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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/4/b/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2014 / 4 / b / Solution by

Codex  0 2026-10-25

First interpret

λ

as the material wavelength

2 π / k

. Since

ω = k v_{p} (k)

,

v_{g} = v_{p} + k \frac{d v _{p}}{d k} = v_{p} - λ \frac{d v _{p}}{d λ} .

For

v_{p} = c / (a - bλ)

this gives

v_{g} = \frac{c ( a - 2 bλ )}{( a - bλ ) ^{2}}, \frac{v _{g}}{v _{p}} = \frac{a - 2 bλ}{a - bλ} .

With the supplied numbers,

bλ = 0.015

and the ratio is

1.470/1.485 = 0.989899 ≃ 0.99

.

There is a common optical convention to distinguish. If the dispersion law is instead parameterized by vacuum wavelength

λ_{0} = 2 π c / ω

, its refractive index is

n (λ_{0}) = a - b λ_{0}

. The group refractive index is then

n_{g} = n - λ_{0} d n / d λ_{0} = a

v_{g} = c / a

and

v_{g} / v_{p} = (a - b λ_{0}) / a = 0.990000

. Both conventions yield the requested approximate ratio, but their exact algebra differs. This is the distinction between group velocity in material and vacuum wavelength coordinates.

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