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If sound sources have same amplitude say $A$ and nearly same angular frequency like say $\omega_1$ and $\omega_2$ then at a point equidistant from them is it correct to assume that the resulting amplitude after superposition will be nearly $2A\cos((\omega_1 -\omega_2)/(2 t))$ ?

If yes,why?

If the waves are $A\sin(\omega_1 t - k_1 x)$ and $A\sin(\omega_2 t - k_2 x)$ then won't the terms $k_1$ and $k_2$ also affect the resulting amplitude?

Asking this because my physics teacher made that assumption in a certain problem but I did'nt receive a satisfactory answer when I asked a reason for it.

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  • $\begingroup$ errata in question: the time t belongs on top with the frequencies not under the division. $\endgroup$ – blanci Oct 7 '20 at 2:56
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Since this is clearly homework and excercises question, I will provide just hints.

  • This kind of treatment is the same how beats are descripted. Study this article, it will help you get that.
  • Since $k=\frac{\omega}{c}$ where $c$ is constant and the distance is the same for both the signals, it will not cause any more uncertain phase shifts.
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Further to Pira: we get the sin of w(t - x/c)=wt - p where p=wx/c is just some constant number which doesn’t change in time and is an extra fixed phase not very important. The main features of adding two oscillations just come from the frequencies and the amplitudes. Any fixed phase is generally unimportant.. corresponding to precise start or zero point of first oscillation. What we really interested in is the approximate resultant frequency and amplitude ... the amplitude slowly varies.. beats.

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