Transcription of 9. Spherical Harmonics
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9. Spherical Harmonics
ORTHOGONAL FUNCTIONS249. Spherical HarmonicsNow we come to some of the most ubiquitous functions in geophysics,used in gravity,geomagnetism and Harmonics are the Fourier series for functions can are used to build solutions to Laplace sequation andother differential equations in a Spherical shall treat Spherical Harmonics as eigensolutions of the surface would be like developing Fourier series as eigensolutions of the operator (d/dx)2on a finite line,but with boundary conditions thatyanddy/dxmatchatthe get some mileage from representing a thing in two ways, onewithin a fixed coordinate system, the other in coordinate-free we need aspherical polar coordinate system: see the originOis alwaysfixed to bethe center of the unit sphere,and all coordinates are referred to that usdefine asurface gra
9. Spherical Harmonics Now we come to some of the most ubiquitous functions in geophysics,used in gravity, geomagnetism and seismology.Spherical harmonics are the Fourier series for the sphere.These functions can are used to build solutions to Laplace’sequation and other differential equations in a spherical setting.
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