The propagation of waves in an inhomogeneous medium

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The propagation of waves in an inhomogeneous medium LL 8 sec 88

The propagation of waves in an inhomogeneous medium LL 8 sec 88

Consider a medium that is electrically inhomogeneous but isotropic • The ionosphere Maxwell’s Equations

Consider a medium that is electrically inhomogeneous but isotropic • The ionosphere Maxwell’s Equations Permittivity changes with position, is a function of the coordinates

Eliminate H Diff. Eq. for E

Eliminate H Diff. Eq. for E

Instead, eliminate E zero Diff. Eq. for H

Instead, eliminate E zero Diff. Eq. for H

Special case where permittivity changes in only one direction, call it z. Orient coordinates

Special case where permittivity changes in only one direction, call it z. Orient coordinates so that propagation is in the x-z plane. All quantities are then independent of y.

Translational invariance in x means that the xdependence of the wave is given by

Translational invariance in x means that the xdependence of the wave is given by exp[ikx] constant For k = 0, the field depends only on z. The wave passes normally through a layer in which e = e(z). For k non-zero, the wave passes obliquely.

First case: “E-waves” Ionosphere has equal numbers of ions and electrons

First case: “E-waves” Ionosphere has equal numbers of ions and electrons

Second case: “H-waves” Equation for “H-waves”

Second case: “H-waves” Equation for “H-waves”

E-wave equation

E-wave equation

In geometrical optics, the wavelength should not change too fast along the ray of

In geometrical optics, the wavelength should not change too fast along the ray of propagation This means that the wavelength hardly changes over distances of order itself. (Equality would mean that wavelength changes by its full value over distances equal to itself, Dl = l when Dz = l)

You cannot see this by simply substituting the solution into the differential equation! The

You cannot see this by simply substituting the solution into the differential equation! The proof is way more difficult! To see it, we go to quantum mechanics, where we have a similar problem!

E-waves Quantum particles 1 D permittivity variation 1 D potential These are the same

E-waves Quantum particles 1 D permittivity variation 1 D potential These are the same equation, just with different constants. The solution of the quantum problem when the de Broglie wavelength is small compared to the characteristic distance for a change in U(z) is known: It’s the “Quasi-classical case” or “WKB approximation”.

E-waves

E-waves

Solution in the quasi-classical case for quantum particles (HW)

Solution in the quasi-classical case for quantum particles (HW)

Conditions of validity for solution Quasi-classical quantum particles Not satisfied near turning point, where

Conditions of validity for solution Quasi-classical quantum particles Not satisfied near turning point, where E = U(z) E-waves Not satisfied at reflection point, where

A function of z. Amplitude decreases as f increases away from reflection point.

A function of z. Amplitude decreases as f increases away from reflection point.

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