In 2-D Cartesian Coordinates, attempt Separation of Variables by writing
|
(1) |
then the Helmholtz Differential Equation becomes
|
(2) |
Dividing both sides by gives
|
(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
|
(8) |
In 3-D Cartesian Coordinates, attempt Separation of Variables by writing
|
(9) |
then the Helmholtz Differential Equation becomes
|
(10) |
Dividing both sides by gives
|
(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