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The Wave Equation

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

$\displaystyle \del \times (\del \times {\bf a}) = \del( \del \cdot {\bf a}) - \nabla^2 {\bf a}$ (11.13)

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

$\displaystyle \nabla^2 u - \frac{1}{v^2} \partialdiv{^2u}{t^2} = 0$ (11.14)

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

$\displaystyle v = \frac{1}{\sqrt{\mu \epsilon}}.$ (11.15)

The wave equation separates11.2 for harmonic waves and we can actually write the following homogeneous PDE for just the spatial part of $ \Vec{E}$ or $ \Vec{B}$ :

$\displaystyle \left(\nabla^2 + \frac{\omega^2}{v^2}\right) \Vec{E} = \left(\nabla^2 + k^2 \right) \Vec{E} = 0$    

$\displaystyle \left(\nabla^2 + \frac{\omega^2}{v^2}\right) \Vec{B} = \left(\nabla^2 + k^2 \right) \Vec{B} = 0$    

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 that11.3:

$\displaystyle \del e^{ik\hat{\bf n} \cdot \sVec{x}} = ik\hat{\bf n} e^{ik\hat{\bf n} \cdot \sVec{x}}$ (11.16)

where $ \hat{\bf n}$ is a unit vector, we can easily see that the wave equation has (among many, many others) a solution on $ \RE^3$ that looks like:

$\displaystyle u({\bf x},t) = u_0 e^{i(k\hat{\bf n} \cdot \sVec{x} - \omega t)}$ (11.17)

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

$\displaystyle k = \frac{\omega}{v} = \sqrt{\mu \epsilon}\omega$ (11.18)

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 2017-07-11