§3.3.2 Separation of spherical variables: zonal harmonics

Christopher CrawfordPHY 416

2014-10-29

Outline• Separation of variables in different coordinate systems

Cartesian, cylindrical, and spherical coordinatesBoundary 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 togetherGeneral solutions to Laplace equation

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

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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:

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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:

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Linear wave functions – exponentials

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Circular waves – Bessel functions

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Polar waves – Legendre functions

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Angular waves – spherical harmonics

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Radial waves – spherical Bessel fn’s

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Solid harmonics

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General solutions to Laplace eq’nor: 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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