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Unformatted text preview: Coordinate Systems and Separation of Variables Revisiting the homogeneous wave equation… 1 2 2 2 2 = ∂ ∂ + ∇ t c ψ ψ where previously in Cartesian coordinates, the Laplacian was given by 2 2 2 2 2 2 2 z y x ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ We are now faced with a spherical polar coordinate system, with the motivation that we might employ the separation of variables technique to solve the wave equation where spherical symmetries are involved. Spherical Polar Coordinates θ φ θ φ θ cos sin sin cos sin r z r y r x = = = [ ] [ ] x y z y x z y x r 1 2 2 1 2 2 2 tan tan − − = + = + + = φ θ ) , , ( θ φ r y φ θ r r 2 2 2 2 2 2 2 2 sin 1 sin sin 1 1 φ θ θ θ θ θ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ = ∇ r r r r r r ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ = ∇ r r r r 2 2 2 1 Assuming variations only in r gives… Thus for a vibrating sphere, we can say immediately that the relevant wave equation is given by the following form of the Helmholtz equation in terms of...
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.067 taught by Professor Davidbattle during the Spring '04 term at MIT.
 Spring '04
 DavidBattle
 Spherical Harmonics

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