Vibrations with Different Frequenc'es Beat

Question

Hello here i have a problem which is following

Say there are two waves which are

$x_1=A_1cos(w_1t)\\x_2=cos(w_2t)$

In my book it is written that $x=x_1+x_2=2Acos\frac{(w_1-W-2)}{2}tcos\frac{(w_1+W-2)}{2}t$ And we know that $f_b=f_2-f_1$ Here $f_b$ is beat frequency.

Because we know that $cos\frac{(w_1+W-2)}{2}t$ make a lot of fluctuation we do not consider it as beats frequency but $cos\frac{(w_1-W-2)}{2}$ makes les we can say it is the frequency for whole motion.

Here $f_b=(w_1-w_2)/2\pi$ we get $cos(f\pi t)$But angular frequency is $w=f2\pi$ I am confused here.

Thank you for all help!!

Answer 1

Here are you two individual waves and the waves combined.

Now the frequency of the envelope is indeed $\dfrac{\omega_1-\omega_2}{2}$ but you will note for every period of that envelope (i to v) there are two maxima and so the beat frequency (number of maxima per second) is $\dfrac{\omega_1-\omega_2}{2}\times 2 = \omega_1-\omega_2$