A plane-stratified ionospheric layer has electron density depending only on height

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Laboratory, industrial and space plasmas

Plasma Physics, Ch Waves in a non-uniform plasma: rays, reflection, mode conversion and radiofrequency heating

Problem

A plane-stratified ionospheric layer has electron density depending only on height zz, with peak plasma frequency fpe,max⁡=8.0 MHzf_{pe,\max} = 8.0\,\mathrm{MHz}. An HF radio wave of frequency f0=10.0 MHzf_0 = 10.0\,\mathrm{MHz} is launched from the ground at an angle θ0=50∘\theta_0 = 50^\circ from the vertical. ((a)) Starting from Hamilton's equations x˙=∂ω/∂k\dot{\mathbf{x}} = \partial\omega/\partial\mathbf{k}, k˙=−∂ω/∂x\dot{\mathbf{k}} = -\partial\omega/\partial\mathbf{x} with ω\omega as Hamiltonian, show that the horizontal wavenumber kyk_y is conserved along the ray, and hence that the wave turns back (its vertical wavenumber vanishes) at the height where the local plasma frequency satisfies fpe(zt)=f0cos⁡θ0f_{pe}(z_t) = f_0\cos\theta_0. ((b)) Determine whether this wave returns to the ground, and find the highest frequency that would return to the ground at this launch angle from the same layer.

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