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

Two waves are

coherentif 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)