\begin {align*} \sin \omega t & =\operatorname {Re}\left [ \frac {1}{i}e^{i\omega t}\right ] =\operatorname {Re}\left [ e^{i\left (\omega t-\frac {\pi }{2}\right ) }\right ] \\ \cos \omega t & =\operatorname {Re}\left [ e^{i\omega t}\right ] \\ \cos \omega t & =\frac {1}{2}\left (e^{i\omega t}+e^{-i\omega t}\right ) \\ \sin \omega t & =\frac {1}{2i}\left (e^{i\omega t}-e^{-i\omega t}\right ) \end {align*}
When 2 harmonics have same amplitude, we can write then as envolope of one in another
\[ A\cos \left (\omega _{1}t-\phi _{1}\right ) +A\cos \left (\omega _{2}t-\phi _{2}\right ) =2A\cos \left (\Delta _{\omega }t-\Delta _{\phi }\right ) \cos \left ( \omega _{av}t-\phi _{av}\right ) \]
Here is an example of the above. We first draw the two signals on their own, then plot the additions of them
f1 = a Cos[w1 t - p1];
f2 = a Cos[w2 t - p2];
parms = {a -> 1, w1 -> 1, p1 -> Pi/3, w2 -> 10, p2 -> Pi/4};
Plot[Evaluate[{f1, f2} /. parms], {t, 0, 10},
PlotStyle -> {Red, Blue}]
Now we add them to see the envelope effect
Plot[Evaluate[{f1, f1 + f2} /. parms], {t, 0, 10},
PlotStyle -> {Red, Blue}]
Now we plot the same signal addition, but using the form after converting to use the mean and delta notation as shown above just to confirm it is the same signal
avW = Mean[{w1, w2}];
avP = Mean[{p1, p2}];
delW = w2 - avW;
delp = p2 - avP;
g = 2 a Cos[delW t - delp] Cos[avW t - avP];
Plot[g /. parms, {t, 0, 10}]
The beat period is \(\frac {\pi }{\Delta \omega }\)(this is the time between each beat to the next beat). The whole signal will be periodic only when \(\frac {\omega _{1}}{\omega _{2}}\) is rational.
Beat shows up when we have 2 harmonics added, that has same amplitude. The beat signal itself will be period when the ratio between the frequencies of the two harmonics is rational. In the context of response of a system, we can think of the steady state response as one signal and the transient response as another singnal. The response will then show a beating signal when the amplitude of the steady state and transient singnals is the same. Here is an example of that from one of my demos
