## Illustration 18.3: Interference in Time and Beats

f = Hz

This animation depicts the superposition of two sound waves. A red wave with λ = 3.43 m and a frequency of f = 100 Hz is added to a green wave, and the resulting wave is shown in blue (position is given in meters and time is given in seconds). Restart.

Now change the frequency of the green wave to 120 Hz. What is the new wavelength of the green wave? You should right-click on the graph to clone it and resize the graph to make it easier to make measurements. The result is 2.86 m. Did you really need to make a measurement to get the wavelength? Since the speed of sound is a constant and v = λf, we also know that λ = 343/f. Notice the animation always does the right thing: As you change the frequency of the green wave, the wavelength also changes to maintain the wave speed of 343 m/s.

Now consider what happens when the two waves add together to create the blue wave. Enter several frequencies for the green wave and see the resulting blue waves. What do you see? When the two frequencies are the same, the resulting blue wave looks like the original waves but with twice the amplitude. But the resulting wave is more interesting when the frequencies (and therefore the wavelengths) do not quite match. Consider the resulting wave when the green wave's frequency is 120 Hz. If you were at x =20 m, you would hear the sound wave getting louder and softer, louder and softer with time. When you hear this pattern, you are hearing beats. The time in between the loud sounds (or conversely the soft sounds) can be measured and is 0.05 seconds. This corresponds to a beat frequency of 20 Hz. This is precisely the difference in the frequencies! What happens when the green wave's frequency is now 80 Hz? We get the same period and therefore the same beat frequency of 20 Hz. Therefore we find that the beat frequency is fbeat = | f1 - f2 |.

So what is going on? Look at the underlying waves. They go in and out of phase with each other as a function of time. At one instant they exactly add together (constructively interfere) at another time they exactly cancel (destructively interfere). We therefore can say that the phenomenon of beats is due to an interference in time.

Try this out for yourself. You may vary the green wave's frequency between 50 and 150 Hz.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.

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