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- Show that the one-dimensional plane wave, (529), is a solution of the one-dimensional wave equation, (536),
provided that

Likewise, demonstrate that the three-dimensional plane wave, (533), is a solution of the three-dimensional wave
equation, (537), as long as

- Demonstrate that for a cylindrically symmetric wavefunction
, where
, the
three-dimensional wave equation (537)
can be re-written

Show that

is an approximate solution of this equation in the limit
, where
.

- Demonstrate that for a spherically symmetric wavefunction
, where
, the
three-dimensional wave equation (537)
can be re-written

Show that

is a solution of this equation, where
.

- Consider an elastic sheet stretched over a rectangular frame that extends from
to
, and from
to
.
Suppose that

Show that the amplitudes and phase angles in the normal mode expansion (556) are given by

where

- The radial oscillations of an ideal gas in a spherical cavity of radius
are governed by the
spherical wave equation

subject to the boundary condition
.
Here,
is a spherical coordinate,
is the radial displacement, and
is the speed of sound. Show that the general
solution of this equation is written

where
,

and
,
are arbitrary constants.

- Show that a light-ray entering a planar transparent plate of thickness
and refractive index
emerges parallel to
its original direction. Show that the lateral displacement of the ray is

where
and
are the angles of incidence and refraction, respectively, at the front side of the plate.

- Suppose that a light-ray is incident on the front (air/glass) interface of a uniform pane
of glass of refractive index
at the Brewster angle. Demonstrate that the refracted ray
is also incident on the rear (glass/air) interface of the pane at the Brewster
angle. [From Fitzpatrick 2008.]

- Show that the Fresnel relations, (603) and (604), for the polarization in which the magnetic intensities of all three waves are
parallel to the interface can be written

where
represents impedance. (Here,
is the impedance of free space, and
the refractive index.)
Demonstrate that the Fresnel relations, (625) and (626), for the other polarization take the form

- Show that the expressions, (695) and (696), for the coefficients of reflection and transmission for a sound wave obliquely incident at
an interface between two immiscible fluids can be written

where
and
are the *acoustic impedances* of the two fluids.
Show that the expression, (697), for the angle of intromission can be written

** Next:** Wave Pulses
** Up:** Multi-Dimensional Waves
** Previous:** Sound Waves in Fluids
Richard Fitzpatrick
2013-04-08