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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
|
(A.34) |
where is the real part of a complex quantity,
the complex field
amplitude and we choose for the time dependency according to the (IEEE) convention in this treatment.
The Maxwell equations can then be simplified to
|
(A.35) |
|
(A.36) |
|
(A.37) |
|
(A.38) |
In this case the wave equation for a source free, non conductive medium
( and ) can be written as
|
(A.39) |
where
is the wave number. For plane waves we impose the
condition that, in a Cartesian coordinate system, and 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
becomes
|
(A.40) |
and by applying (A.23) and (A.21) we find for the H-field
|
(A.41) |
where
is the intrinsic impedance of the medium,
is a unit vector in propagation (z) direction and and the complex amplitudes.
Similarly we find for a plane wave propagating in an arbitrary direction
|
(A.42) |
|
(A.43) |
where
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 the direction and the magnitude of the
electric field vector
is given by the real part of the complex harmonic field expression
|
(A.44) |
According to Poynting's theorem, the power flux density of the wave equals the direction and
magnitude of the Poynting vector
|
(A.45) |
The time average over one period
of the power flux density of a harmonically varying
field is therefore given by
|
(A.46) |
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