Analytical method to determine beat frequency

Consider two sound waves of the SAME amplitude a but slightly different frequencies $n_1$ and $n_2$, assumed (for simplicity) to arrive exactly in phase with each other at a chosen listening point x = 0: $y_1=a\sin(2\pi n_1t)$ and $y_2=a\sin(2\pi n_2t)$. By the principle of superposition, $y=y_1+y_2$; applying the sum-to-product identity $\sin C+\sin D=2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$ (Eq. 6.37) and rearranging gives $y=\left[2a\cos\left(2\pi\dfrac{n_1-n_2}{2}t\right)\right]\sin(2\pi nt)$, where $n=\dfrac{n_1+n_2}{2}$ is the MEAN of the two original frequencies (Eq. 6.38). Writing the bracketed, slowly-varying factor as an amplitude $A=2a\cos\left[2\pi\left(\frac{n_1-n_2}{2}\right)t\right]$, the resultant $y=A\sin(2\pi nt)$ is itself a progressive wave oscillating at the mean frequency n, but with an amplitude A that itself varies periodically and comparatively SLOWLY with time (since $n_1-n_2$ is small).

Since the ear's perceived loudness (intensity) tracks the square of this amplitude, maximum loudness (WAXING) occurs whenever $A=\pm2a$, i.e. $\cos\left[2\pi\left(\frac{n_1-n_2}{2}\right)t\right]=\pm1$; working through the successive instants at which this condition is met shows that consecutive waxings are spaced exactly $\dfrac{1}{n_1-n_2}$ apart in time. Minimum loudness (WANING, $A=0$) occurs when the same cosine factor is zero, and working through those instants shows consecutive wanings are ALSO spaced exactly $\dfrac{1}{n_1-n_2}$ apart -- as expected, since waxing and waning alternate at the same regular rate. The BEAT FREQUENCY -- the number of loud maxima heard per second -- is therefore simply the reciprocal of this spacing: $N=n_1-n_2$ (Eq. 6.39), the (positive) difference between the two original frequencies. …