Chasing Rainbows

“C’mon, Sy, Newton gets three cheers for tying numbers to the rainbow’s colors and all that, but what’s it got to do with that three speeds of light thing which is where we started this discussion?”

“Vinnie, they weren’t just numbers, they were angles. The puzzle was why each color was bent to a different degree when entering or leaving the prism. That was an inconvenient truth for Newton.”

“Inconvenient? There’s a loaded word.”

“Indeed. A little context — Newton was in a big brouhaha about whether light was particles or waves. Newton was a particle guy all the way, battling wave theory proponents like Euler and Descartes and their followers on the Continent. Even Hooke in London had a wave theory. Newton’s problem was that his beam deflections happened right at the prism’s air‑glass interfaces.”

“What difference does that … wait, you mean that there’s no bending inside the prism? Light inside still goes straight but in a different direction?”

“That’s it, exactly. The deflection angles are the same, whether the beam hits the prism near the short‑path tip or the long‑path base. No evidence of further deviation inside the prism unless it has bubbles — Newton had to discard or mask off some bad prisms. Explaining the no‑curvature behavior is difficult in a particle framework, easy in a wave framework.”

“Really? I don’t see why.”

Left: faster medium, right: slower medium
Credit: Ulflund, under Creative Commons 1.0

“Suppose light is particles, which by definition are local things affected only by local forces. The medium’s effects on a particle would happen in the bulk material rather than at the interface. The effect would accumulate as the particles travel further through the medium. The bend should be a curve. Unfortunately for Newton, that’s not what his observations showed.”

“OK, scratch particles. Why not scratch waves, too?”

“Waves have no problem with abrupt variation at an interface, They flip immediately to a new stable mode. For example. here’s an animation showing an abrupt speed change at the interface between a fast‑travel medium like air and a slow‑travel medium like glass or water. See how one end of each bar gets slowed down while the other end is still moving at speed? By the time the whole bar is inside, its path has slewed to the refraction angle.”

“Like a car sliding on ice when a rear wheel sticks for an moment, eh Sy?”

“That was not a fun ride, Vinnie.”

“I enjoyed it. Whatever, I get how going air‑to‑glass or vice‑versa can change a beam’s direction. But if everything’s going through the same angle, how do rainbows happen?”

“Everything doesn’t go through the same angle. Frequencies make a difference. Go back to the video and keep your eye on one bar as it sweeps up the interface. See how the sweep’s speed controls the deflection angle?”

“Yeah, if the sweep went slower the beam would get a chance to bend further. Faster sweeps would bend it less. But what could change the sweep speed?

“Two things. One, change the medium to one with a different transmission speed. Two, change the wave itself so it has a different speed. According to Snell’s Law, the important parameter for a pair of media is their ratio of fast‑speed divided by slow‑speed. If the fast medium is a vacuum that ratio is the slow medium’s index of refraction. The greater the index, the greater the bend.”

“Changing the medium doesn’t apply. I got one prism, it’s got one index, but I still get a whole rainbow.”

“Right, rainbows are about how one prism treats a bunch of waves with different time and space frequencies.”

“Space frequency?”

“If you measure a wave in meters it’s cycles per meter.”

“Wavelength upside down. Got it.”

“Whether you figure in frequencies or intervals, the wave speed works out the same.”

“Speed of light, finally.”

“Point is, when a wave goes through any medium, its time frequency doesn’t change but its space frequency does. Interaction with local charge shortens the wavelength. Short‑wavelength blue waves are held back more than long‑wavelength red ones. The different angles make your rainbow. The hold‑back is why refraction indices are usually greater than one.”

“Usually?”

~~ Rich Olcott