In 2-D Cartesian Coordinates, attempt Separation of Variables by writing
![\begin{displaymath}
F(x, y) = X(x)Y(y),
\end{displaymath}](h_906.gif) |
(1) |
then the Helmholtz Differential Equation becomes
![\begin{displaymath}
{d^2X\over dx^2}Y + {d^2Y\over dy^2}X +k^2 XY= 0.
\end{displaymath}](h_907.gif) |
(2) |
Dividing both sides by
gives
![\begin{displaymath}
{1\over X}{d^2X\over dx^2}+ {1\over Y}{d^2Y\over dy^2}+k^2= 0.
\end{displaymath}](h_909.gif) |
(3) |
This leads to the two coupled ordinary differential equations with a separation constant
,
where
and
could be interchanged depending on the boundary conditions. These have solutions
The general solution is then
![\begin{displaymath}
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)].
\end{displaymath}](h_920.gif) |
(8) |
In 3-D Cartesian Coordinates, attempt Separation of Variables by writing
![\begin{displaymath}
F(x, y, z) = X(x)Y(y)Z(z),
\end{displaymath}](h_921.gif) |
(9) |
then the Helmholtz Differential Equation becomes
![\begin{displaymath}
{d^2X\over dx^2}YZ + {d^2Y\over dy^2}XZ + {d^2Z\over dz^2}XY+k^2XY = 0.
\end{displaymath}](h_922.gif) |
(10) |
Dividing both sides by
gives
![\begin{displaymath}
{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.
\end{displaymath}](h_924.gif) |
(11) |
This leads to the three coupled differential equations
where
,
, and
could be permuted depending on boundary conditions. The general solution is therefore
|
|
|
(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.
© 1996-9 Eric W. Weisstein
1999-05-25