In order for the occurence of beat, why is it compulsory that $|{\omega_2 - \omega_1}| \ll \omega_1 + \omega_2$?

Question

As A.P.French in Vibrations and Waves writes,

The beating effect is most easily analyzed if we consider the addition of two SHM's of equal amplitude: $$ \mathbf{x_1} = \mathrm{A} \cos{\omega_1 .t} \& \mathbf{x_2} = \mathrm{A} \cos{\omega_2 .t}$$ . . . Clearly their addition,as a purely mathematical result,can be carried out for any values of $\omega_1$ & $\omega_2$. But its description as a beat phenomenon is physically meaningful only if $|{\omega_2 - \omega_1}| \ll \omega_1 + \omega_2.$

Why is it so? What does the author want to say??

Answer 1

It's not, other than perceptually. If you plot the sum and differences for a variety of $\omega_1 , \omega_2$ pairs, you only "see" a "beat" when the two frequencies are close to each other.

Over in my life as a musician, we hear a "beat" when two instruments are close, but not exactly, in tune (difference frequency). We also hear overtones, often as a result of deliberate design by the composer, when two notes differ in frequency by a small integer multiple of the fundamental. For example, tonic + fifth produces an octave overtone (of the fifth).

Answer 2

He means that the beats will be unrecognizable without that condition. Imagine two piano strings, one tuned to 440 Hz, the other to 441 Hz. You will perceive a beat of 1 Hz, a warbling of the tone. If one was tuned to 440 Hz, and the other to 100 Hz, the beat frequency will be 340 Hz, and it won't sound like warbling.

That's an example from human perception, but the principle applies to beats of any nature. What constitutes an interpretation as beats from not is a judgment call that will vary from situation to situation. I wouldn't use the word "compulsory".

Generally, you want to have several/many periods of the fundamental frequencies within one beat period.