Physics – Bears, Beats, Battlestar Galactica

Algebra of wave superposition (AKA interference)

Example 1: Creation of Beats

Open the Excel file from Edline called “Bears, Beats, Battlestar Galactica.” Locate in cells B2 and E2 where you can modify the frequencies of two component sine waves, whose names are W1 and W2. Give the two waves slightly different frequencies: 50Hz for one and 60Hz for the other. Then look at their interference pattern by clicking on the tab in the lower left of the screen called Interference Pattern. Doing that puts you on the worksheet that represents the SUM of W1 and W2.

You can also play with the phase number for the waves by making them 1 or -1. As long as you use no other values for these besides 1 or -1, things will be clean.

Assignment:

You must find your name in the chart on Pages3 and 4 and only report on the results obtained from the two frequencies that I have assigned for you individually in this chart.

Back in the Excel file, observe the interference pattern created when W1 and W2 have the frequencies that I have assigned you. The mathematical activity is to use observation of what is onscreen to figure out what is called the interference pattern’s frequency. This is also called the Beat Frequency. This is not supposed to be obvious so observeyour on-screen interference pattern graph to calculate the beat frequency by the following steps:

  • Observe the interference pattern graph to report the beat period. Any period is the amount of time it takes for a repeating pattern to repeat itself exactly once. The units of period is second.You can find time values on the horizontal axis. By reading the horizontal axis, you can tell how much time passes between major repeating sections of the interference pattern. For example, the major sections of the interference pattern below:

are such that 4 sections occupy the time interval of duration 0.1 s. From this, 4T = 0.1 s.

  • The units of any frequency is Hz which is s-1 by definition. A period represents how many seconds per cycle. A frequency represents how many cycles per second. Anyone who has cared about dimensional analysis can figure out that the frequency is the reciprocal of the period. In the numerical example shown, T was 0.025 s. Since that time is the Beat Period, the reciprocal of it is the Beat Frequency. Reporting any frequency without units is as meaningless as reporting a time without units, and second’s reciprocal is s-1, whose other name is Hz.

After observing your personalized Interference Pattern graph, report your assigned frequencies and your Beat Period carefully with proper units. Submit a paper printout of your Interference Pattern graph and on that graph, have marked in pencil or pen which points on the axis you used to define your time measuring in order to conclude your Beat Period value.

Once your assigned frequencies and Beat Period are clearly recorded for your reader, show the work clearly that calculates your Beat Frequency from your Beat Period. If you don’t show this work or if the Beat Frequency does not come as calculated from the Beat Period or if the Beat Period had not come from a paper-printed Interference Pattern graph, this assignment will not receive credit. (And why should a person expect credit for submitting undefended work, and could this someday expected to be the norm instead of the teacher reminding these things?) Also, if I receive irrelevant graphs that are not the Interference Pattern Graph (comes from people not completely paying attention to the main idea), then my collections will be cluttered with too many pages, it slows me down, and I will not correct such over-cluttered papers. I need to go through this correction quickly to serve the students who are paying attention to detail.

After you’ve submitted the givens and results described above, there are numbered questions to answer.

To prepare for the questions, change the original two component frequencies to 1000 Hz and 1050 Hz. Repeat the determination of Beat Frequency from Beat Period. You don’t have to submit the printed graph for this one.

  1. Guess what the formula for beat frequency is in terms of the original two frequencies of the original two component waves.
  2. What must be true of your original component waves’ frequency values (in relation to each other) if you want to get a low frequency beat?
  3. Can the beat frequency ever be greater than BOTH of the original component waves’ individual frequencies?
  4. What about the last question if the component waves are just a few percent different from each other.
  5. An ultrasonic sound is one that has a frequency that is higher than the highest frequency that humanscan hear. Give an example of two component waves at frequencies that are both ultrasonic, but that produce a beat that is within the range of human hearing. The human hearing range is supposedly from about 20 Hz to 20,000 Hz, but those two numbers are really pushing it.

There are physicists who believe that two ultrasonic sound waves could interfere with each other, and even though the two waves are each individually impossible to hear, it is believed that the beat they create would be possible to hear. This is what question #5 is about. I hope you like it.

Find your name:

W1 Frequency / W2 Frequency
(Hz) / (Hz)
Ahmadi, Nilaab Z. / 200 / 220

Belzberg, Michael
/ 200 / 225

Bidondo-Yore, Dominic R.
/ 200 / 230

Choi, Ryan J.
/ 200 / 235

Colomer, Griffin A.
/ 200 / 240

Dodge, Michelle I.
/ 200 / 245

Ewald, Tyler K.
/ 200 / 250

Fujimoto, Ryan A.
/ 300 / 320

Hautau, Jeremy L.
/ 300 / 325

Heidelberg, Christopher M.
/ 300 / 330

Huang, Zachary C.
/ 300 / 335

Karmelich, Matthew B.
/ 300 / 340

Lin, Noah
/ 300 / 345

/ 300 / 350

Littman, Kyle G.
/ 400 / 420

Marshall, Robert D. (Robbie)
/ 400 / 425

Mohl, Kai Tristan L.
/ 400 / 430

Oster, Christian J.
/ 400 / 435

Pope, Natalia L.
/ 400 / 440

Smith, Makena A.
/ 400 / 445

Stiebel, Brooke H.
/ 400 / 450

Stiebel, Holly A.
/ 500 / 520

Thattai, Vidyuth B.
/ 500 / 525

Van Gieson, Henry
/ 500 / 530

Yi, Joonseon
/ 500 / 535