Preliminaries

What's a beat?

If two harmonic waves have similar frequencies, a phenomena known as beats, that is, a repetitive variation of the sound intensity, occurs. For example, when a musical instrument is being tuned to match the pitch of another instrument, the characteristic "wa-wa" sound is often produced as the two frequencies approach each other.

Frequency

Mathematically, we can represent oscillations with the trigonometric functions sine or cosine. For example, the cosine function

Input := 

Plot[Cos[t], {t, 0, 8 Pi}]

Output =

-Graphics-

vibrates (or oscillates) 1 time per every 2 Pi time units and in fact the function Cos(2 Pi t)

Input := 

Plot[Cos[2 Pi t], {t, 0, 10}]

Output =

-Graphics-

has a frequency of 1 Hz or 1 vibration per second. If we want to increase the number of vibrations per second, we increase the frequency f in the expression Cos(2 Pi f t). For example, the wave

Input := 

Plot[Cos[2 Pi 4.2 t], {t, 0, 1}]

Output =

-Graphics-

undergoes precisely 4.2 oscillations per second.

Period and Amplitude

The period of oscillation is the length of time it takes to complete one full cycle (vibration.) The period of Cos[2 Pi f / t] is inversely related to the frequency by the relation Period = 1 / f.

The amplitude of a wave is the distance from the average value to the peak value of the oscillation. In all of the oscillations above, the amplitude has a constant value of 1. However, some functions oscillate with varying amplitudes. The curve below has a decreasing amplitude starting at 1 and decaying to zero.

Input := 

Plot[ Exp[-t/2] Cos[2 Pi t], {t, 0, 5},
 PlotPoints -> 50]

Output =

-Graphics-

Its worth remarking that functions such as the one above are not periodic, and yet we still have a feeling for how long it takes to oscillate through a full cycle. In this case, we would speak of the quasi-period instead of period.