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


(2.6a)

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


(2.6b)

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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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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