Noise Cancellation

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Notice that the 'zone of cancellation' is pretty small. This works OK for headphones, but does not work well if we attempt to generate 'antisound' for large areas (like cancelling the sound of your neighbour's hi-fi, in your own front room or bedroom).

So far, all our superpositions have been conherent. What happens if we superimpose two sine waves which are incoherent?

Beats

We have noted that two sine waves of the same frequency are coherent – they always have the same phase relationship. Where this phase = 0° they interfere constructively, and where the phase = 180° they interfere destructively.

When we have two incoherent waves with two slightly different frequencies, then the phase between them shifts over time. If you connect two sine generators to a dual-beam oscilloscope and set them up to generate slightly different frequencies, one will appear to move relative to the other in time.

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What happens if you change the frequency of the second wave to be greater or less than the frequency of the first?
Let's see. We'll also add a trace representing the sum of the two waves, to see the result of this shifting phase. The slider on the animation changes the frequency of one of the waves. Change the slider and see what effect it has on the resulting combined wave.

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If we plot the sum waveform over time, we can better see what is happening:

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The combined wave appears to have two frequencies. There is a sine wave pattern which is at a similar frequency to the original waves, but there is also a more slowly varying wave pattern which gradually changes the amplitude.

Oscilloscope showing two sine waves combining

The diagram above shows the time between two maximum amplitudes in the sum waveform -this is called the beat period Tb. During this period the higher frequency tone (f1) completes exactly one more cycle than the lower frequency tone (f2).

f1Tb - f2Tb = 1

The frequency of the beats (fb) equals the difference between the frequencies f1 and f2:

f1 - f2 = 1/Tb = fb

Beats are often used by guitar players when tuning up. If you listen for beats between two notes played on different strings, which are fretted (or plucked using harmonics) so as to have the same pitch, then by adjusting the tension in the string until the beats slow down to zero, you can bring the instrument into tune.

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