velocity of red light vs. blue light in water - qualitative reasoning?

vvarma

Every source I have referred to says red light is faster in water than blue light. However, nearly all descriptions/explanations depend on refractive indices and 'bending' to show that blue light is slowed to a greater extent. Instead, I'm wondering if a more precise explanation based on dispersion is possible b/c every time I attempt to reason out this fact based on group velocities, I get the opposite conclusion.

My reasoning: Water is a dispersive material so frequency w is non-linear wrt wave number k. Group velocity is dw/dk and always less than c [sanity check] but the slope of the graph increases with k so group velocity is higher at lower wavelengths [k inversely proportional to lambda] which seems to imply that wave packets of lower wavelength should have a faster speed in water...

Am I using the dispersion relationship of w-k wrongly or is the group velocity not the same as the velocity of propagation [I'm fairly confident it is since number sources cite it as the speed that energy/information of the wave travel]?

Should I use phase velocity w/k instead? I have strong doubts given a) it always exceeds c and b) that dispersive wave means that w/k not equal to dw/dk = v. However, if I do use it, I get the correct connection of greater phase velocity for great wavelength.

I have a feeling that though I understand the terms and concepts to some degree, I can't draw the connection I want to b/c I'm lacking some other tool. Can you even draw the conclusion that vred > vblue in water knowing only w-k relationship?

vanhees71

The classical dispersion theory of em. waves in media is enough to understand the principle properties of light in media. A very good source about this is

A. Sommerfeld, Lectures on Theoretical Physics, Vol. IV (Optics)

There also wave propagation in the case of anomalous dispersion, where the phase velocity is greater than the in vacuo speed of light, is thoroughly discussed in a very elegant manner.

jambaugh

The ratio of omega to k is the phase velocity.

Just remember that phase and group velocities have inverse relationship relative to c. Thus your reasoning when applied to phase velocity -- -red has lower phase velocity-- reverses to --red has higher group velocity--.

vvarma

@vanhees71 I'll definitely check out that source. Thank you.

@jambaugh Thanks for the reciprocal argument. It seems possible to me that I might be using the wrong w-k relationship to show this result due to a mislabeling on the part of my professor. Just to verify: is the graph of w(k) for water [and transparent materials in general] at all similar to w²=w_p²+c²k² because this is what my [incorrect] analysis is based on. If so, do you potentially have the correct w(k) from which to draw the wanted results?