Deriving Wave Equation - Electric Field Inside Metal

[H12504106]

Homework Statement

Consider an electromagnetic wave hitting a metallic surface with conductivity σ
at normal incidence.
a) Derive the wave equation describing this situation. Hint: Use Ohm’s law, J = σE to
eliminate the current.
b) Solve the wave equation for the electric field to obtain the electric field inside the metal.
How far into the metal does the field propagate?

Homework Equations

The Maxwell Equations in matter:
$\epsilon\nabla \cdot\vec{E} = \rho_f$
$\nabla \times \vec{E} = -\mu\dfrac{\partial \vec{H}}{\partial t}$
$\nabla \cdot \vec{H} = 0$
$\nabla \times \vec{H} = \sigma\vec{E} + \epsilon \dfrac{\partial \vec{E}}{\partial t}$

The Attempt at a Solution

By manipulating the maxwell's equations above and using vector calculus, i can obtain the following:

$\nabla^2\vec{E} = \mu\sigma\dfrac{\partial\vec{E}}{\partial t}+\mu\epsilon\dfrac{\partial^2 \vec{E}}{\partial t^2}$ and
$\nabla^2\vec{H} = \mu\sigma\dfrac{\partial\vec{H}}{\partial t}+\mu\epsilon\dfrac{\partial^2 \vec{H}}{\partial t^2}$.

But i can't proceed on with part (b). How do i solve the wave equation for the electirc field? Is the solution to this wave equation exponential?

Thanks!

 

[HMT]

Hi I solved past year in the course of partial differential equations, I rembember tha we use the separation of variables methods, I will search it, and if a I get I will tell you.

 

[HMT]

Sorry, I forget to say that the way we solved was a Fourier Series, (boundary conditions included)