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Helmholtz Differential Equation--Circular Cylindrical Coordinates

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In cylindrical coordinates, the scale factors are h_r=1, h_theta=r, h_z=1, so the Laplacian is given by

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

Attempt separation of variables in the Helmholtz differential equation

del ^2F+k^2F=0
(2)

by writing

F(r,theta,z)=R(r)Theta(theta)Z(z),
(3)

then combining (1) and (2) gives

(d^2R)/(dr^2)ThetaZ+1/r(dR)/(dr)ThetaZ+1/(r^2)(d^2Theta)/(dtheta^2)RZ+(d^2Z)/(dz^2)RTheta+k^2RThetaZ=0.
(4)

Now multiply by r^2/(RThetaZ),

((r^2)/R(d^2R)/(dr^2)+r/R(dR)/(dr))+1/Theta(d^2Theta)/(dtheta^2)+(r^2)/Z(d^2Z)/(dz^2)+k^2r^2=0,
(5)

so the equation has been separated. Since the solution must be periodic in theta from the definition of the circular cylindrical coordinate system, the solution to the second part of (5) must have a negative separation constant

1/Theta(d^2Theta)/(dtheta^2)=-m^2,
(6)

which has a solution

Theta(theta)=C_mcos(mtheta)+D_msin(mtheta).
(7)

Plugging (7) back into (5) gives

(r^2)/R(d^2R)/(dr^2)+r/R(dR)/(dr)-m^2+(r^2)/Z(d^2Z)/(dz^2)+k^2r^2=0,
(8)

and dividing through by r^2 results in

1/R(d^2R)/(dr^2)+1/(rR)(dR)/(dr)-(m^2)/(r^2)+1/Z(d^2Z)/(dz^2)+k^2=0.
(9)

The solution to the second part of (9) must not be sinusoidal at +/-infty for a physical solution, so the differential equation has a positive separation constant

1/Z(d^2Z)/(dz^2)=n^2,
(10)

and the solution is

Z(z)=E_ne^(-nz)+F_ne^(nz).
(11)

Plugging (11) back into (9) and multiplying through by R yields

(d^2R)/(dr^2)+1/r(dR)/(dr)+(n^2+k^2-(m^2)/(r^2))R=0
(12)

But this is just a modified form of the Bessel differential equation, which has a solution

R(r)=A_(mn)J_m(rsqrt(n^2+k^2))+B_(mn)Y_m(rsqrt(n^2+k^2)),
(13)

where J_n(x) and Y_n(x) are Bessel functions of the first and second kinds, respectively. The general solution is therefore

F(r,theta,z)=sum_(m=0)^inftysum_(n=0)^infty[A_(mn)J_m(rsqrt(k^2+n^2))+B_(mn)Y_m(rsqrt(k^2+n^2))] ×[C_mcos(mtheta)+D_msin(mtheta)](E_ne^(-nz)+F_ne^(nz)).
(14)

In the notation of Morse and Feshbach (1953), the separation functions are f_1(r)=r, f_2(theta)=1, f_3(z)=1, so the Stäckel determinant is 1.

The Helmholtz differential equation is also separable in the more general case of k^2 of the form

k^2(r,theta,z)=f(r)+(g(theta))/(r^2)+h(z)+k^('2).
(15)

See also

Cylindrical Coordinates, Helmholtz Differential Equation, Helmholtz Differential Equation--Polar Coordinates

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References

Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 15-17, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 656-657, 1953.

Cite this as:

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

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