Are two waves coherent iff they have the same frequency?

Question

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

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

?

Answer 1

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)

Answer 2

This is actually much more subtle, the question is very deep and whole books are written about theory of coherence in optics, quantum mechanics etc. What is important is that you might have an ensemble of sources (say in a lightbulb) which all emit light of the same frequencies but in random moments, making the relative phases between different light pulses effectively random (and rapidly varying). Properties of such incoherent radiation are very different than those of coherent radiation (one with stable phase).

EDIT: Another way of saying what I said is following. When speaking of coherent or incoherent radiation, we never mean one single light pulse of the form $\sin ( \omega t - \delta )$. Instead we mean large amount of light waves. Each of them has its own $\omega$ and $\delta$. It is not physically possible that all $\omega$'s are the same. Every light source has its specific bandwidth $\Delta \omega$, so it emits radiation in whole spectrum $(\omega _0 -\frac{\Delta \omega}{2},\omega _0 +\frac{\Delta \omega}{2})$. This means that even if phases are correlated at the moment of emission, they will stop being correlated after a while. Spread of frequencies $\Delta \omega$ is a quantitative measure of rapidity of this process. However, very frequently it happens that phases are uncorrelated from the very beggining. For example collding atoms emit energy through radiation. In this case phases will be uncorrelated because times of collision are random.

Light is really en electromagnetic field. However, what we usually measure is intensity of the field rather than field itself. Intensity of the field (or in optical words, brightness) is proportional to square of the field. Now imagine that you have two light waves, $E_1=A \sin(\omega t - \delta_1)$ and $E_2 =A\sin(\omega t -\delta_2)$. Intensity of superposition is

$I=|E_1+E_2|^2=E_1^2+E_2 ^2 +2E_1 E_2=I_1+I_2+2A^2 \sin (\omega t - \delta_1)\sin (\omega t - \delta_2)$.

If you have coherent radiation, then last term (so called interference term) is very important. However, if you have many light waves and phase relation between them is random it will be sometimes positive, sometimes negative but zero on average. Therefore you can neglect it and have

$I=I_1 + I_2$

This is the fundamental difference between incoherent and coherent radiation. The same phenomenon actually governs the transition between quantum and classical world. Classical world emerges from quantum world when you discard the phases, which are rapidly varying and can't be observed in macroscopic situations.