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- Verify Equations (701)-(703). Derive Equations (704) and (705) from Equation (699) and
Equations (701)-(703).
- Suppose that

Demonstrate that

where
is the square-root of minus one, and
. [Hint: You
will need to complete the square of the exponent of
, transform the variable of
integration, and then make use of the standard result that
.] Hence, show from *Euler's theorem*,
, that

- Demonstrate that

- Verify Equations (728) and (729).
- Derive Equations (708) and (709) directly from Equation (707) using the results (727)-(729).
- Verify directly that Equation (746) is a solution of the wave equation, (730),
for arbitrary pulse shapes
and
.
- Verify Equation (750).

- Consider a function
that is zero for negative
, and takes the
value
for
. Find its Fourier transforms,
and
, defined in

[Hint: Use Euler's theorem.]

- Let
be zero, except in the interval from
to
. Suppose that in this
interval
makes exactly one sinusoidal oscillation at the angular frequency
, starting and ending with the value zero. Find the previously defined Fourier transforms
and
.

- Demonstrate that

where the relation between
,
, and
is as previously defined.
This result is known as *Parseval's theorem*.

- Suppose that
and
are both even functions of
with the cosine
transforms
and
, so that

Let
, and let
be the cosine transform of
this even function, so that

Demonstrate that

This result is known as the *convolution theorem*, since the above type of
integral is known as a convolution integral. Suppose that
.
Show that

- Demonstrate that

where
is an arbitrary function, and
a constant unit vector, is a solution of the three-dimensional wave equation, (537).
How would you interpret this solution?
- Demonstrate that

where
is an arbitrary function,
is a solution of the spherical wave equation, (540).
How would you interpret this solution?

** Next:** Dispersive Waves
** Up:** Wave Pulses
** Previous:** Bandwidth
Richard Fitzpatrick
2013-04-08