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§3.3. 2 Separation of spherical variables: zonal harmonics

§3.3. 2 Separation of spherical variables: zonal harmonics. Christopher Crawford PHY 416 2014-10-29. Outline. Separation of variables in different coordinate systems Cartesian , cylindrical , and spherical coordinates Boundary conditions: external and internal

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§3.3. 2 Separation of spherical variables: zonal harmonics

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  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 systemsCartesian, cylindrical, and spherical coordinatesBoundary conditions:external and internal • Plane wave functions in different coordinatesLinear 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 togetherGeneral solutions to Laplace equation

  3. Helmholtz equation: free wave • k2 = curvature of wave; k2=0[Laplacian]

  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:

  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:

  6. Linear wave functions – exponentials

  7. Circular waves – Bessel functions

  8. Polar waves – Legendre functions

  9. Angular waves – spherical harmonics

  10. Radial waves – spherical Bessel fn’s

  11. Solid harmonics

  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

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