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4. Beats

So far we have dealt with waves of the same frequency, which leads to a significant simplification: if the waves start in phase, they stay in phase. We can then get constructive interference at all times. We could also start the waves out of phase;  they'll stay out of phase and give destructive interference at all times. The way this arises mathematically is that the time-dependent term \(2 \pi t/T\) is the same for both waves and so contributes nothing to the change in total phase. The type of interference we get is independent of time.

What if the frequencies and periods are different? Consider an example where one wave has a period of 5 seconds, and the other has a period of 6 seconds.  If we measure the displacement of the medium at a particular location against time. After 5 seconds, one wave has completed a cycle, but the other has not. If they started in phase (giving constructive interference) they are now not quite in phase so the interference is only partial.

After 15 seconds, one of the waves has completed 3 cycles (\((5 s \times 3 = 15 s\)) but the other wave has only completed 2.5 cycles (\(6 s \times 2.5 = 15 s\)). If the waves were constructive initially, they are now destructive!  The type of interference that leads to constructive and then destructive interference is termed beats. This term comes from how beats sound:  sound waves that exhibit beats oscillate between loud (constructive interference) to quiet (destructive interference) over and over.

Now that we have described this phenomena, let us actually plot out two waves and observe beats visually. Recall we are taking two waves that reach the same location, then plotting \(\Delta y\) against time.

The dashed vertical lines show that we are adding together the displacement caused by wave 1 to the displacement caused by wave at the same time. Remember that throughout this section on superposition, this is the only way we superpose waves!

Let us make a few comments about this graph. We can see that the waves go gradually from in step to out of step and back again (the vertical dashed lines at points A, B and C can serve as a guide to your eyes). The dashed lines can roughly show us how far in step or out of step the two waves are. If the two wave frequencies are \(f_1\) and \(f_2\) then the waves go from constructive to destructive and back to constructive interference an \(|f_1 - f_2|\) times per second. We call this the beat frequency: \[f_{beat} = |f_1 - f_2| \]  If we were listening to the beat, we would hear the sound become loud and then soft again an \(f_{beat}\) number of times per second.  There is also beat period \(T_{beat} = 1/f_{beat}\) which is simply the amount of time in between consecutive pieces of constructive interference.

The last relevant feature of this graph is how the combined wave oscillates at a different frequency than our initial waves. The frequency of this oscillation is called the carrier frequency, and it is the average of the frequencies of wave 1 and wave 2: \[f_{carrier} = \dfrac{f_1 + f_2}{2}\]  If we listened to this wave, the carrier frequency would determine the pitch (i.e. note) that we hear. If we are dealing with electromagnetic waves, the carrier frequency determines the color that we see (or the part of the spectrum that the light is found in).