- Write the traveling wave
as a
superposition of two standing waves. Write the standing wave
as a superposition of two traveling waves propagating in
opposite directions. Show that the following superposition of
traveling waves,
- Show that the solution of the wave equation,

for , can be written*d'Alembert solution*. - Demonstrate that for a transverse traveling wave propagating on a stretched
string,
- A transmission line of characteristic impedance
occupies the region
, and is terminated at
.
Suppose that the current carried by the line takes the form
- Suppose that the transmission line in the previous exercise is short
circuited, such that the line is effectively terminated by a negligible resistance.
Find the relationship between
and
. Show that the current and voltage oscillate
radians out of phase everywhere along the line. Demonstrate that there is zero net flux of
electromagnetic energy along the line.
- A lossy transmission line has a resistance per unit length
,
in addition to an inductance per unit length
, and a capacitance
per unit length
. The resistance can be considered to be in series with the
inductance. Demonstrate that the Telegrapher's equations generalize to

where and are the voltage and current along the line. Derive an energy conservation equation of the form

where , , , and . Show that : that is, the decay length of the signal is much longer than its wavelength. Estimate the maximum useful length of a low resistance, high frequency, lossy transmission line. - Suppose that a transmission line consisting of two uniform parallel
conducting strips of width
and perpendicular distance apart
, where
, is terminated by a strip of material of uniform resistance per square
meter
. Such material is known
as
*spacecloth*. Demonstrate that a signal sent down the line is completely absorbed, with no reflection, by the spacecloth. Incidentally, the resistance of a uniform strip of material is proportional to its length, and inversely proportional to its cross-sectional area. - At normal incidence, the mean radiant power from the Sun illuminating one square meter of the Earth's surface is
kW. Show that the amplitude of the
electric component of solar electromagnetic radiation at the
Earth's surface is
. Demonstrate that the
corresponding amplitude of the magnetic component is
. [From Pain 1999.]
- According to Einstein's famous formula,
, where
is energy,
is mass, and
is the velocity of light in vacuum. This formula implies that
anything that possesses energy also has an effective mass. Use this idea to show
that an electromagnetic wave of mean intensity (energy per unit time per unit area)
has an associated mean pressure (momentum per unit
time per unit area)
. Hence,
estimate the pressure due to sunlight at the Earth's surface (assuming that the sunlight is completely absorbed).
- A glass lens is coated with a non-reflecting coating of thickness
one quarter of a wavelength (in the coating) of light whose
wavelength in air is
. The index of refraction of the glass is
, and that
of the coating is
. The refractive index of air can be taken to be unity. Show that the coefficient of reflection for light normally incident on the lens from air is
- A glass lens is coated with a non-reflective coating whose thickness is one quarter of
a wavelength (in the coating) of light whose frequency is
. Demonstrate that the
coating also suppresses reflection from light whose frequency is
,
, et cetera, assuming that the
refractive index of the coating and the glass is frequency independent.
- A plane electromagnetic wave, linearly polarized in the
-direction, and propagating in the
-direction through an electrical
conducting medium of conductivity
, is governed by

where and are the electric and magnetic components of the wave. (See Appendix C.) Derive an energy conservation equation of the form

where , , and . Demonstrate that : that is, the wave penetrates many wavelengths into the medium. Estimate how far a high frequency electromagnetic wave penetrates into a low conductivity conducting medium. - Sound waves travel horizontally from a source to a receiver. Let the source have the speed
, and the receiver the speed
(in the same
direction). In addition, suppose that a wind of speed
(in the same direction) is blowing from the source to the receiver. Show that if the source
emits sound whose frequency is
in still air then the frequency recorded by the
receiver is