- Demonstrate that the phase
velocity of traveling waves on an infinitely long beaded string is
- 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 (9.27) and (9.29) for propagating electromagnetic waves in a plasma from Equations (9.19) and (9.23)–(9.26).
- Derive expressions (9.39) and (9.40) for evanescent electromagnetic
waves in a plasma from Equations (9.24), (9.25), and (9.36)–(9.38).
- Derive Equations (9.49)–(9.52) from Equations (9.47) and
(9.48).
- Derive Equations (9.60) and (9.61) from Equations (9.54)–(9.59).
- Derive Equations (9.207) and (9.208) from Equations (9.203)–(9.206).
- Derive Equations (9.216)–(9.219) from Equations (9.214) and
(9.215), in the limit
.
- A medium is such that the product of the phase and group
velocities of electromagnetic waves is equal to at all wave
frequencies, where is the velocity of light in vacuum. Demonstrate that the dispersion relation for
electromagnetic waves takes the form
- 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 - Show that the general eigenmode equation, (9.168), yields the
following dispersion relation for electromagnetic waves propagating through a
magnetized plasma:
- Show that the solution to the dispersion relation derived in the previous
exercise can be written
- Show that the dispersion relation derived in Exercise 11 can also
be written in the form
- Show that the cutoff frequencies for an electromagnetic wave propagating in a
general direction through a magnetized plasma occur when
- Show that the resonant frequencies for an electromagnetic wave propagating in a
general direction through a magnetized plasma occur when

- Show that the cutoff frequencies for an electromagnetic wave propagating in a
general direction through a magnetized plasma occur when
- Consider an electromagnetic wave propagating in the positive -direction
through a conducting medium of conductivity . Suppose that the
wave electric field is
- The aluminum foil used in cooking has an electrical conductivity
,
and a typical thickness
(Wikipedia contributors 2018). Show that such foil can be used to shield a region from electromagnetic
waves of a given frequency, provided that the skin-depth of the waves in the foil is less than about a third of its thickness.
Because skin-depth 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, vacuum-filled, rectangular waveguide that runs parallel to the -axis, and has perfectly conducting 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
- Consider a hollow, vacuum-filled, rectangular waveguide that runs parallel to the -axis, and has perfectly conducting 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
- Deduce that, for a hollow, vacuum-filled, rectangular waveguide, the mode with the lowest cutoff frequency is
a TE mode.
- Consider a vacuum-filled 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 denotes path-length. Hence, deduce that the pulse travels in the arc of a circle, of radius
- 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
*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 right-angles 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 non-propagating 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
non-propagating 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
- Derive the dispersion relation (9.328), and
show that it generalizes to
- 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
- 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 9.14 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 9.6), 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).]