Transcription of 1 Solutions in cylindrical coordinates: Bessel functions
{{id}} {{{paragraph}}}
Example: biology
1 Solutions in cylindrical coordinates: Bessel functions
Bessels 20201 Solutions in cylindrical coordinates: Bessel functionsLaplace s equation in cylindrical coordinates is:1 +1 1 + 2 z2=0 Separate variables: Let =R( )W( )Z(z).Then wefind:1R R +1W 2 2W 2+1Z 2Z z2=0 The last term is a function ofzonly, while the sum of thefirst two terms is afunction of and only. Thus we take each part to be a constant 2Z z2=k2 Zand the Solutions areZ=e kzThis is the appropriate solution outside of a charge distribution, say above aplane, ( 0asz ),or inside a cylinder with grounded walls andnon-zero potential on one remaining equation is:1R R +1W 2 2W 2+k2=0 Now multiply through by 2: R R +k2 2+1W 2W 2=0 Here the last term is a function of only and thefirst two terms are functionsof only. Again we often want a solution that is periodic with period2 ,sowe choose a negative separation constant: 2W 2= m2W W=e im Finally we have the equation for the function of : R +k2 2R m2R=01To see that this equation is of Sturm-Liouville form, divide through by : R +k2 R m2 R=0(1)Now we have a Sturm-Liouville equation (slreview notes eqn.)
is Bessel’s equation. The solutions are orthogonal functions. Since f (0) = 0, we do not need to specify any boundary condition at ρ=0if our range is 0 ≤ρ≤a, as is frequently the case. (We do specify that R remain finite.) We do need a boundary condition at ρ= a. It is simpler and more elegant to solve Bessel’s equation if we change ...
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: