The Wave Equation

After a little work (take the curl of the curl equations, using the identity:

$$\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$\nabla$}\ti... ...( \mbox{\boldmath$\nabla$}\cdot {\bf a}) - \nabla^2 {\bf a}$$ (9.16)

and using Gauss's source-free Laws) we can easily find that $\mbox{\boldmath$E$}$ and $\mbox{\boldmath$B$}$ in free space satisfy the wave equation:

$$\nabla^2 u - \frac{1}{v^2} \frac{\partial ^2u}{\partial t^2} = 0$$ (9.17)

(for $u = \mbox{\boldmath$E$}$ or $u = \mbox{\boldmath$B$}$) where

$$v = \frac{1}{\sqrt{\mu \epsilon}}.$$ (9.18)

The wave equation separates^9.2 for harmonic waves and we can actually write the following homogeneous PDE for just the spatial part of $\mbox{\boldmath$E$}$ or $\mbox{\boldmath$B$}$:

$$\left(\nabla^2 + \frac{\omega^2}{v^2}\right) \mbox{\boldmat... ...left(\nabla^2 + k^2 \right) \mbox{\boldmath$E$} = 0 \nonumber$$

$$\left(\nabla^2 + \frac{\omega^2}{v^2}\right) \mbox{\boldmat... ...left(\nabla^2 + k^2 \right) \mbox{\boldmath$B$} = 0 \nonumber$$

where the time dependence is implicitly $e^{-i\omega t}$ and where $v = \omega/k$.

This is called the homogeneous Helmholtz equation (HHE) and we'll spend a lot of time studying it and its inhomogeneous cousin. Note that it reduces in the $k \to 0$ limit to the familiar homogeneous Laplace equation, which is basically a special case of this PDE.

Observing that^9.3:

$$\mbox{\boldmath$\nabla$}e^{ik\hat{\bf n} \cdot \mbox{\scrip... ...\bf n} e^{ik\hat{\bf n} \cdot \mbox{\scriptsize\boldmath$x$}}$$ (9.19)

where $\hat{\bf n}$ is a unit vector, we can easily see that the wave equation has (among many, many others) a solution on $\rm I \hspace{-.180em} R^3$ that looks like:

$$u({\bf x},t) = u_0 e^{i(k\hat{\bf n} \cdot \mbox{\scriptsize\boldmath$x$} - \omega t)}$$ (9.20)

where the wave number $\mbox{\boldmath$k$}= k\hat{\bf n}$ has the magnitude

$$k = \frac{\omega}{v} = \sqrt{\mu \epsilon}\omega$$ (9.21)

and points in the direction of propagation of this plane wave.