What does the multiplication of sines in a wave $\sin(\omega_1t)\sin(\omega_2t)$ mean?

Question

I have the following exercise, and I don't understand, I can't find what this multiplication of sines means for a wave, I understand that it occurs in waves with similar amplitude.

Two vehicles are approaching along the same route, traveling in opposite directions at speeds $v_1$, and $v_2$ respectively. Both vehicles have their horns activated which, when at rest, emit a pure sound of $f_0 = 440hz$ and similar intensity.The speed of sound is $v_s = 343m/s$. A microphone records the sound of the horns in a time interval around instant $t_0$, when both vehicles are equidistant from the microphone. Around $t_0$, the microphone measures an amplitude described approximately by $$A(t) = \sin(2π\frac{5}{4}f_0 t)\sin(\frac{π}{100}f_0 t).$$

Find the speed $v_1$ and $v_2$ of the cars.

Answer 1

It's acoustic beating, i.e. the way how a pair of signals makes superposition with each other. Your task is from general law of amplitudes extrapolate each sound wave frequencies $f_1, f_2$, using trigonometric relationship :

$$ {\displaystyle {\cos(2\pi f_{1}t)+\cos(2\pi f_{2}t)}={2\cos \left(2\pi {\frac {f_{1}+f_{2}}{2}}t\right)\cos \left(2\pi {\frac {f_{1}-f_{2}}{2}}t\right)}} \tag 1$$

I've done a sample Desmos chart to illustrate this $\sin/\cos$ wave addition principle and mapping to $f_1,f_2$.

Then when you will know these freqencies,- from the Doppler effect:

$$ {\displaystyle \Delta f={\frac {\Delta v}{c}}f_{0}} \tag 2$$

you will find both car's speed which is $v = v_0 + \Delta v$, assume $v_0=0$.