Physics · Ch 6 — Optics
Proof for Laws of Refraction Using Huygens' Principle
Proof for Laws of Refraction Using Huygens' Principle
The same Huygens' construction, applied at a refracting rather than a reflecting boundary, proves Snell's law. A plane wavefront in medium (1), perpendicular to incident rays , strikes a plane refracting surface obliquely and enters medium (2), where the wave travels at a different speed. Because the two media give different speeds , the two secondary wavelets from and grow by different distances in the same elapsed time: . Working through the resulting right-triangle ratio gives ; substituting the refractive indices and recovers Snell's law, . Because light travels faster in a rarer medium and slower in a denser one, this same construction shows the wavelength itself is longer in the rarer medium and shorter in th …
What this figure shows. A plane wavefront AB in medium (1), perpendicular to incident rays L and M, strikes a plane refracting boundary XY obliquely, entering medium (2) where the wave travels at a different speed. As in the reflection proof, point A reaches the boundary first; by the time B reaches it at B', the secondary wavelet from A has expanded into medium (2), but now by a different distance AA' than BB' (since the two media give the two rays different speeds v1 and v2 over the very same elapsed time). Setting up the resulting right-triangle ratio sin i/sin r = (BB'/AB')/(AA'/AB') = v1/v2, and substituting the refractive indices n1 = c/v1 and n2 = c/v2, recover …
Worked out. Building directly on the wavelength/speed relations derived from Huygens' construction (lambda1/lambda2 = n2/n1), the accompanying worked calculation for sodium light entering water (vacuum wavelength 5893 angstrom, n=1.33) computes the frequency two independent ways -- once from v(vacuum)/lambda(vacuum) and once from v(water)/lambda(water) -- and finds exactly the same value both times, about 5.091 times 10 to the 14 hertz. This numerically confirms the general rule already stated in words in the section: a light wave's frequency is fixed by its source and never changes as the wave crosses from one medium into another, no matter how different the two media's refractive indices are; only the wave's speed …