1. §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29

2. Outline Separation of variables in different coordinate systems Cartesian, cylindrical, and spherical coordinates Boundary conditions: external and internal Plane wave functions in different coordinates Linear waves: Circular harmonics (sin, cos, exp) (x,y,z) Azimuthal waves: Cylindrical (sectoral) harmonics (φ) Polar waves: Legendre poly/fns: zonal harmonics (θ) Angular waves: Spherical (tesseral) harmonics (θ,φ) Radial waves: 2d Bessel (s), 3d spherical Bessel (r) Laplacian: planar (s,φ), solid harmonics (r,θ,φ) Putting it all together General solutions to Laplace equation 2

3. Helmholtz equation: free wave k 2 = curvature of wave; k 2 =0 [Laplacian] 3

4. Review: external boundary conditions Uniqueness theorem – difference between any two solutions of Poisson’s equation is determined by values on the boundary External boundary conditions: 4

5. Internal boundary conditions Possible singularities (charge, current) on the interface between two materials Boundary conditions “sew” together solutions on either side of the boundary External: 1 condition on each side Internal: 2 interconnected conditions General prescription to derive any boundary condition: 5

6. Linear wave functions – exponentials 6

7. Circular waves – Bessel functions 7

8. Polar waves – Legendre functions 8

9. Angular waves – spherical harmonics 9

10. Radial waves – spherical Bessel fn’s 10

11. Solid harmonics 11

12. General solutions to Laplace eq’n or: All I really need to know I learned in PHY311 Cartesian coordinates – no general boundary conditions! Cylindrical coordinates – azimuthal continuity Spherical coordinates – azimuthal and polar continuity 12