Helmholtz Differential Equation—Circular Cylindrical Coordinates

Author

Eric W. Weisstein
November 16, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/HelmholtzDifferentialEquationCircularCylindricalCoordinates.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HelmholtzDifferentialEquationCircularCylindricalCoordinates.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Solution

Angular

DSolve[(1/(Θ[θ]))Θ''[θ]==-(m^2),Θ[θ],θ,GeneratedParameters->(Subscript[C, θ])]

Axial

DSolve[(1/(Z[z]))Z''[z]==n^2,Z[z],z,GeneratedParameters->(Subscript[C, z])]

Radial

DSolve[R''[r]+(1/r)R'[r]+((n^2)+(k^2)-((m^2)/(r^2)))R[r]==0,R[r],r,GeneratedParameters->(Subscript[C, r])]

Simplify[%,k>0&&n>0]

Check

With[{f=(Cos[m θ] (Subscript[C, θ])[1]+Sin[m θ] (Subscript[C, θ])[2])((E^(n z)) (Subscript[C, z])[1]+(E^(-n z)) (Subscript[C, z])[2])(BesselJ[m,(Sqrt[(k^2)+(n^2)]) r] (Subscript[C, r])[1]+BesselY[m,(Sqrt[(k^2)+(n^2)]) r] (Subscript[C, r])[2])},
(1/r)(∂_r)(r (∂_r)f)+(1/(r^2))(∂_(θ,θ))f+(∂_(z,z))f+(k^2)f
]//FullSimplify