# PHYS20171 MATHEMATICS OF WAVES AND FIELDS

## Web Resources

An extremely useful resource is the HELM project which is available from the School of Mathematics. (Follow the HELM link in the menu on the left. You will need to login using your university username and password to access HELM.) It provides a set of workbooks covering a wide range of mathematical topics. The most relevant for this module are Sections 23, 24 and 25. Note that in some areas these workbooks provide more detail than is covered in this module, whereas in others this module goes into greater depth.

Fourier Series

Fourier Transforms and Convolution

Waves

Applets and movies

Many of these links are taken directly from Mike Birse's course page.

### Spherical Harmonics

• Figure showing the real spherical harmonics Yl,m for l=0 to 4 (top to bottom) and m=0 to 4 (left to right). The negative order harmonics Yl,-m are rotated about the z axis by 90o/m.
• Schematic representation of Y_{\ell m} on the unit sphere and its nodal lines. Y_{\ell m} is equal to 0 along m great circles passing through the poles, and along \ell-m circles of equal latitude. The function changes sign each time it crosses one of these lines.
• 3D color plot of the spherical harmonics of degree n = 5
• A java applet which displays the wave functions of the rigid rotor (rigid rotator), which are the spherical harmonics.
• Wikipedia entry (source of the figures above)
• Video showing the 3D standing waves in a sphere
• Video waves in a sphere of water. An experiment from the International Space Station.

Figures, Images and Diagrams Used in Lecture

### Examples of applications

• An image of the temperature fluctuations in the cosmic microwave background from the WMAP mission after 5 years of observations.

This image has been decomposed using spherical harmonics to produce a power spectrum to show the temperature fluctuations as a function of angular size scale. The image below shows this decomposition. The points are the measured values and the curve the predictions of a model.

(These images are from the WMAP website.)
• The science of cooking: