= Solution
The <Drude model> treats carriers as classical independent particles, accelerated between collisions by the electric force. Collisions occur at constant rate $\tau^{-1}$ and randomize the directed velocity to zero mean; the ions are an effectively stationary background. Over a short time $dt$, directed velocity gains $(e/m)\mathbf E\,dt$ while the fraction $dt/\tau$ that collides loses its mean directed motion. Therefore
$$
\frac{d\langle\mathbf v\rangle}{dt}=-\frac{\langle\mathbf v\rangle}{\tau}+\frac em\mathbf E.
$$
Here $e$ is the signed carrier charge. For the stated complex harmonic convention,
$$
\mathbf v_0=\frac{e\tau/m}{1-i\omega\tau}\mathbf E_0,
\qquad
\boxed{\mathbf J=ne\langle\mathbf v\rangle=\sigma\mathbf E,\quad
\sigma=\frac{ne^2\tau/m}{1-i\omega\tau}.}
$$
The conductivity is positive in its dissipative real part regardless of the sign of $e$.
For zero charge density, Gauss's law makes the electric amplitude transverse. Faraday's and Ampère-Maxwell's laws, with displacement current retained, give
$$
\nabla\times\nabla\times\mathbf E
=-\mu_0\sigma\partial_t\mathbf E-\mu_0\epsilon_0\partial_t^2\mathbf E.
$$
Substitution of $e^{i\mathbf k\cdot\mathbf x-i\omega t}$ yields
$$
\boxed{k^2=\frac{\omega^2}{c^2}\epsilon_r,\qquad
\epsilon_r=1+\frac{i\sigma}{\omega\epsilon_0}
=1-\frac{\omega_p^2}{\omega(\omega+i/\tau)}.}
$$
For propagation in a fixed direction, $k$ is the possibly complex scalar wave number; equivalently the spatial equation uses $\mathbf k\cdot\mathbf k$, without complex conjugation. The printed real-vector notation $|\mathbf k|$ must be analytically continued in an absorbing or evanescent medium.
In the <high-frequency Drude plasma cutoff> regime, $\omega\tau\gg1$ gives $\epsilon_r\simeq1-\omega_p^2/\omega^2$. Below the plasma frequency, the physical decaying branch is
$$
\boxed{k\simeq i\sqrt{\omega_p^2-\omega^2}/c,\qquad
\mathbf E\propto e^{-x\sqrt{\omega_p^2-\omega^2}/c}.}
$$
Finite collisions add a small phase component while maintaining positive attenuation. Very near the threshold one should retain the full complex permittivity rather than the purely imaginary approximation.
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