# 5.3: Beating between Tones

Perhaps you have heard two slightly mistuned musical instruments play pure tones whose frequencies are close but not equal. If so, you have sensed a beating phenomenon wherein a pure tone seems to wax and wane. This waxing and waning tone is, in fact, a tone whose frequency is the average of the two mismatched frequencies, amplitude modulated by a tone whose “beat” frequency is half the difference between the two mismatched frequencies. The effect is illustrated in the Figure. Let's see if we can derive a mathematical model for the beating of tones.

We begin with two pure tones whose frequencies are $$ω_0+ν$$ and $$ω_0−ν$$ (for example, $$ω_0=2π×10^3\mathrm{rad}/\mathrm{sec}$$ and $$ν=2π \mathrm{rad}/\mathrm{sec}$$). The average frequency is $$ω_0$$, and the difference frequency is $$2ν$$. What you hear is the sum of the two tones:

$x(t)=A_1\cos[(ω_0+ν)t+φ_1]+A_2\cos[(ω_0−ν)t+φ_2]$

The first tone has amplitude $$A_1$$ and phase $$φ_1$$; the second has amplitude $$A_2$$ and phase $$φ_2$$. We will assume that the two amplitudes are equal to $$A$$. Furthermore, whatever the phases, we may write them as

$φ_1=φ+ψ;\mathrm{and};φ_2=φ−ψ$

$φ=\frac 1 2 (φ_1+φ_2);\mathrm{and}ψ=\frac 1 2 (φ_1−φ_2)$

Recall our trick for representing $$x(t)$$ as a complex phasor:

$x(t)=A\mathrm{Re}\{e{j[(ω_0+ν)t+φ+ψ]},+,e^{j[(ω_0−ν)t+φ−ψ]}\}$

$=A\mathrm{Re}\{e^{j(ω_0t+φ)},[e^{j(νt+ψ)}+e^{−j(νt+ψ)]}\}$

$=2A\mathrm{Re}\{e^{j(ω_0t+φ)},\cos(νt+ψ)\}$

$=2A\cos(ω_0t+φ)\cos(νt+ψ)$

This is an amplitude modulated wave, wherein a low frequency signal with beat frequency $$ν$$ rad/sec modulates a high frequency signal with carrier frequency $$ω_0$$ rad/sec. Over short periods of time, the modulating term $$\cos{νt+ψ}$$ remains essentially constant while the carrier term $$\cos{ω_0t+φ}$$ turns out many cycles of its tone. For example, if $$t$$ runs from 0 to $$\frac {2π} {10ν}$$ (about 0.1 seconds in our example), then the modulating wave turns out just 1/10 cycle while the carrier turns out $$10νω_Δ$$ cycles (about 100 in our example). Every time $$νt$$ changes by $$2π$$ radians, then the modulating term goes from a maximum (a wax) through a minimum (a wane) and back to a maximum. This cycle takes

$νt=2π⇔t=\frac {2π} {ν} \mathrm{seconds}$

which is 1 second in our example. In this 1 second the carrier turns out 1000 cycles.

Exercise $$\PageIndex{1} Find out the frequency of A above middle C on a piano. Assume two pianos are mistuned by \(±1\mathrm{Hz}(±2π/mathrm{rad/sec})$$. Find their beat frequency $$ν$$ and their carrier frequency $$ω_0$$.

Exercise $$\PageIndex{2}$$

(MATLAB) Write a MATLAB program to compute and plot

$$A\cos[(ω_0+ν)t+φ_1],A\cos[(ω_0−ν)t+φ_2]$$, and their sum. Then compute and plot $$2A\cos(ω_0t+φ)\cos(νt+ψ)$$.

Verify that the sum equals this latter signal.