** Next:** Traveling Waves
** Up:** Longitudinal Standing Waves
** Previous:** Fourier Analysis

- Estimate the highest possible frequency (in hertz), and the smallest possible wavelength, of a longitudinal wave in a thin Aluminium rod, due to the discrete atomic structure of this material. The mass density, Young's modulus, and atomic weight
of Aluminium are
,
, and
, respectively. (Assume, for the sake of simplicity, that Aluminium has a simple cubic lattice.)

- A simple model of an ionic crystal consists of a linear array of a great many equally spaced atoms of alternating masses
and
, where
. The masses are connected by identical chemical bonds that are
modeled as springs of spring constant
. Show that the frequencies of the system's longitudinal modes of vibration
either lie in the band 0
to
or in the band
to
.
Show that, in the long wavelength limit, modes whose frequencies lie in the lower band are such that neighboring
atomics move in the same direction, whereas modes whose frequencies lie in the upper band are such that
neighboring atoms move in opposite directions. The lower band is known as the
*acoustic branch*, whereas
the upper band is known as the *optical branch*.

- Consider a linear array of
identical simple pendula of mass
and
length
that are suspended from equal height points, evenly
spaced a distance
apart. Suppose that each pendulum bob is attached to
its two immediate neighbors by means of light springs of unstretched
length
and spring constant
. The following figure shows a small part of
such an array. Let
be the equilibrium position of the
th bob, for
, and let
be its horizontal displacement. It is assumed that
for
all
. Demonstrate that the equation of motion of the
th pendulum bob is

Consider a general normal mode of the form

Show that the associated dispersion relation is

Suppose that the first and last pendulums in the array are attached to immovable
walls, located a horizontal distance
away, by means of light springs of unstretched length
and spring constant
. Find the normal modes of the system.
Suppose, on the other hand, that the first and last pendulums are not attached to anything on their
outer sides. Find the normal modes of the system.

- Consider a periodic waveform
of period
, where
for
all
, which is represented as a Fourier series,

where
. Demonstrate that

where
and
, and

where

- Demonstrate that the periodic square-wave

where
for all
, has the Fourier representation

Here,
. Plot the reconstructed waveform, retaining the first 4, 8, 16, and
32 terms in the Fourier series.

- Show that the periodically repeated pulse waveform

where
for all
, and
, has the Fourier representation

Demonstrate that if
then the most significant terms in the preceding series have
frequencies (in hertz) that range from the fundamental frequency
to a frequency
of order
.

** Next:** Traveling Waves
** Up:** Longitudinal Standing Waves
** Previous:** Fourier Analysis
Richard Fitzpatrick
2013-04-08