Path Difference Decides Bright or Dark
Place two point sources humming at the same rhythm side by side, and every place their waves meet grows a pattern of bright and dark bands. What decides which is which? A single quantity, the path difference Δ, the gap between the distances from the two sources to that point. When Δ is a whole number of wavelengths, crest meets crest and it is bright; when it is off by half a wavelength, the waves erase each other and it goes dark.
Circular wavefronts spread out from the two point sources. Join the places where crest overlaps crest and you trace a bright antinodal line; join where crest meets trough and you trace a dark nodal line. The fringe spacing is inversely proportional to the gap between the sources, so the farther apart you set them the more tightly the antinodal and nodal lines pack together. Drag the separation between the sources wider and watch the pattern crowd together.
Drag the probe to any point on the field and the difference Δ between its distances to the two sources reads out at once. Where Δ is a whole number of wavelengths (mλ) the waves reinforce and it is bright; where it is off by half ((m+½)λ) they cancel and it is dark. The points that share a constant Δ trace a hyperbola with the two sources as its foci, so the bright and dark marks are curves, and on a distant screen those curves look like almost straight fringes. Move between a bright curve and a dark curve and watch Δ change.
With the two sources in step (Δφ=0) the central line is bright. Give them a phase difference Δφ and the whole pattern of bright and dark slides sideways. The fringes only hold still if this phase relationship stays fixed, which is what coherence means. If the phase wandered at random the pattern would smear away. This is why a laser, whose phase relationship stays locked, makes crisp fringes, while two ordinary bulbs, with phases drifting on their own, leave no interference pattern no matter how you line them up.
Now change only the wavelength λ. As the wavelength shrinks, more crests pack into the same space, so the bright and dark curves crowd together. As it grows, the pattern spreads out. You can see the rule that fringe spacing scales with wavelength. So red light makes wider-spaced fringes than blue, and in white light the differently spaced colors overlap so that only the central band at zero path difference stays white.
Switch the two sources on in phase and the center (Δ=0) is the brightest antinode, crest meeting crest. Flip them to opposite phase and the same spot becomes a fully dark node, crest meeting trough. The path difference has not changed at all, yet flipping only the starting phase swaps bright for dark. Bright or dark is set by the sum of the phase from the path difference and the starting phase difference of the sources, so flipping the starting phase by 180 degrees does exactly what adding half a wavelength of path would.