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Helmholtz Differential Equation--Elliptic Cylindrical Coordinates

In elliptic cylindrical coordinates, the scale factors are h_u=h_v=sqrt(sinh^2u+sin^2v), h_z=1, and the separation functions are f_1(u)=f_2(v)=f_3(z)=1, giving a Stäckel determinant of S=(sin^2v+sinh^2u). The Helmholtz differential equation is

1/(sinh^2u+sin^2v)((partial^2F)/(partialu^2)+(partial^2F)/(partialv^2))+(partial^2F)/(partialz^2)+k^2F=0.
(1)

Attempt separation of variables by writing

F(u,v,z)=U(u)V(v)Z(z),
(2)

then the Helmholtz differential equation becomes

Z/(sinh^2u+sin^2v)(V(d^2U)/(du^2)+U(d^2V)/(dv^2))+UV(d^2Z)/(dz^2)+k^2UVZ=0.
(3)

Now divide by UVZ to give

1/(sinh^2u+sin^2v)(1/U(d^2U)/(du^2)+1/V(d^2V)/(dv^2))+1/Z(d^2Z)/(dz^2)+k^2=0.
(4)

Separating the Z part,

1/Z(d^2Z)/(dz^2)=-(k^2+m^2) 1/(sinh^2u+sin^2v)(1/U(d^2U)/(du^2)+1/V(d^2V)/(dv^2))=m^2,
(5)

so

(d^2Z)/(dz^2)=-(k^2+m^2)Z,
(6)

which has the solution

Z(z)=A_(km)cos(sqrt(k^2+m^2)z)+B_(km)sin(sqrt(k^2+m^2)z).
(7)

Rewriting (◇) gives

(1/U(d^2U)/(du^2)-m^2sinh^2u)+(1/V(d^2V)/(dv^2)-m^2sin^2v)=0,
(8)

which can be separated into

1/U(d^2U)/(du^2)-m^2sinh^2u=c
(9)
c+1/V(d^2V)/(dv^2)-m^2sin^2v=0,
(10)

so

(d^2U)/(du^2)-(c+m^2sinh^2u)U=0
(11)
(d^2V)/(dv^2)+(c-m^2sin^2v)V=0.
(12)

Now use

sinh^2u=1/2[cosh(2u)-1]
(13)
sin^2v=1/2[1-cos(2v)]
(14)

to obtain

(d^2U)/(du^2)-{c+1/2m^2[cosh(2u)-1]}U=0
(15)
(d^2V)/(dv^2)+{c-1/2m^2[1-cos(2v)]}V=0.
(16)

Regrouping gives

(d^2U)/(du^2)-[(c-1/2m^2)+1/2m^2cosh(2u)]U=0
(17)
(d^2V)/(dv^2)+[(c-1/2m^2)+1/2m^2cos(2v)]V=0.
(18)

Let a=c-m^2/2 and q=-m^2/4, then these become

(d^2V)/(dv^2)+[a-2qcos(2v)]V=0
(19)
(d^2U)/(du^2)-[a-2qcosh(2u)]U=0.
(20)

Here, (19) is the mathieu differential equation and (20) is the modified mathieu differential equation. These solutions are known as mathieu functions.

See also

Elliptic Cylindrical Coordinates, Helmholtz Differential Equation, Mathieu Differential Equation, Mathieu Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Mathieu Functions." Ch. 20 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 721-746, 1972.Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 17-19, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 657, 1953.

Referenced on Wolfram|Alpha

Helmholtz Differential Equation--Elliptic Cylindrical Coordinates

Cite this as:

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

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