Helmholtz Differential Equation--Confocal Ellipsoidal Coordinates

Using the Notation of Byerly (1959, pp. 252-253), Laplace's Equation can be reduced to

$$\nabla^2F=(\mu^2-\nu^2){\partial^2F\over\partial\alpha^2}+(\... ...ta^2} +(\lambda^2-\mu^2){\partial^2F\over\partial\gamma^2}=0,$$ (1)

where

$$\alpha = c\int_c^\lambda {d\lambda\over\sqrt{(\lambda^2-b^2)(\lambda^2-c^2)}}$$

$$= F\left({{b\over c}, {\pi\over 2}}\right)-F\left({{b\over c}, \sin^{-1}\left({c\over\lambda}\right)}\right)$$ (2)

$$\beta = c\int_b^\mu {d\mu\over\sqrt{(c^2-\mu^2)(\mu^2-b^2)}}$$

$$= F\left({\sqrt{1-{b^2-c^2}}, \sin^{-1}\left({\sqrt{1-{b^2\over\mu^2}\over 1-{b^2\over c^2}}\,}\right)}\right)$$ (3)

$$\gamma = c\int_0^\nu{d\nu\over\sqrt{(b^2-\nu^2)(c^2-\nu^2)}}$$

$$= F\left({{b\over c}, \sin^{-1}\left({\nu\over b}\right)}\right).$$ (4)

In terms of $\alpha$, $\beta$, and $\gamma$,

$$\lambda = c\mathop{\rm dc}\nolimits \left({\alpha, {b\over c}}\right)$$ (5)

$$\mu = b\mathop{\rm nd}\nolimits \left({\beta,\sqrt{1-{b^2\over c^2}}\,}\right)$$ (6)

$$\nu = b\mathop{\rm sn}\nolimits \left({\gamma,{b\over c}}\right).$$ (7)

Equation (1) is not separable using a function of the form

$$F=L(\alpha)M(\beta)N(\gamma),$$ (8)

but it is if we let

$${1\over L}{d^2L\over d\alpha^2} = \sum a_k\lambda^k$$ (9)

$${1\over M}{d^2M\over d\beta^2} = \sum b_k\mu^k$$ (10)

$${1\over N}{d^2N\over d\gamma^2} = \sum c_k\nu^k.$$ (11)

These give

$$a_0 = -b_0=c_0$$ (12)

$$a_2 = -b_2=c_2,$$ (13)

and all others terms vanish. Therefore (1) can be broken up into the equations

$${d^2L\over d\alpha^2} = (a_0+a_2\lambda^2)L$$ (14)

$${d^2M\over d\beta^2} = -(a_0+a_2\mu^2)M$$ (15)

$${d^2N\over d\gamma^2} = (a_0+a_2\nu^2)N.$$ (16)

For future convenience, now write

$$a_0 = -(b^2+c^2)p$$ (17)

$$a_2 = m(m+1),$$ (18)

then

$${d^2L\over d\alpha^2}-[m(m+1)\lambda^2-(b^2+c^2)p]L = 0$$ (19)

$${d^2M\over d\beta^2}+[m(m+1)\mu^2-(b^2+c^2)p]M = 0$$ (20)

$${d^2N\over d\gamma^2}-[m(m+1)\nu^2-(b^2+c^2)p]N = 0.$$ (21)

Now replace $\alpha$, $\beta$, and $\gamma$ to obtain

$$(\lambda^2-b^2)(\lambda^2-c^2){d^2L\over d\lambda^2}+\lambda(\lambda^2-b^2+\lambda^2-c^2){dL\over d\lambda}-[m(m+1)\lambda^2-(b^2+c^2)p]L = 0$$ (22)

$$(\mu^2-b^2)(\mu^2-c^2){d^2M\over d\mu^2}+\mu(\mu^2-b^2+\mu^2-c^2){dM\over d\mu}-[m(m+1)\mu^2-(b^2+c^2)p]M = 0$$ (23)

$$(\nu^2-b^2)(\nu^2-c^2){d^2N\over d\nu^2}+\nu(\nu^2-b^2+\nu^2-c^2){dN\over d\nu}-[m(m+1)\nu^2-(b^2+c^2)p]N = 0.$$ (24)

Each of these is a Lamé's Differential Equation, whose solution is called an Ellipsoidal Harmonic. Writing

$$L(\lambda) = E_m^p(\lambda)$$ (25)

$$M(\lambda) = E_m^p(\mu)$$ (26)

$$N(\lambda) = E_m^p(\nu)$$ (27)

gives the solution to (1) as a product of Ellipsoidal Harmonics $E_m^p(x)$.

$$F=E_m^p(\lambda)E_m^p(\mu)E_m^p(\nu).$$ (28)

References

Arfken, G. ``Confocal Ellipsoidal Coordinates $(\xi_1, \xi_2, \xi_3)$.'' §2.15 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 117-118, 1970.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 251-258, 1959.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.