning forks whose frequencies vary by 5 to 50 Hz, Vernier microphone and computer (logger pro, Wolfram Alpha, and online simulation).


Playing with Beats

(not to be confused with Beets, which I would never assign to anyone)

For this preview activity, you will need two different cars next to each other, and a good digital timer [If you cannot find your own cars, google “blinkers beat frequency” to find my Youtube video].

  1. Turn on the blinkers of one car. Watch it blink a whole bunch of times. Record the time it takes for 20 blinks. (This needs to be pretty precise to work out!)

Time for 20 blinks: / Frequency of blinking:
(blinks/sec or Hz)
  1. Repeat for a second car.

Time for 20 blinks: / Frequency of blinking:
(blinks/sec or Hz)
  1. These blinking rates are sometimes close, but almost never the same. What is the difference between them?

Δf = ______Hz

  1. Now, with the cars next to each other, turn on the blinkers for both cars at the same time, stand back and watch for a few minutes.
    Describe what you are seeing.
    Explain why you think this is happening.
  2. Time how long it takes for the blinkers to go from being in synch, to out of synch, to back in synch again. This is known as a “Beat”. The time is usually between 10 seconds and 2 minutes, depending on the cars. Record the time and repeat for a second trial.

trial 1 / trial 2
Time for one beat
(sec/beat)


In class, we’ve already seen two waves interfering on a single slinky. This superposition principle can also happen with light, sound, and lots of other things.

Car blinkers are only an analogy, since “off” isn’t really a trough. But it helps us understand some key principles.

  1. Thinking about superposition of waves, what happens when two crests meet?
  2. What happens when a crest and a trough meet?
  3. What would these two examples mean if we were talking about sound waves instead of slinky waves?
  4. On a slinky, what pattern do we get when a crest and trough meet at the same location all the time? (recall we generally did this by making a slinky wave collide with the reflected slinky wave, thus ensuring the waves were the same frequency and out of phase).
  5. On a slinky, what do you think would happen if the two colliding waves were slightly different frequencies?
  6. What would these two examples mean if we were talking about sound waves instead of slinky waves?
  7. Write down again the times for your blinker beats below. Then take the inverse to find the beat frequency.

trial 1 / trial 2 / average
Time for one beat
(sec/beat)
beat frequency
(beats/sec)
  1. If you conducted your experiment carefully, your beat frequency should be the same as the difference between the frequencies of the two blinkers (Δf from problem #4). Calculate your percent difference between the average of your beat frequency measurements and your Δf.
  2. Explain why you think the frequency of the interference pattern should be dependent on how different the frequencies of the waves are.


Beats lab: Seeing and Hearing waves colliding.

Your teacher will provide you with a pair of tuning forks that have similar frequencies.

  1. Strike the one fork on a soft surface and listen to the tone. Record the sound wave on the computer. (Use 0.1 sec at a rate of at least 50 kHz). Ensure that you have created a nice/clean sound wave.
  2. Use a curve fit to determine the equation of this sound wave. Record the full equation below.

Y=

  1. From your data, determine the frequency of the wave f= ______Hz
    (this should be similar to the frequency stamped onto the tuning fork)
  1. We want to save this dataset and graph it along with the sound of the second fork. From the Experiment menu, choose “Store Latest run”.
  1. Now with the second tuning fork, strike the fork and listen to the sound. Can you tell, without looking at the tuning fork, if it is a higher or lower pitch sound than the first fork?
  2. Use the computer to collect the sound waves of this second sound.
  3. If you don’t get a nice clean sinusoidal wave, of similar amplitude to the first, repeat the data collection.
  1. What does it look like when both datasets are graphed at the same time?
  2. I recommend at this point saving your data file.
  3. Use a curve fit to determine the equation of this second sound wave. Record the equation below.
  4. Y=
  1. From your data, determine the frequency of the wave f= ______Hz

Part 2: Combining the data:

