Wave equation in cylindrical and spherical coordinates

From SEG Wiki
Jump to navigation Jump to search

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

Continue reading

Previous section Next section
General solutions of the wave equation Sum of waves of different frequencies and group velocity
Previous chapter Next chapter
Introduction Partitioning at an interface

Table of Contents (book)

Also in this chapter

External links

find literature about
Wave equation in cylindrical and spherical coordinates
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png