Helmholtz Differential Equation

A Partial Differential Equation which can be written in a Scalar version

$$\nabla^2\psi+k^2\psi = 0,$$ (1)

or Vector form,

$$\nabla^2{\bf A}+k^2{\bf A} = 0,$$ (2)

where $\nabla^2$ is the Laplacian. When $k=0$, the Helmholtz differential equation reduces to Laplace's Equation. When $k^2<0$, the equation becomes the space part of the diffusion equation.

The Helmholtz differential equation can be solved by Separation of Variables in only 11 coordinate systems, 10 of which (with the exception of Confocal Paraboloidal Coordinates) are particular cases of the Confocal Ellipsoidal system: Cartesian, Confocal Ellipsoidal, Confocal Paraboloidal, Conical, Cylindrical, Elliptic Cylindrical, Oblate Spheroidal, Paraboloidal, Parabolic Cylindrical, Prolate Spheroidal, and Spherical Coordinates (Eisenhart 1934). Laplace's Equation (the Helmholtz differential equation with $k=0$) is separable in the two additional Bispherical Coordinates and Toroidal Coordinates.

If Helmholtz's equation is separable in a 3-D coordinate system, then Morse and Feshbach (1953, pp. 509-510) show that

$${h_1h_2h_3\over{h_n}^2}=f_n(u_n)g_n(u_i,u_j),$$ (3)

where $i\not=j\not=n$. The Laplacian is therefore of the form

$\nabla^2 = {1\over h_1h_2h_3} \left\{{g_1(u_2,u_3){\partial\over\partial u_1} \left[{f_1(u_1){\partial\over\partial u_1}}\right]}\right.$
$+g_2(u_1,u_3){\partial\over\partial u_2} \left[{f_2(u_2){\partial\over\partial u_2}}\right]$
$\left.{+g_3(u_1,u_2){\partial\over\partial u_3} \left[{f_3(u_3){\partial\over\partial u_3}}\right]}\right\},\quad$
	(4)

which simplifies to

$$\nabla^2 = {1\over{h_1}^2f_1}{\partial\over\partial u_1} \le... ...rtial u_3} \left[{f_3(u_3){\partial\over\partial u_3}}\right].$$ (5)

Such a coordinate system obeys the Robertson Condition, which means that the Stäckel Determinant is of the form

$$S={h_1h_2h_3\over f_1(u_1)f_2(u_2)f_3(u_3)}.$$ (6)

Coordinate System	Variables	Solution Functions
Cartesian	$X(x)Y(y)Z(z)$	exponential, Circular Functions, Hyperbolic Functions
Circular Cylindrical	$R(r)\Theta(\theta)Z(z)$	Bessel Functions, Exponential Functions, Circular Functions
Conical		Ellipsoidal Harmonics, Power
Ellipsoidal	$\Lambda(\lambda )M(\mu)N(\nu)$	Ellipsoidal Harmonics
Elliptic Cylindrical	$U(u)V(v)Z(z)$	Mathieu Function, Circular Functions
Oblate Spheroidal	$\Lambda(\lambda )M(\mu)N(\nu)$	Legendre Polynomial, Circular Functions
Parabolic		Bessel Functions, Circular Functions
Parabolic Cylindrical		Parabolic Cylinder Functions, Bessel Functions, Circular Functions
Paraboloidal	$U(u)V(v)\Theta(\theta)$	Circular Functions
Prolate Spheroidal	$\Lambda(\lambda )M(\mu)N(\nu)$	Legendre Polynomial, Circular Functions
Spherical	$R(r)\Theta(\theta )\Phi(\phi)$	Legendre Polynomial, Power, Circular Functions

See also Laplace's Equation, Poisson's Equation, Separation of Variables, Spherical Bessel Differential Equation

References

Eisenhart, L. P. ``Separable Systems in Euclidean 3-Space.'' Physical Review 45, 427-428, 1934.

Eisenhart, L. P. ``Separable Systems of Stäckel.'' Ann. Math. 35, 284-305, 1934.

Eisenhart, L. P. ``Potentials for Which Schroedinger Equations Are Separable.'' Phys. Rev. 74, 87-89, 1948.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 125-126 and 509-510, 1953.