The scale factors are
,
and the separation functions are
,
,
, given a Stäckel
determinant of
. The Laplacian is
 |
(1)
|
Attempt separation of variables by writing
 |
(2)
|
then the Helmholtz differential equation
becomes
![1/(u^2+v^2)[VTheta(1/u(dU)/(du)+(d^2U)/(du^2))+UTheta(1/v(dV)/(dv)+(d^2V)/(dv^2))]
+(UV)/(u^2v^2)(d^2Theta)/(dtheta^2)+k^2UVTheta=0.](/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation3.svg) |
(3)
|
Now multiply through by
,
![(u^2v^2)/(u^2+v^2)[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))]
+1/Theta(d^2Theta)/(dtheta^2)+k^2u^2v^2=0.](/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation4.svg) |
(4)
|
Separating the
part gives
 |
(5)
|
which has solution
 |
(6)
|
Plugging (5) back into (4) and multiplying by
gives
![[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))]
-m^2(u^2+v^2)/(u^2v^2)+k^2(u^2+v^2).](/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation7.svg) |
(7)
|
Rewriting,
![[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))]
-m^2(1/(v^2)+1/(u^2))+k^2(u^2+v^2).](/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation8.svg) |
(8)
|
This can be rearranged into two terms, each containing only
or
,
![[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+k^2u^2-(m^2)/(u^2)]
+[1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))+k^2v^2-(m^2)/(v^2)]](/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation9.svg) |
(9)
|
and so can be separated by letting the first part equal
and the second equal
, giving
 |
(10)
|
 |
(11)
|
See also
Helmholtz Differential
Equation,
Parabolic Coordinates
Explore with Wolfram|Alpha
References
Arfken, G. "Parabolic Coordinates
." §2.12 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-111,
1970.Moon, P. and Spencer, D. E. Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, p. 36, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York McGraw-Hill, pp. 514-515 and
660, 1953.
Cite this as:
Weisstein, Eric W. "Helmholtz Differential Equation--Parabolic Coordinates." From MathWorld--A Wolfram
Web Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationParabolicCoordinates.html
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