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lect_7_31

# lect_7_31 - Coordinate Systems and Separation of Variables...

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Coordinate Systems and Separation of Variables 2 Revisiting the wave equation… 2 ψ + 1 ψ = 0 2 c t 2 where previously in Cartesian coordinates, the Laplacian was given by 2 2 2 2 = + + x 2 y 2 z 2 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.

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Spherical Polar Coordinates ) , , ( θ φ r φ θ r x = r sin θ cos φ y = r sin θ sin φ z = r cos θ 2 r = x + y 2 + z 2 2 θ = tan 1 [ x + y 2 z ] y 1 φ = tan [ y x ] 2 2 1 r 2 1 1 2 2 2 = r r r + r 2 sin θ θ sin θ θ + r 2 sin θ φ
Separation of Variables Objective is to restate problems in alternative orthogonal coordinate systems such that for the particular boundary conditions in force, the solutions can be assumed to separate such that… ( ( ψ ( r , θ , φ , t ) = r R ) Θ ( θ ) Φ ( φ ) t T ) 2 Azimuth dependence: d Φ + m 2 = Φ 0 2 d φ 1 d m 2 sin θ d Θ + n n + 1) sin 2 θ = Θ 0 Elevation dependence: sin θ d θ d θ

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