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Helmholtz Differential Equation Polar Coordinates

In 2-D Polar Coordinates, attempt Separation of Variables by writing

\begin{displaymath}
F(r, \theta) = R(r)\Theta(\theta),
\end{displaymath} (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} (2)

Divide both sides by $R\Theta$
\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} (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} (4)

which has solutions
$\displaystyle \Theta(\theta)$ $\textstyle =$ $\displaystyle c_1e^{i\sqrt{k^2+m^2}\,\theta}+c_2e^{-i\sqrt{k^2+m^2}\,\theta}$  
  $\textstyle =$ $\displaystyle c_3\sin(\sqrt{k^2+m^2}\,\theta)+c_4\cos(\sqrt{k^2+m^2}\,\theta).$  
      (5)

Plug (4) back into (3)
\begin{displaymath}
r^2R''+rR'-m^2R = 0.
\end{displaymath} (6)

This is an Euler Differential Equation with $\alpha \equiv 1$ and $\beta \equiv -m^2$. The roots are $r = \pm m$. So for $m = 0$, $r = 0$ and the solution is
\begin{displaymath}
R(r) = c_1+c_2\ln r.
\end{displaymath} (7)

But since $\ln r$ blows up at $r = 0$, the only possible physical solution is $R(r) = c_1$. When $m > 0$, $r = \pm m$, so
\begin{displaymath}
R(r) = c_1r^m+c_2r^{-m}.
\end{displaymath} (8)

But since $r^{-m}$ blows up at $r = 0$, the only possible physical solution is $R_m(r) = c_1r^m$. The solution for $R$ is then
\begin{displaymath}
R_m(r) = c_mr^m
\end{displaymath} (9)

for $m = 0$, 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} (10)


References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 502-504, 1953.



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© 1996-9 Eric W. Weisstein
1999-05-25