In 2-D Polar Coordinates, attempt Separation of Variables by writing
![\begin{displaymath}
F(r, \theta) = R(r)\Theta(\theta),
\end{displaymath}](h_1082.gif) |
(1) |
then the Helmholtz Differential Equation becomes
![\begin{displaymath}
{d^2R\over dr^2}\Theta + {1\over r}{dR\over dr}\Theta + {1\over r^2} {d^2\Theta \over d\theta^2}R +k^2R\Theta = 0.
\end{displaymath}](h_1083.gif) |
(2) |
Divide both sides by
![\begin{displaymath}
\left({{r^2\over R}{d^2R\over dr^2}+ {r\over R}{dR\over dr}}...
...({{1\over \Theta } {d^2\Theta \over d\theta^2}+k^2}\right)= 0.
\end{displaymath}](h_1085.gif) |
(3) |
The solution to the second part of (3) must be periodic, so the differential equation is
![\begin{displaymath}
{d^2\Theta\over d\theta^2}{1\over\Theta}= -(k^2+m^2),
\end{displaymath}](h_941.gif) |
(4) |
which has solutions
Plug (4) back into (3)
![\begin{displaymath}
r^2R''+rR'-m^2R = 0.
\end{displaymath}](h_1089.gif) |
(6) |
This is an Euler Differential Equation with
and
. The roots are
.
So for
,
and the solution is
![\begin{displaymath}
R(r) = c_1+c_2\ln r.
\end{displaymath}](h_1095.gif) |
(7) |
But since
blows up at
, the only possible physical solution is
. When
,
, so
![\begin{displaymath}
R(r) = c_1r^m+c_2r^{-m}.
\end{displaymath}](h_1099.gif) |
(8) |
But since
blows up at
, the only possible physical solution is
. The solution for
is
then
![\begin{displaymath}
R_m(r) = c_mr^m
\end{displaymath}](h_1102.gif) |
(9) |
for
, 1, ...and the general solution is
![\begin{displaymath}
F(r, \theta) = \sum_{m=0}^\infty [a_mr^m\sin (\sqrt{k^2+m^2}\,\theta)+b_mr^m\cos(\sqrt{k^2+m^2}\,\theta)].
\end{displaymath}](h_1103.gif) |
(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