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measuring frequency






Beats is a phenomenon associated with sound waves, though the effect applies to all waves.

Essentially, when two similar frequencies ( f1 , f2) are sounded, a third much lower frequency is heard at the same time.

This third frequency is called the beat frequency ( fB ).

The beat frequency is simply the difference between the two original frequencies.



beat frequency equation



The beat frequency is measured from the rise and fall in the loudness/volume.


There is yet another frequency called the combined frequency( fC).


This is the result of superposition of the two original frequencies. The combined frequency is simply the average of these frequencies.



the combined frequency



Since the frequencies f1 , f2 are almost the same, the change in frequency to fC is hardly noticeable.


An example of the effect is the sound from a twin engined prop. aircraft. There is a periodic 'wow' or 'drone' noise produced as a result of the change in r.p.m. from the different propeller blades.



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beats - waves in phase



The effect is a result of superposition of two sound wave frequencies producing a succession of constructive and distructive interference.

When the two frequencies are in phase they add, producing a wave with double the amplitude.

When the two waves are out of phase, they destroy eachother.



beats - amplitude envelope




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Consider our two original frequencies f1 and f2.


In time t the number of cycles completed by each frequency is f1t and f2t .



no. cycles = (no cyles per second) x (no. seconds)



Let us choose the time t such that the first wave completes one more cycle than the second.



beats - theory equation #1


beats theory equation #3



From the first of two images (above), t is the time interval between the waves being in phase with each other.


So t is the beat period T (time for one complete 'beat' wave).



beats theory equation #4



For any wave, period and frequency are inversely proportional to one another.


So for beat period T and beat frequency fB ,


beats theory equation #5


hence, by similarity between the last two equations,



beats theory equation #6



assuming f1 > f2    




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Measuring an unknown frequency


The method is to use a frequency( fU ), where only an approximate value is known.

This is used with a known frequency( fK ) close to the approximate value of fU to produce beats.


The beat frequency ( fB ) is given by:


beats theory - equation #7


or (depending on the relative magnitudes of fK and fU )


beats theory - equation #8


bringing the two equations together,


beats theory - equation #9


This is quite an accurate method, achieving results of 0.01% accuracy.




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