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I believe I found the solution for my question, it was actually fairly simple (no calculus or explicit use of the Doppler effect!). In the received signal $r(t)$, the value at any time $t$ is equal to the sum of the perturbations that would reach the observer at time $t$. Assuming that the speed of the source is significantly lower than the speed of a wave ...


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I would assume only the distance between the source and the observer, $x(t)$, has any effect on the frequency scaling. Wave Dispersion There are multiple effects that matter here if the medium is dispersive (i.e., the wave frequency depends upon the wave number or another way of stating this is that the phase speed depends upon the wave frequency ...



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