§3.3. 2 separation of spherical variables: zonal harmonics christopher crawford phy 416 2014-10-29
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§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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