We know that for electromagnetic waves, according to Maxwell's Theory
$$v=\frac{1}{\sqrt{\mu\epsilon}}$$
Now consider an opaque object like say Gold. It has a particular value of permittivity and permeability, which implies we will have a finite defined value of $v$. But how is light travelling in gold? I understand the fact that other more energetic waves may travel through gold like $\gamma$ ray. But it specifies nothing about wavelength or frequency.
Feynman chapter 23 Refractive Index of Dense Materials in his Lectures Volume 2 explains the fact that the permittivity of a material is frequency dependent (and can be imaginary!) as is its refractive index.
Here is an example for gold which shown quite clearly that permittivity is not constant over a range of frequencies.
The imaginary nature of the permittivity is a Mathematical convenience so that components of the properties of a material can be described in a compact form. The real part gives information about the speed of an electromagnetic wave through the medium and the imaginary part gives information about the attenuation of an electromagnetic wave due to energy loss as it passes through the medium.
The gold (or whatever) atoms/ions experience the radiation as an oscillating electric field. The atomic electrons oscillate in response. The size (and phase) of their response depends on the detail of the atomic structure, particularly resonances, and how they match the frequency of the original, and follows the mathematics of the forced simple harmonic oscillator. All these oscillating atomic electrons act like dipole antennae to re-radiate EM radiation, at the same frequency as the original. So down the line, an observer will see the combined effect of the original wave and all the re-radiated waves. Standard trig manipulatios $\sin(\omega t)+a \sin(\omega t+b)=(1+a\cos b)\sin(\omega t) + a \cos(\omega t)=A\sin(\omega t + \delta) $ shows that this is still a sine wave but with the phase (usually) behind thr original, i.e. the wave travels more slowly.
