This is because the two waves are interacting and causing acoustic interference. The times when we are hearing the peak sound of the beat is when the amplitudes of the two waves add up maximally - additive effect. When there is no sound, or lesser sound, the amplitudes are not adding up to the maximum, and hence the produced is less intense than the peak of each beat. When the sound is minimal or absent, the two waves are canceling themselves out - subtractive effect. Thus we get the waxing and waning beat sounds.
Schematic explaining the mathematics and the mechanism of production of beats in a musical instrument in terms of the acoustic interference pattern of two notes sounded simultaneously.
(Click to enlarge)
When the two waves are nearly of the same frequency (within 20Hz of each other), then they produce distinctively audible beats. If the difference be larger, then the ear perceives that beat as a new distinct sound of the frequency |f1 - f2|. Thus we hear that the two notes are not the same but we can not hear any beats individually.
We will also not hear beats when the frequencies are exact multiples of each other, i.e.
f1 = n X f2, where n is a whole number and f1 is greater than f2.
There is a simpler way of referring to this occurrence: Harmonics
Those frequencies that do not comply with the above mathematical criteria (i.e. n is not a whole number), are referred to as being Inharmonic (non harmonic).
How To Tune A Guitar Scientifically:
- Physics of Guitar Tuning
- Beats - Why Do We Get Beats? (reading)
- Phase Difference and Harmonics
- Importance of Phase Difference
- The Bend Test - Beats
- Steps of a thorough Tuning
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