[Physics] Are two waves coherent iff they have the same frequency


The essential property that two waves must own in order to interfere with each other is to be coherent.

Two waves are coherent if their phase difference $\phi_2-\phi_1$ does not change in time

The phase of a one dimensional wave is $\phi=kx-\omega t+\delta$

Does saying that $\phi_2-\phi_1$ does not depend on time imply that $\omega_1=\omega_2$ (That is the two waves, to be coherent, must surely have the same frequency) ?

If this true does the reverse holds, and therefore

$$\phi_2-\phi_1 \mathrm{indipendent \space from \space time} \iff \omega_1=\omega_2$$


Best Answer

Yes. Omega is the time derivative of phi. Phi1 dot = phi2 dot means omegas are the same.

See my other answer a couple days ago on the subtleties of coherency. There is phase noise on any transmitter and freq as a result has drift and random noise. It depends on the time proof, it could be coherent to 1 part in 10^6 for milliseconds and 1 in 10^5 for a sec. It depends on the stability and sometimes actively synchronizing the oscillators and few other circuits that provide the freq sources

That assumes the same x (or you have phase offset you could use to geolocate), and the same initial phase (or you have a constant phase difference, sync them or do phase difference detection)

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