The first attempt to find the formula for the sum of two sine waves of different frequencies produced an erroneous result but we were able to determine that the amplitude was a function of half the difference in frequencies and the effective frequency of the combination was the average to the two original frequencies. Using a formula from trigonometry we were able to get an exact solution for the case where the original amplitudes were equal. Finally we were able to express the sum of the two frequencies as the product of the two cosine functions and a remainder for one of the frequencies.
The original attempt shows that although a false assumption usually gives erroneous results it can in some cases yield a useful approximation. And we found that we can check to see if two mathematical expressions are equal to one another by evaluating them over a set of values.
The formula that was found is quite useful in explaning the beats that result when two nearby tones are combined. Besides its use for sound the formula also has practical application in electronics where it is used for such things as modulating radio frequencies, converting one frequency to another and detecting radio frequencies. Light is another example where two waves can interfere with each other. And the wave functions associated with moving particles in quantum mechanics exhibit interference phenomena which is analogous to the beating of two waves.
For a discussion on the vector addition of sine waves see A Course in Electrical Engineering (1922) by Chester Laurens Dawes, p. 19.