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Up: Dispersive Waves
Previous: Capillary Waves
 Demonstrate that the phase
velocity of traveling waves on an infinitely long beaded string is
where
,
is the tension in the string,
the
spacing between the beads,
the mass of the beads, and
the wavenumber
of the wave. What is the group velocity?
 A uniform rope of mass per unit length
and length
hangs vertically.
Determine the tension
in the rope as a function of height from the bottom
of the rope. Show that the time required for a transverse wave pulse to
travel from the bottom to the top of the rope is
.
 Derive expressions (797) and (799) for propagating electromagnetic waves in a plasma from Equations (789) and (793)(796).
 Derive expressions (809) and (810) for evanescent electromagnetic
waves in a plasma from Equations (794), (795), and (806)(808).
 Derive Equations (819)(822) from Equations (817) and
(818).
 Derive Equations (830) and (831) from Equations (824)(829).
 Derive Equations (869) and (870) from Equations (865)(868).
 Derive Equations (878)(881) from Equations (876) and
(877), in the limit
.
 The number density of free electrons in the ionosphere,
, as a
function of vertical height,
, is measured by timing how long it takes a radio pulse
launched vertically upward from the ground (
) to return to ground level again, after
reflection by the ionosphere, as a function of the pulse frequency,
.
It is conventional to define the equivalent height,
, of the reflection layer
as the height it would need to have above the ground if the pulse always traveled
at the velocity of light in vacuum. Demonstrate that
where
, and
. Show that if
then
.
 Consider an electromagnetic wave propagating in the positive
direction
through a conducting medium of conductivity
. Suppose that the
wave electric field is
where
is the skindepth. Demonstrate that the mean electromagnetic energy
flux across the plane
matches the mean rate at which electromagnetic
energy is dissipated, per unit area, due to Joule heating in the region
. [The rate
of Joule heating per unit volume is
(Fitzpatrick 2008).]
 The aluminium foil used in cooking has an electrical conductivity
,
and a typical thickness
(Wikipedia contributors 2012). Show that such foil can be used to shield a region from electromagnetic
waves of a given frequency, provided that the skindepth of the waves in the foil is less than about a third of its thickness.
Since skindepth increases as frequency decreases, it follows that the foil can only shield waves whose frequency exceeds a critical
value.
Estimate this critical frequency (in hertz). What is the corresponding wavelength?
 Consider a hollow, vacuumfilled, rectangular waveguide that runs parallel to the
axis, and has perfectly walls located at
and
.
Maxwell's equations for a TE mode (which is characterized by
) are (see Appendix C)
subject to the boundary conditions
at
and
at
. Show that the problem reduces to
solving
subject to the boundary conditions
at
, and
at
. Demonstrate that
the various TE modes satisfy the dispersion relation
where
is the
component of the wavevector,
and
are nonnegative integers, one of which must be nonzero.
 Consider a hollow, vacuumfilled, rectangular waveguide that runs parallel to the
axis, and has perfectly walls located at
and
.
Maxwell's equations for a TM mode (which is characterized by
) are (see Appendix C)
subject to the boundary conditions
at
,
at
, and
at
and
. Show that the problem reduces to
solving
subject to the boundary conditions
at
and
. Demonstrate that
the various TM modes satisfy the dispersion relation
where
is the
component of the wavevector,
and
are positive integers.
 Deduce that for a hollow, vacuumfilled, rectangular waveguide the mode with the lowest cutoff frequency is
a TE mode.
 Consider a vacuumfilled rectangular waveguide of internal dimensions
. What is the frequency
(in MHz) of the lowest frequency TE mode that will propagate along the waveguide without attenuation?
What are the phase and group velocities (expressed as multiples of
) of this mode when its
frequency is
times the cutoff frequency?
 A wave pulse propagates in the

plane through an inhomogeneous medium with the
linear dispersion relation
where
Here,
and
are positive constants.
Show that if
at
then the equations of motion of the pulse can be written
where
denotes pathlength. Hence, deduce that the pulse travels in the arc of a circle, of radius
whose center lies at
.
 The speed of sound in the atmosphere decreases approximately linearly with increasing altitude (at relatively
low altitude) due to an approximately linear decrease in the temperature of the atmosphere with height.
Assuming that the sound speed varies with altitude,
, above the Earth's surface as
where
and
are positive constants, show that sound generated by a source located a height
above the
ground is refracted upward by the atmosphere such that it never reaches ground level at points whose radial distances from the
point lying directly beneath the source exceed the value
This effect is known as acoustic shadowing.
 A low amplitude sinusoidal gravity wave travels through shallow water of gradually decreasing depth
toward the shore. Assuming that the
wave travels at rightangles to the shoreline, show that its wavelength and vertical amplitude vary as
and
, respectively.
 Demonstrate that a small amplitude gravity wave, of angular frequency
and wavenumber
, traveling over the surface of a lake of uniform depth
causes an individual water volume element located at a depth
below the surface to execute a nonpropagating elliptical orbit
whose major and minor axes are horizontal and vertical, respectively. Show that
the variation of the major and minor radii of the orbit with depth is
and
, respectively, where
is a constant. Demonstrate that
the volume elements are moving horizontally in the same direction as the wave
at the top of their orbits, and in the opposite direction at the bottom.
Show that a gravity wave traveling over the surface of a very deep lake causes water volume elements to execute
nonpropagating circular orbits whose radii decrease exponentially with depth.
 Water fills a rectangular tank of length
and breadth
to a depth
.
Show that the resonant frequencies of the water are
where
and
,
are nonnegative integers that are not both zero. Neglect surface tension.
 Derive the dispersion relation (990), and
show that it generalizes to
in water of arbitrary depth.
 Show that in water of uniform depth
the phase velocity of surface waves can only attain a stationary (i.e., maximum or minimum) value
as a function of wavenumber,
, when
where
. Hence, deduce that the phase velocity has just one stationary value (a minimum) for any
depth greater than
, but no stationary values for lesser depths.
 Unlike gravity waves in deep water, whose group velocities are half their phase velocities, the group velocities of capillary waves are
times their phase velocities. Adapt the analysis of
Section 10.9 to investigate the generation of capillary waves by a very small object traveling across the surface of the water at
the constant speed
. Suppose that the unperturbed surface corresponds to the

plane.
Let the object travel in the minus
direction, such that it is instantaneously found at the origin. Find the present
position of waves that were emitted with wavefronts traveling at an angle
to the object's direction of motion (see Figure 57), when it was located at
,
.
Show that along a given interference maximum the quantities
and
vary in such a manner that
takes a constant value,
(say). Deduce that the interference maximum is given parametrically by the equations
Sketch the pattern of capillary waves generated by the object. [Modified from Lighthill (1978).]
Next: Wave Optics
Up: Dispersive Waves
Previous: Capillary Waves
Richard Fitzpatrick
20130408