Problem 2.6a
Show that the wave equation (2.5a) can be written in cylindrical coordinates (see Figure 2.6a) as
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(2.6a)
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Solution
Figure 2.6a. Cylindrical coordinates.
We shall solve by direct substitution. We have
,
,
, and
,
. The following solution is lengthy, so we use subscripts to denote partial derivatives and write
We shall require the derivatives:
To replace
and
with derivatives with respect to
and
, we write:
Then,
Thus
so
Problem 2.6b
Transform the wave equation into spherical coordinates (see Figure 2.6b), showing that it becomes
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(2.6b)
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Solution
Spherical coordinates
and rectangular coordinates are related as follows (see Figure 2.6b):
We continue to use subscripts to denote derivatives and letters to denote sines and cosines:
The derivatives of
,
, and
now become:
Figure 2.6b Spherical coordinates.
Summarizing these results, we have
We now calculate the derivatives
, etc.:
Adding the three derivatives, we get
Substituting the values of
,
,
, and
, we get for the wave equation
This is often written in the more compact form
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Also in this chapter
External links
find literature about Wave equation in cylindrical and spherical coordinates
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