Difference between revisions of "Template:Separation of variables in cylindrical coordinates"
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<center> | <center> | ||
<math> | <math> | ||
− | \phi(r,\theta,z)= | + | \phi(r,\theta,z)=\frac{\cos k(z+h)}{\cos kh}\sum_{n=-\infty}^{\infty}\rho_{n}(r)e^{i n \theta} |
</math> | </math> | ||
</center> | </center> | ||
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\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r} | \frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r} | ||
\frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left( | \frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left( | ||
− | \frac{n^{2}}{r^{2}}+ | + | \frac{n^{2}}{r^{2}}+k{2}\right) \rho_{n}=0. |
</math> | </math> | ||
</center> | </center> | ||
− | + | We can convert this equation to the | |
− | + | standard form by substituting <math>y=k r</math> (provided that | |
− | |||
− | standard form by substituting <math>y= | ||
<math>\mu\neq 0</math>to obtain | <math>\mu\neq 0</math>to obtain | ||
<center> | <center> | ||
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modified Bessel functions of order <math>n</math>, <math>I_{n}(y)</math> and | modified Bessel functions of order <math>n</math>, <math>I_{n}(y)</math> and | ||
<math>K_{n}(y)</math> ([[Abramowitz and Stegun 1964]]). | <math>K_{n}(y)</math> ([[Abramowitz and Stegun 1964]]). | ||
+ | |||
+ | Therefore | ||
+ | <center> | ||
+ | <math> | ||
+ | \rho_n(r) = C_1 I_{n}(kr) + C_2 K_{n}(kr) | ||
+ | </math> | ||
+ | </center> | ||
+ | for some constants <math>C_1</math> and <math>C_2</math> |
Revision as of 05:52, 26 August 2008
Separation for Cylindrical Coordinates
We now separate variables, noting that since the problem has circular symmetry we can write the potential as
[math]\displaystyle{ \phi(r,\theta,z)=\frac{\cos k(z+h)}{\cos kh}\sum_{n=-\infty}^{\infty}\rho_{n}(r)e^{i n \theta} }[/math]
We now solve for the function [math]\displaystyle{ \rho_{n}(r) }[/math]. Using Laplace's equation in polar coordinates we obtain
[math]\displaystyle{ \frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r} \frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left( \frac{n^{2}}{r^{2}}+k{2}\right) \rho_{n}=0. }[/math]
We can convert this equation to the standard form by substituting [math]\displaystyle{ y=k r }[/math] (provided that [math]\displaystyle{ \mu\neq 0 }[/math]to obtain
[math]\displaystyle{ y^{2}\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}y^{2}}+y\frac{\mathrm{d}\rho_{n} }{\rm{d}y}-(n^{2}+y^{2})\rho_{n}=0 }[/math]
The solution of this equation is a linear combination of the modified Bessel functions of order [math]\displaystyle{ n }[/math], [math]\displaystyle{ I_{n}(y) }[/math] and [math]\displaystyle{ K_{n}(y) }[/math] (Abramowitz and Stegun 1964).
Therefore
[math]\displaystyle{ \rho_n(r) = C_1 I_{n}(kr) + C_2 K_{n}(kr) }[/math]
for some constants [math]\displaystyle{ C_1 }[/math] and [math]\displaystyle{ C_2 }[/math]