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Plane Waves

For remote sensing radar applications the case of time-harmonic-fields may be assumed [Kostinski86]. For time-harmonic-fields the instantaneous field vectors varies sinusoidally in time with a single angular frequency $ \omega$

$\displaystyle \vec{X}(\vec{r},t) = \Re \{\vec{X}(\vec{r})e^{iwt}\}$ (A.34)

where $ \Re$ is the real part of a complex quantity, $ \vec{X}(\vec{r})$ the complex field amplitude and we choose $ e^{iwt}$ for the time dependency according to the (IEEE) convention in this treatment.

The Maxwell equations can then be simplified to

$\displaystyle \nabla \times \vec{E}(\vec{r}) = - i\omega\vec{B}(\vec{r})$ (A.35)

$\displaystyle \nabla \times \vec{H}(\vec{r}) = i\omega\vec{D}(\vec{r}) +\vec{J}(\vec{r})$ (A.36)

$\displaystyle \nabla \cdot \vec{B}(\vec{r}) = 0$ (A.37)

$\displaystyle \nabla \cdot \vec{D}(\vec{r}) = \rho(\vec{r})$ (A.38)

In this case the wave equation for a source free, non conductive medium ($ \rho=0$ and $ J=0$) can be written as

$\displaystyle \nabla^2 \vec{E}(\vec{r},t) + k^2 \vec{E}(\vec{r},t) = 0$ (A.39)

where $ k = \omega \sqrt{\mu \epsilon}$ is the wave number. For plane waves we impose the condition that, in a Cartesian coordinate system, $ \vec{E}$ and $ \vec{H}$ are not a function of x and y, if z is the propagation direction, which means that the field vector oscillates in a plane perpendicular to the propagation direction. Such waves are called transversal electromagnetic (TEM) waves. In this case $ \vec{E}(\vec{r},t)$ becomes

$\displaystyle \vec{E}(z)= \hat{E}_0e^{ikz}$ (A.40)

and by applying (A.23) and (A.21) we find for the H-field

$\displaystyle \vec{H}(z)= \hat{H}_0e^{ikz} \quad \quad with  \hat{H}_0 = \frac{1}{\eta}\vec{e}_z \times \hat{E}_0$ (A.41)

where $ \eta=\sqrt{\mu/\epsilon}$ is the intrinsic impedance of the medium, $ \vec{e}_z$ is a unit vector in propagation (z) direction and $ \hat{E}_0$ and $ \hat{H}_0$ the complex amplitudes. Similarly we find for a plane wave propagating in an arbitrary direction $ \vec{e}_{p}$

$\displaystyle \vec{E}(r)= \hat{E}_0e^{ik\vec{e}_{p} \vec{r}}$ (A.42)

$\displaystyle \vec{H}(r)= \hat{H}_0e^{ik\vec{e}_{p} \vec{r}} \quad \quad with  \hat{H}_0 = \frac{1}{\eta}\vec{e}_p \times \hat{E}_0$ (A.43)

where $ \vec{e}_{p}$ is a unit vector parallel to the propagation direction.
As shown above the electromagnetic wave system is made up of a set of coupled, time varying and mutually orthogonal electric and magnetic vector fields. The field vectors are perpendicular to the propagation direction.
For a specific time t and point in space $ \vec{r}$ the direction and the magnitude of the electric field vector $ \vec{E}(r,t)$ is given by the real part of the complex harmonic field expression

$\displaystyle \vec{E}{re}(r,t) = \Re(\hat{E}_{0}e^{i(\omega t + \vec{k}\vec{r})})$ (A.44)

According to Poynting's theorem, the power flux density of the wave equals the direction and magnitude of the Poynting vector $ \vec{S}$

$\displaystyle \vec{S}=\vec{E}_{re}(r,t) \times \vec{H}_{re}(r,t)$ (A.45)

The time average over one period $ t \in [0,T]$ of the power flux density of a harmonically varying field is therefore given by

$\displaystyle \frac{1}{T}\int_0^T \vec{S} dt = \frac{1}{T}\int_0^T \Re \{\vec{E} \} \times\Re\{\vec{H}\} dt \frac{1}{2\eta}\vert\vec{E}_0\vert^2\vec{e}_p$ (A.46)


next up previous contents
Next: How to do a Up: From the Maxwell Equations Previous: Electromagnetic Waves   Contents