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Helmholtz Differential Equation--Polar Coordinates

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In two-dimensional polar coordinates, the Helmholtz differential equation is

1/rpartial/(partialr)(r(partialF)/(partialr))+1/(r^2)(partial^2F)/(partialtheta^2)+k^2F=0.
(1)

Attempt separation of variables by writing

F(r,theta)=R(r)Theta(theta),
(2)

then the Helmholtz differential equation becomes

(d^2R)/(dr^2)Theta+1/r(dR)/(dr)Theta+1/(r^2)(d^2Theta)/(dtheta^2)R+k^2RTheta=0.
(3)

Multiply both sides by r^2/(RTheta) to obtain

((r^2)/R(d^2R)/(dr^2)+r/R(dR)/(dr)+k^2r^2)+(1/Theta(d^2Theta)/(dtheta^2))=0.
(4)

The solution to the second part of (4) must be periodic, so the differential equation is

(d^2Theta)/(dtheta^2)1/Theta=-m^2,
(5)

which has solutions

Theta(theta)=A_mcos(mtheta)+B_msin(mtheta).
(6)

Plug (5) back into (4)

((r^2)/R(d^2R)/(dr^2)+r/R(dR)/(dr)+k^2r^2)=m^2.
(7)

This has solution

R(r)=C_mJ_m(kr)+D_mY_m(kr),
(8)

where J_m(x) and Y_m(x) are Bessel functions of the first and second kinds, respectively. Combining the solutions gives the general solution

F(r,theta)=sum_(m=0)^infty[A_mcos(mtheta)+B_msin(mtheta)]×[C_mJ_m(kr)+D_mY(kr)].
(9)

See also

Helmholtz Differential Equation--Circular Cylindrical Coordinates

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References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York McGraw-Hill, pp. 502-504, 1953.

Cite this as:

Weisstein, Eric W. "Helmholtz Differential Equation--Polar Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationPolarCoordinates.html

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