There are the following two conditions for the interference of two light waves:
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The sources of the waves must be coherent, which means they emit identical waves with a constant phase difference.
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The waves should be monochromatic – they should be of a single wavelength.
Here is the question, If two sources are coherent it means they are producing waves with the same frequency and since the interference takes place at a common location it means that the waves should have the same speed in there. This implies that they must have the same wavelength which automatically is the second condition. It means if the first condition is met, the second follows immediately. Why do then we have them separately? I mean the first condition was enough for interference.
Best Answer
Monochromaticity makes interference easier to observe, but it is not necessary for interference. "Coherence" between two sources usually means "temporal coherence", which in turn means that there is a constant phase relationship between the two sources. Note that I didn't say "constant phase difference". For example if the phase difference between the two sources is changing at a constant rate the phase relationship is still constant. There will be temporal interference between them. As @flippiefanus said (but in different words), a detector sees the changing phase difference as a beat frequency. If the two are directed at an imaging array, the changing phase difference shows up as moving fringes (each detector in the array sees a beat frequency, but adjacent detectors see the beat frequency out of phase with each other).
If your question is simply, "Does satisfying the first condition imply that the second condition is satisfied?", then the answer is "no". It is possible for waves from two sources to be coherent without being monochromatic. For example, a very short laser pulse is not monochromatic; it is comprised of a superposition of wavelengths. Two pulses, formed by using a beamsplitter to split the pulse, will certainly interfere because they are identical. If two separate sources produce identical pulses, the pulses will interfere.
Note that those two pulses meet the condition I stated above: there is a constant phase relationship between them (because they are identical). They would also have a constant phase relationship (with constantly changing phase difference) if one was bounced off a moving mirror; but if interfered the two pulses would produce a beat frequency proportional to the speed of the moving mirror. Of course the beat frequency would itself be in a pulse, and would only be produced when the two pulses overlap.