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

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.

• please use mathjax Apr 4, 2023 at 2:33
• sorry if it seems like homework, but it's not, it's a test I already solved, I couldn't do it. Apr 4, 2023 at 3:34
• I've hidden a number of comments that should have been posted as answers, and replies to them. To provide a partial answer to a question, post an answer. To answer an off-topic question, please edit the question so that it is on-topic.
– rob
Apr 4, 2023 at 4:03

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 $$f_1,f_2$$.
$${\displaystyle \Delta f={\frac {\Delta v}{c}}f_{0}} \tag 2$$
you will find both car's speed which is $$v = v_0 + \Delta v$$, assume $$v_0=0$$.