This is quoted from Concepts of Physics by H.C.Verma, chapter "Light Waves", page 370, under the topic "17.9 Coherent and Incoherent Sources":
In order to obtain a fairly distinct interference pattern, the path difference between the two waves originating from coherent sources should be kept small. This is so because the wavetrains are finite in length & hence with large difference in path, the waves do not overlap at the same instant in the same region of space. The second wavetrain arrives well after the first train has already passed & hence, no interference takes place.
Why did the author write the wavetrains are finite in length? What is the cause for the finiteness of the wavetrain? Even if it is finite, why doesn't interference occur when the distance between the two waves is large?
Also, in another stanza, he writes:
Because of the incoherent nature of the be=asic process of light emission in ordinary sources, these sources can't emit highly monochromatic light. … The light emitted by an ordinary source always has a spread in wavelength. An ordinary sodium vapour lamp emits light of wavelength $589.0~\text{nm} \, \& \, 589.6~\text{nm}$ with a spread of about $\pm 0.01~\text{nm}$ in each line.
Now, what does the author mean by "spread in wavelength"? How does it hamper/disturb interference effect?
Best Answer
The first quote is about what is called temporal coherence. As the light emission processes in sources other than lasers are due to spontaneous emission the phases of "wave trains" of different emission processes are not aligned (as the different "wave trains" begin randomly at different instances). On the time-scale of a single decay process (perhaps $10^{-10}\,\mathrm{s}$), however, the phase of the wave will be stable, only when waiting longer the phase correlation will vanish. This coherence is quantified by the coherence time $\tau_c$ and the corresponding coherence length $\lambda_c = c \cdot \tau_c$.
If you now have a setting with a optical path difference above the coherence length, the phases of the to beams will not be correlated, as they where emitted with a time difference larger than the coherence time, so no interference patterns will be observed.
This is one of the reasons you do not see interference fringes on a window pane in natural light (as the pane is thicker than the coherence length), but you do see fringes on thin films in natural light.
The spread in wavelength means, that even in a lamp that (incoherently, that is due to spontaneous emission) emits light due to the relaxation of a single well defined atomic orbital, the photons vary in wavelength. This has several reasons (listed here for the case of a gas discharge lamp):
Why does this broadening impede interference? Because the shifted frequencies blur the phase relations on a time-scale even shorter than one emission process.