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Next: Collisional Damping Up: Wave Propagation Through Inhomogeneous Previous: Resonances


Resonant Layers

Consider the situation, studied in the previous section, in which a plane wave, polarized in the $ y$ -direction, is launched along the $ z$ -axis, from an antenna located at large positive $ z$ , and absorbed at a resonance located at $ z=0$ . In the vicinity of the resonant point, the electric component of the wave satisfies

$\displaystyle \frac{d^2 E_y}{dz^2} + \frac{k_0^2\,b}{z+{\rm i}\,\epsilon} E_y = 0,$ (6.55)

where $ b>0$ and $ \epsilon<0$ .

The time-averaged Poynting flux in the $ z$ -direction is written

$\displaystyle P_z = - \frac{(E_y\,B_x^{\,\ast} + E_y^{\,\ast}\,B_x)}{4 \,\mu_0}.$ (6.56)

Now, the Faraday-Maxwell equation yields

$\displaystyle {\rm i}\,\omega\,B_x = -\frac{d E_y}{dz}.$ (6.57)

Thus, we have

$\displaystyle P_z = -\frac{{\rm i}}{4\,\mu_0\,\omega} \left(\frac{d E_y}{dz}\, E_y^{\,\ast} - \frac{d E_y^{\,\ast}}{dz} \,E_y\right).$ (6.58)

Let us ascribe any variation of $ P_z$ with $ z$ to the wave energy emitted by the plasma. We then obtain

$\displaystyle \frac{d P_z}{dz} = W,$ (6.59)

where $ W$ is the power emitted by the plasma per unit volume. It follows that

$\displaystyle W = -\frac{{\rm i}}{4\,\mu_0\,\omega}\left(\frac{d^2 E_y}{dz^{\,2}}\,E_y^{\,\ast} - \frac{d^2 E_y^{\,\ast}}{dz^{\,2}}\,E_y\right).$ (6.60)

Equations (6.55) and (6.60) yield

$\displaystyle W = \left(\frac{k_0^{\,2}\,b}{2\,\mu_0\,\omega}\right)\left( \frac{\epsilon}{z^{\,2} + \epsilon^{\,2}}\right)\vert E_y\vert^{\,2}.$ (6.61)

Note that $ W<0$ , because $ \epsilon<0$ , so wave energy is absorbed by the plasma. It is clear, from the previous formula, that the absorption takes place in a narrow layer, of thickness $ \vert\epsilon\vert$ , centered on the resonance point, $ z=0$ .


next up previous
Next: Collisional Damping Up: Wave Propagation Through Inhomogeneous Previous: Resonances
Richard Fitzpatrick 2016-01-23