Helmholtz Differential Equation--Cartesian Coordinates
المؤلف:
Morse, P. M. and Feshbach, H
المصدر:
Methods of Theoretical Physics, Part I. New York: McGraw-Hill
الجزء والصفحة:
...
13-7-2018
2093
Helmholtz Differential Equation--Cartesian Coordinates
In two-dimensional 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
![F(x,y)=sum_(m=1)^infty(A_me^(mx)+B_me^(-mx))
×[E_msin(sqrt(m^2+k^2)y)+F_mcos(sqrt(m^2+k^2)y)].](http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationCartesianCoordinates/NumberedEquation4.gif) |
(9)
|
In three-dimensional Cartesian coordinates, attempt separation of variables by writing
 |
(10)
|
then the Helmholtz differential equation becomes
 |
(11)
|
Dividing both sides by 
gives
 |
(12)
|
This leads to the three coupled differential equations
where 
, 
, and 
could be permuted depending on boundary conditions. The general solution is therefore
 |
(16)
|
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.
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