How to Render Spherical Harmonics

The different possible configurations of a vibrating, resonant sphere are described by spherical harmonics.

Spherical harmonics represent a set of solutions for Laplace's Equation described in spherical coordinates. Conceptually, this equates to a description of the different vibrational configurations capable of resonance (referred to as "normal modes" or "eigenmodes") within a spherical, three-dimensional system. Many different science-oriented software packages contain modules for rendering spherical harmonics. Wolfram Research, however, allows users to render spherical harmonics through a Mathematica-powered web applet with no required purchase of Wolfram software.

Instructions

Using Wolfram Online Function Plotting

1. Navigate to the Spherical harmonic function page on The Wolfram Functions Site (see Resources).

2. Specify values for l, m, theta and phi. Note that l and m are indices that define which spherical mode will be displayed, while theta and phi are coordinates that specify how that mode will be plotted.

3. Select the type of plot desired. In general, higher dimension plots and contour plots will provide a more complete description of the operation of the spherical harmonic you are examining, but may include superfluous information depending on your goal.

4. Specify how complex numbers are to be incorporated into the answer. For many problems, ignoring the imaginary component of complex numbers is acceptable. However, some research and engineering applications will require that all components of complex numbers being included.

5. Specify boundary conditions for theta and phi. Multiples of Pi are often intuitive as these represent integer periods of the harmonic's behavior. Click "Evaluate" at the bottom left of the screen. A separate window will appear with your plot.

Using Wolfram Mathematica

1. Open Wolfram Mathematica on your computer.

2. Represent the function using the following notation:

   SphericalHarmonicY[l,m,theta,phi]

   Note that l and m are indices that define which spherical mode will be displayed, while theta and phi are coordinates that specify how that mode will be plotted.

3. Render as desired using "Plot," "Plot3D" or another Mathematica plotting command of your choice.

Tips & Warnings

• While the Wolfram online interface does not contain a "Print" command, any plots created can be printed through browser commands as desired. Alternatively, plots can also be downloaded as image files by right-clicking on them and selecting "Save Image As..." (or local browser equivalent).

References

• Wolfram Research: Spherical harmonic function
• Wolfram MathWorld: Spherical Harmonic

Resources

• The Wolfram Functions Site