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The Wave Equation

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

\begin{displaymath}
\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$\nabla$}\ti...
...( \mbox{\boldmath$\nabla$}\cdot {\bf a})
- \nabla^2 {\bf a}
\end{displaymath} (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:
\begin{displaymath}
\nabla^2 u - \frac{1}{v^2} \frac{\partial ^2u}{\partial t^2} = 0
\end{displaymath} (9.17)

(for $u = \mbox{\boldmath$E$}$ or $u = \mbox{\boldmath$B$}$) where
\begin{displaymath}
v = \frac{1}{\sqrt{\mu \epsilon}}.
\end{displaymath} (9.18)

The wave equation separates9.2 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$}$:


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


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

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 that9.3:

\begin{displaymath}
\mbox{\boldmath$\nabla$}e^{ik\hat{\bf n} \cdot \mbox{\scrip...
...\bf n} e^{ik\hat{\bf
n} \cdot \mbox{\scriptsize\boldmath$x$}}
\end{displaymath} (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:
\begin{displaymath}
u({\bf x},t) = u_0 e^{i(k\hat{\bf n} \cdot \mbox{\scriptsize\boldmath$x$} - \omega t)}
\end{displaymath} (9.20)

where the wave number $\mbox{\boldmath$k$}= k\hat{\bf n}$ has the magnitude
\begin{displaymath}
k = \frac{\omega}{v} = \sqrt{\mu \epsilon}\omega
\end{displaymath} (9.21)

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


next up previous contents
Next: Plane Waves Up: The Free Space Wave Previous: Maxwell's Equations   Contents
Robert G. Brown 2013-01-04