waves - Confusion in understanding the concept of beats

. . .it is the combined vibration or disturbance basically having the average of the combining frequencies, but with an amplitude that varies periodically with time-one cycle of this variation including many cycles of the basic vibration.

Now, what does the author want to meany by the above bolded statement?

Also another:

. . .description as a beat phenomenon is physically meaningful only if $|\omega_1 - \omega_2| \ll \omega_1 + \omega_2$; ie. if, over some substantial number of cycles, the vibration approximates to sinusoidal vibration with constant amplitude and with angular frequency $\dfrac{{\omega_1} + {\omega_2}}{2}$.

At first part, it was said that amplitude does vary with time; however here it is mentioning about constant amplitude! What does this mean? And also difference of two numbers is always less than their sum; what is so special about that? Why does the author emphasize on $|\omega_1 - \omega_2| \ll \omega_1 + \omega_2$? Plz help.

Answer

but with an amplitude that varies periodically with time - one cycle of this variation including many cycles of the basic vibration.

There the author is talking about this kind of situation:

Amplitude of vibration goes down slowly, then amplitude of vibration goes up slowly, then amplitude of vibration goes down slowly, and so on. Slowly means slowly compared to how quickly the basic vibration goes up and down.

over some substantial number of cycles, the vibration approximates to sinusoidal vibration with constant amplitude

There the author repeats that amplitude must go up and down slowly, by saying that amplitude must stay approximately constant over a few cycles of basic vibration.

Why does the author emphasize on $|\omega_1 − \omega_2| \ll \omega_1 + \omega_2$?

The idea there is that difference of frequencies should be small compared to frequencies.

A more clear way to say that: $|\omega_1 − \omega_2| < \omega_1 \& |\omega_1 − \omega_2| <\omega_2$.