 Absolute and signed beat envelopes

For $x_2=(a/2)[\cos {\omega }_{1}t-\cos {\omega }_{2}t]$, the slowly varying signed factor is $a\sin [({\omega }_2-{\omega }_1)t/2]$. The absolute amplitude envelope repeats in $2\pi /|{\omega }_2-{\omega }_1|$, half the period of that signed factor.