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- Thread starter dspampi
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pasmith

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If you separate variables in Laplace's equation in such a way that the axial dependence is exponential, then the equation for the radial dependence is Bessel's Equation:

[tex]\nabla^2(R(r)e^{\pm kz}e^{\pm in\theta}) = 0 \quad\Rightarrow\quad r^2R'' + rR' + (k^2r^2 - n^2)R = 0[/tex]

If you separate variables in Laplace's equation in such a way that the axial dependence is sinusoidal, then the equation for the radial dependence is the modified Bessel's Equation:

[tex]

\nabla^2(R(r)e^{\pm ikz}e^{\pm in\theta}) = 0 \quad \Rightarrow \quad r^2R'' + rR' - (k^2r^2 + n^2)R = 0

[/tex]

If you're working in a region where [itex]z[/itex] is unbounded then you will only need [itex]e^{\pm kz}[/itex]. If you're working in a region where [itex]z[/itex] is bounded then you will generally need both [itex]e^{\pm kz}[/itex] and [itex]e^{\pm ikz}[/itex].

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jasonRF

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http://en.wikipedia.org/wiki/Bessel_function

Here is a summary (see the link for more of the story).

regular Bessel:

J ~ cos(x-a)/sqrt(x)

Y ~ sin(x-a)/sqrt(x)

Modified bessel:

I ~ exp(x)/sqrt(x)

K ~ exp(-x)/sqrt(x)

So regular bessel functions oscillate, and modified do not.

The hankel functions (H = J +/i Y) thus go like exp(+/- i (x-a))/sqrt(x), which is why they are useful for cylindrical wave problems.

The behavior at zero matters for some applications, especially those for which your solution must be bounded at zero. J and I are bounded, while Y and K (and hence the hankel functions H) are not.

In light of what pasmith wrote, when the Z-dependence oscillates, the R does not, and vice versa.

jason

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