In 2-D Polar Coordinates, attempt Separation of Variables by writing
|
(1) |
then the Helmholtz Differential Equation becomes
|
(2) |
Divide both sides by
|
(3) |
The solution to the second part of (3) must be periodic, so the differential equation is
|
(4) |
which has solutions
Plug (4) back into (3)
|
(6) |
This is an Euler Differential Equation with
and
. The roots are .
So for , and the solution is
|
(7) |
But since blows up at , the only possible physical solution is . When , , so
|
(8) |
But since blows up at , the only possible physical solution is
. The solution for is
then
|
(9) |
for , 1, ...and the general solution is
|
(10) |
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 502-504, 1953.
© 1996-9 Eric W. Weisstein
1999-05-25