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# Exercises

1. 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?

2. 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 .

3. Derive expressions (797) and (799) for propagating electromagnetic waves in a plasma from Equations (789) and (793)-(796).

4. Derive expressions (809) and (810) for evanescent electromagnetic waves in a plasma from Equations (794), (795), and (806)-(808).

5. Derive Equations (819)-(822) from Equations (817) and (818).

6. Derive Equations (830) and (831) from Equations (824)-(829).

7. Derive Equations (869) and (870) from Equations (865)-(868).

8. Derive Equations (878)-(881) from Equations (876) and (877), in the limit .

9. 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 .

10. 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 skin-depth. 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).]

11. 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 skin-depth of the waves in the foil is less than about a third of its thickness. Since 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?

12. Consider a hollow, vacuum-filled, 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 non-negative integers, one of which must be non-zero.

13. Consider a hollow, vacuum-filled, 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.

14. Deduce that for a hollow, vacuum-filled, rectangular waveguide the mode with the lowest cut-off frequency is a TE mode.

15. 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 cut-off frequency?

16. 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 path-length. Hence, deduce that the pulse travels in the arc of a circle, of radius

whose center lies at .

17. 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.

18. 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.

19. 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.

20. Water fills a rectangular tank of length and breadth to a depth . Show that the resonant frequencies of the water are

where

and , are non-negative integers that are not both zero. Neglect surface tension.

21. Derive the dispersion relation (990), and show that it generalizes to

in water of arbitrary depth.

22. 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.

23. 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 2013-04-08