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

First interpret $\lambda$ as the material wavelength $2\pi /k$. Since $\omega =k{v}_p(k)$,For $v_p=c/(a-b\lambda )$ this givesWith the supplied numbers, $b\lambda =0.015$ and the ratio is $1.470/1.485=0.989899\simeq 0.99$.

There is a common optical convention to distinguish. If the dispersion law is instead parameterized by vacuum wavelength ${\lambda }_0=2\pi c/\omega$, its refractive index is $n({\lambda }_0)=a-b{\lambda }_0$. The group refractive index is then $n_g=n-{\lambda }_{0}dn/d{\lambda }_0=a$, giving $v_g=c/a$ and $v_g/v_p=(a-b{\lambda }_0)/a=0.990000$. 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.