29:106/186 RADIO ASTRONOMY
Second Homework Set...February 3, 2000
Due: February 9, 2000
(1) Show by graphical means that the superposition of several sinusoidal
signals with slightly different periods produces a temporally-modulated
sinusoidal signal. What is the relationship between the spread in period
and the modulation scale?
(2) Show that if one includes the quadratic term in a Taylor Series expansion
of a dispersion relationship, a wave packet will spread out with time as it
propagates.
(3) Here's a long one. Get a cup of coffee then read it through! The dispersion relation for a certain class of waves is
The direction of propagation is z. The + sign is for radio waves polarized in the
x direction, and the - sign is for waves polarized in the y direction. You may
assume that 
for all frequencies
of interest.
A signal consisting of both x and y polarized waves is transmitted at z=0. The
x and y polarized waves are in phase at transmission. What is the phase difference
at distance z=L?
(4) Show that if
The vector 
is a constant.
You may use a specific coordinate system (i.e. Cartesian) in your demonstration.
(5) Given the dispersion relation for radio waves in a plasma,
derive the expression for the group velocity of the radio waves. You may assume that
.