Beats

In this part of the activity you will study the phenomenon of beats using an Excel spreadsheet. The spreadsheet is already programmed to plot the sum of two sinusoidal waves of different frequencies. You will be able to change the frequencies and amplitudes of each of the waves, and observe the results. A picture of the spreadsheet is provided below. The frequencies of the two signals are entered in cells B4 and B5, and the amplitudes in cells B7 and B8. Everything else is calculated and graphed automatically.

A link to the spreadsheet is provided at the Physics II website, under "Class activities". The graph of x1 + x2 as a function of time clearly shows the variation of amplitude known as beats. Open the program and follow the instructions below.

1. With f1 = 350 Hz and f2 = 375 Hz, and both A1 and A2 equal to one, measure the length of time that one beat lasts; in other words, the period of one beat (Tbeat). Then take the reciprocal of the period (1/Tbeat ) to find the beat frequency. Record the results. Does the beat frequency seem to equal the frequency difference between the component signals?

2. With f1 = 350 Hz and f2 = 390 Hz, and both A1 and A2 equal to one, measure the length of time that one beat lasts; in other words, the period of one beat (Tbeat). Then take the reciprocal of the period (1/Tbeat ) to find the beat frequency. Record the results. Does the beat frequency seem to equal the frequency difference between the component signals?

3. With f1 = 350 Hz and f2 = 330 Hz, and both A1 and A2 equal to one, measure the length of time that one beat lasts; in other words, the period of one beat (Tbeat). Then take the reciprocal of the period (1/Tbeat ) to find the beat frequency. Record the results. Does the beat frequency seem to equal the frequency difference between the component signals?

4. With f1 = 350 Hz and f2 = 390 Hz, and A1 = 2 and A2 = 1, you should still get beats, but the amplitude will never go to zero. Measure the length of time that one beat lasts; in other words, the period of one beat (Tbeat). Then take the reciprocal of the period (1/Tbeat ) to find the beat frequency. Record the results. Does the beat frequency seem to equal the frequency difference between the component signals?

The Doppler Effect

You have a summer job with a team of marine biologists studying dolphin communication off the coast of Hawaii. Massive boulders on the ocean floor can interrupt the reception of underwater sound waves from the dolphins. To reduce these disruptions, your team has decided to put several "transceivers" (a device that receives a signal, amplifies the signal, and then transmits it) at strategic locations on the ocean floor. A transceiver will receive sound waves from a dolphin and then retransmit them to the researchers on the ship. The ship's receiver is on a long cable so that it is at approximately the same depth as the dolphins. Because of your physics background, you worry that the frequency received at the moving ship will be different than that emitted by the dolphin. To determine the size of this effect, you assume that the ship is moving at 35km/h away from the stationary transceiver. Meanwhile, the dolphin is moving at 60km/h towards the transceiver when it emits a sound frequency of 660Hz. The speed of sound in sea water is 1520 m/s.

5. Will the wavelength of the signal received by the transceiver be greater than, or less than, the wavelength of the original signal emitted by the dolphin as it approaches the transceiver?

6. What will be the frequency of the signal received by the transceiver? Show your work.

7. Will the wavelength of the signal received by the ship be greater than, or less than, the wavelength of the signal emitted by the transceiver as the ship moves away from the transceiver?

8. What will be the frequency of the signal received by the ship from the transceiver? Show your work.

9. What percentage of the original frequency is the change in frequency? (i.e., how significant of an effect does the motion have on the signal?)

10. Compare the frequency you obtained in question 8 to the frequency obtained if you hadn't used a transceiver. (i.e., use the equation for when both the source and the detector are moving to find the frequency measured by the ship). Do your answers make sense?