interference of light explained
Set two point sources that tremble in the same rhythm side by side, and wherever the two waves meet a pattern of bright and dark appears. What splits bright from dark is the difference in distance from the two sources to that point, the path difference Δ. When the path difference is a whole number of wavelengths, crest meets crest and the spot is bright. When it is off by half a wavelength, the waves erase each other and the spot is dark. The difference in distance is the whole pattern.
Circular wavefronts spread from the two point sources. Join the places where crest meets crest and you get a bright line (antinodal). Join the places where crest meets trough and you get a dark line (nodal). The spacing of the pattern is inversely related to the gap between the two sources. Spread the sources farther apart and the bright lines and dark lines pack tighter. Widen the gap and the pattern growing tighter shows itself. The farther apart, the more lines.
Drop a probe on any point of the screen and the difference in distance Δ to the two sources reads off at once. A place where the path difference is a whole number of wavelengths (mλ) reinforces and is bright, and a place off by half a wavelength, (m+½)λ, cancels and is dark. Points of constant path difference draw hyperbolas with the two sources as foci, so the bright and dark traces appear as curves, and on a distant screen those curves look almost like straight stripes. Move between a bright curve and a dark curve and you can follow how Δ changes. Far away, the curve becomes a stripe.
Line up the phases of the two sources, that is set Δφ=0, and the center line is bright. Give them a phase difference and the whole pattern of bright and dark slides sideways. The phase relation of the two sources has to stay fixed or the pattern will not hold still, and that property is called coherence. If the phase wanders on its own, the pattern smears and disappears. A laser, whose phase relation is tightly bound, makes a crisp pattern, while two lamps whose phases wander leave no interference pattern no matter how you set them side by side. A bound phase is what holds the pattern.
Now change only the wavelength. Shorten it and more crests pack into the same gap, so the bright and dark curves grow tighter. Lengthen it and the pattern grows sparse. The rule that the spacing is proportional to the wavelength shows itself plainly. Red light therefore has a sparser pattern than blue light, and in light of many colors the patterns of the colors slip a little against one another, so only the center stripe, where the path difference is 0, stays white. Only the center, where the colors overlap, stays white.
Turn the two sources on in the same phase and the center, where Δ=0, is crest meeting crest, the brightest antinode. Flip them to the opposite phase and, at that same place, crest meets trough and it becomes a completely dark node. The path difference is unchanged, yet changing only the starting phase swaps bright and dark. Bright and dark are fixed by the phase the path difference makes, added to the starting phase, so flipping the starting phase by 180 degrees is the same as walking an extra half wavelength along the path. Flipping the start equals half a wavelength.