Helmholtz Differential Equation--Cartesian Coordinates

In 2-D Cartesian Coordinates, attempt Separation of Variables by writing

$$F(x, y) = X(x)Y(y),$$ (1)

then the Helmholtz Differential Equation becomes

$${d^2X\over dx^2}Y + {d^2Y\over dy^2}X +k^2 XY= 0.$$ (2)

Dividing both sides by $XY$ gives

$${1\over X}{d^2X\over dx^2}+ {1\over Y}{d^2Y\over dy^2}+k^2= 0.$$ (3)

This leads to the two coupled ordinary differential equations with a separation constant $m^2$,

$${1\over X}{d^2X\over dx^2} = m^2$$ (4)

$${1\over Y}{d^2Y\over dy^2} = -(m^2+k^2),$$ (5)

where $X$ and $Y$ could be interchanged depending on the boundary conditions. These have solutions

$$X = A_me^{mx}+B_me^{-mx}$$ (6)

$$Y = C_me^{i\sqrt{m^2+k^2}\,y}+D_me^{-i\sqrt{m^2+k^2}\,y}$$

$$= E_m\sin (\sqrt{m^2+k^2}\,y)+F_m\cos (\sqrt{m^2+k^2}\,y).$$ (7)

The general solution is then

$$F(x, y) = \sum_{m=1}^\infty (A_me^{mx}+B_me^{-mx})[E_m\sin(\sqrt{m^2+k^2}\,y)+F_m\cos(\sqrt{m^2+k^2}\,y)].$$ (8)

In 3-D Cartesian Coordinates, attempt Separation of Variables by writing

$$F(x, y, z) = X(x)Y(y)Z(z),$$ (9)

then the Helmholtz Differential Equation becomes

$${d^2X\over dx^2}YZ + {d^2Y\over dy^2}XZ + {d^2Z\over dz^2}XY+k^2XY = 0.$$ (10)

Dividing both sides by $XYZ$ gives

$${1\over X}{d^2X\over dx^2}+ {1\over Y}{d^2Y\over dy^2}+{1\over Z}{d^2Z\over dz^2}+k^2= 0.$$ (11)

This leads to the three coupled differential equations

$${1\over X}{d^2X\over dx^2} = l^2$$ (12)

$${1\over Y}{d^2Y\over dy^2} = m^2$$ (13)

$${1\over Z}{d^2Z\over dz^2} = -(k^2+l^2+m^2),$$ (14)

where $X$, $Y$, and $Z$ could be permuted depending on boundary conditions. The general solution is therefore

$F(x, y, z) = \sum_{l=1}^\infty \sum_{m=1}^\infty (A_le^{lx}+B_le^{-lx})(C_me^{my}+D_me^{-my})$
$\times (E_{lm}e^{-i\sqrt{k^2+l^2+m^2}\,z}+F_{lm}e^{i\sqrt{k^2+l^2+m^2}\,z}).\quad$	(15)

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 501-502, 513-514 and 656, 1953.