Wave equation in cylindrical and spherical coordinates

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Problem 2.6a

Show that the wave equation (2.5a) can be written in cylindrical coordinates (see Figure 2.6a) as



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:




Problem 2.6b

Transform the wave equation into spherical coordinates (see Figure 2.6b), showing that it becomes



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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General solutions of the wave equation Sum of waves of different frequencies and group velocity
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Wave equation in cylindrical and spherical coordinates
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