  1. Adjust your zoom so that both waves are clearly visible on the screen at once and you can see around twenty cycles of each wave. Observe how the waves overlap. If we were allowing these waves to superimpose instead of just overlap, what would the pattern look like?
  2. Create a new third calculated column in the data. Go to the Data menu, choose “New Calculated Column”. Name the new column “superposition”, and using the menu buttons, build the following equation to add the two values: “Latest|Sound Pressure”+”Run1|Sound Pressure”
  3. Click on the vertical axis label to display the new “superposition” data.
  4. Explain what your data shows, and what it means.
  1. Set your graph to display all three sets of data in a reasonably sized and scaled window.
  2. Copy this graph to the clipboard, then open Word and paste it into a document.
  3. Then go to a web browser and launch Wolfram Alpha.
  4. Enter into the search bar the addition of the two sine functions that you wrote down from the computer. Include only the frequency component of the equation to keep it simple (amplitude of 1, no shifts). The result should be a more “ideal” version of what you saw on the computer.
  5. Copy this graphic into your word document also.
  6. When we think about this ideal superposition, what does it represent in an actual sound wave?
  1. If you were to play both tuning forks simultaneously, what do you predict you will hear?
  2. Give this a try. Strike them both so you hear the same volume of sound from each. Place them both up to one ear so you hear them equally loud. Explain what you hear.
  3. Play this sound for the computer and collect the data in a new dataset.
  4. What does it look like?
  5. Copy this graph onto your Word document, size all three graphs appropriately, and print it out to turn in with the lab.
  1. Just like the turn signal blinkers, the difference in the frequencies gives us the frequency of the beat. Explain why, AND show numerically that this is true. Calculate a percent error, which should be less than 10%.


Follow-up simulation with interfering waves.

The following questions use a simulation found here: www.mta.ca/faculty/science/physics/suren/Beats/Beats.html

(you can type “beat frequency applet” in google to find it, or go to Edmodo).

  1. Set the frequency of each wave to 15 Hz. Notice the white dot on each wave. The red and the green waves add up to make the yellow wave (it is the superposition of the other two waves).
  2. The red and green waves both have an amplitude of 1 unit. What is the amplitude of the yellow wave?
  3. Change the “phase” so they are out of synch. Explain why the yellow wave disappears.
  4. Now change the green wave to 16 Hz, while keeping the red at 15 Hz. You’ll notice that suddenly there is a pattern within the yellow waves. The large “wave envelope” shows the “beat”.
  5. Explain what is happening to create the pattern of large/small waves. (hint: watch the little dots moving, and press the “stop” button and try advancing frame by frame.)
  6. Stop the simulation. If you count, you’ll find that 15 red waves require 21 squares of “time”. So, 21 squares must be one second in this slow-motion simulation (because the red wave is set to 15 Hz). Silly scale. Anyway, based on that, how long does it take for one full yellow wave envelope?
  7. Repeat this measurement for several different combinations of waves, as shown in the table below.

Red wave / Green wave / Time for yellow envelope / Frequency of yellow envelope
15 Hz / 16 Hz / 21 squares = 1 second / 1 wave/1 sec = 1 Hz
15 Hz / 18 Hz
15 Hz / 19 Hz / 5.25 squares = 0.25 seconds / 1 wave/0.25 sec = 4 Hz
15 Hz / 13 Hz
10 Hz / 13 Hz
10 Hz / 9 Hz
  1. The last column is called the “beat frequency”. If you were listening to these red and green waves interacting, the yellow wave would be what you would hear. Because it gets loud/quiet/loud/quiet, it makes a beating sound. The frequency of the beat should correspond to the relationship between the original waves. Explain what you see in the data, and how this relationship works.
  2. If you had two waves, let’s say 30 Hz and 36 Hz, what would the beat frequency be?


Teacher Notes:

This activity combines some out-of-class observation of car blinkers with slightly different frequencies cycling in and out of phase, and uses this as a preview for understanding beat frequencies. The lab has lots of steps explained because it uses some precise technology. Yet it uses this to lead students to discover, on their own, the relationships important for wave interference, specifically as we see in a beat frequency application. It also gives them some in-depth, yet straightforward, opportunities to use math, and several software tools, to help understand this physics concept.

The goal of the inquiry is to understand beat frequencies and uncover the idea that the frequency of the beat is the difference in the frequencies of the two sources.

As a follow-up, this can be applied quite readily to tuning of stringed instruments, as many students have experience with.

Questions or suggestions? Contact Aaron Schuetz at Yorktown