Want to know if the frequency of wave can be changed with respect to distance, is it possible either by using multiple interfering waves or any other way?
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$\begingroup$ Hi! Why do you think using multiple interfering waves can do the trick? Oh, I can still say Happy (or unhappy) new year! $\endgroup$– Deschele SchilderCommented Jan 7, 2020 at 7:23
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1$\begingroup$ "Multiple interfering waves" suggests you might be thinking about beat frequencies. That makes me start to think about period-doubling (used to build digital clocks) and frequency-doubling (a common nonlinear effect). Is that what you have in mind? The title suggests you're imagining a modest distance- based frequency shift, like a musical instrument that is in tune when you stand next to it but flat when you're a mile away. $\endgroup$– rob ♦Commented Jan 7, 2020 at 12:31
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$\begingroup$ Yes, check cosmological redshift, basically expanding space stretches EM radiation wavelengths (lowers frequency). But this effect is not part of acoustics. $\endgroup$– Agnius VasiliauskasCommented Dec 18, 2023 at 8:09
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3 Answers
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Yes, it is possible for a wave to have a frequency change along its propagation direction; famously, this was demonstrated by an experiment with gravity causing a frequency change in gamma rays, by Robert Pound and Glen Rebka. The change isn't large, but the effect is real.
In non-relativistic theory, this would not happen; it relies on the nature of time being rather complex.
It is also possible for an accelerating wave source to create, from a fixed-frequency generator, a field of wave motion with different frequencies due to the Doppler shift and time-of-propagation variation, but that isn't a true function of position, rather depends on position and time.
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Since this has the tag of acoustics I believe it is meant to be for sound waves, so I'll just cite what I know about those.
As far as I am aware there can be no change in the frequency of a "monochromatic" sound wave. Instead, the non-linearities associated with its propagation can indeed change the waveform of a travelling wave. Due to the difference in velocity between the different "parts" of the wave the various "parts" (with the most prominent being the extremes) will have a different net velocity (flow velocity + sound velocity). This will distort the waveform introducing additional frequencies (check Fourier theorem - https://en.wikipedia.org/wiki/Fourier_transform - for more info). Nevertheless, the fundamental (or the initial excitation's frequency) will not cease to exist.
The phenomenon rarely concerns the acoustics engineers because the higher attenuation of high frequencies acts against it and the final result is of extremely minor importance for practical applications. In the absence of attenuation, the phenomenon would result in the creation of a triangular-like wave with abrupt changes in pressure. In Acoustics - An Introduction book by Heinrich Kuttruff you can find a brief introduction of the concept.
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Not sure if I am right:
For wave equation: $\nabla^2 \phi = 1/c^2 \frac{\partial^2\phi}{\partial t^2}$, the solution is $\phi =A(r) e^{i(\omega t-r \cdot k(r))}$.
If $\omega$ is a function of r, then $\nabla^2 \phi$ would yield a term of $t^2 \nabla^2\omega$, which looks not a proper solution.
