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# Exercises

1. The electric polarization, , in a linear dielectric medium is related to the electric field-strength, , according to

where is the refractive index. Any divergence of the polarization field is associated with a bound charge density

whereas any time variation generates a polarization current whose density is

Consider an electromagnetic wave propagating through a quasi-neutral, linear, dielectric medium. Assuming a common time variation of the wave fields, demonstrate from Maxwell's equations that

where .

2. Consider an electromagnetic wave, polarized in the -direction, that propagates in the -direction through a medium of refractive index . Assuming that

demonstrate that

where .

3. Consider an electromagnetic wave, polarized in the -direction, that propagates in the - plane through a medium of refractive index . Assuming that

demonstrate that

where

and , and .

Show that the WKB solutions take the form

and that the criterion for these solutions to be valid is

4. Consider an electromagnetic wave, polarized in the - -plane, that propagates in the - plane through a medium of refractive index . Assuming that

demonstrate that

where

and , and .

Show that the WKB solutions take the form

and that the criterion for these solutions to be valid is

5. An electromagnetic wave pulse of frequency is launched vertically from ground level, travels upward into the ionosphere, is reflected, and returns to ground level. If is the net travel time of the pulse then the so-called equivalent height of reflection is defined . It follows that is the altitude of the reflection layer calculated on the assumption that the pulse always travels at the velocity of light in vacuum. Let be the ionospheric plasma frequency, where measures altitude above the ground. Neglect collisions and the Earth's magnetic field.
1. Demonstrate that

where .
2. Show that if is a monotonically increasing function of then the previous integral can be inverted to give

or, equivalently,

(Hint: This is a form of Abel inversion. See Budden 1985.)

3. Demonstrate that if

where , , and are positive constants, then for , and

for . Here, is a Gamma function (Abramowitz and Stegun 1965a).

6. Suppose that the refractive index, , of the ionosphere is given by for , and for , where and are positive constants, and the Earth's magnetic field and curvature are both neglected. Here, measures altitude above the Earth's surface.
1. A point transmitter sends up a wave packet at an angle to the vertical. Show that the packet returns to Earth a distance

from the transmitter. Demonstrate that if then for some values of the previous equation is satisfied by three different values of . In other words, wave packets can travel from the transmitter to the receiver via one of three different paths. Show that the critical case corresponds to and .
2. A point radio transmitter emits a pulse of radio waves uniformly in all directions. Show that the pulse first returns to the Earth a distance from the transmitter, provided that .

Next: Magnetohydrodynamic Fluids Up: Wave Propagation Through Inhomogeneous Previous: Ionospheric Radio Wave Propagation
Richard Fitzpatrick 2016-01-23