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Physical derivation of Bessel functions

Here I'll go through a more physical way of viewing Bessel functions. Bessel functions occur often in the study of problems with cylindrical symmetry. So when you see cylindrical symmetry think ``Bessel functions", spherical symmetry think ``Legendre Polynomials" and when you see Cartesians think ``sine and cosine".

Suppose we want to solve tex2html_wrap_inline121 in cylindrical coordinates. Write tex2html_wrap_inline123 . This substitution, a la separation of variables, leads to the equations




The tex2html_wrap_inline125 in the last equation is just 2 dimensional, different from the original tex2html_wrap_inline125 used above which is three dimensional. Eqn. (2) is often refered to as Helmoltz's equation. To solve it we could use two methods. The first is to seperate variables into polar coordinates tex2html_wrap_inline129 . This gives


which has solutions tex2html_wrap_inline131 where n is an integer. This is the same as Boas chapter 13 equation (5.6). The equation for R, Boas(5.7) is


We'd like to know how to solve this equation, which is closely related to Bessel's equation. We don't know how to solve it so we have two choices. One is to do a power series expansion as is done in chapter 12 of Boas. Instead we can backtrack to eqn. (2) and solve it in Cartesian coordinates. Doing separation of variables again with tex2html_wrap_inline137 , we obtain


with tex2html_wrap_inline139 as a condition on the two constants tex2html_wrap_inline141 and tex2html_wrap_inline143 that is obtained when you go through separation of variables. So F(x,y) can be written in a rather nice form:


So the general solution can be written


The physical interpretation of this is as follows. tex2html_wrap_inline147 is a plane wave travelling in the tex2html_wrap_inline149 direction. Its magnitude is restricted to be K. So the general solution to eqn. (2) is the sum of plane waves all with the same wavelength (or wave-vector), travelling in any arbitrary direction. The coefficient tex2html_wrap_inline153 indicates the amplitude and phase of a wave travelling in the direction of tex2html_wrap_inline155 .

Since there are a continuous range of angles that the wave could go in, we should actually write eqn. (7) as an integral over all possible angles. So writing tex2html_wrap_inline157 and tex2html_wrap_inline159 we can rewrite eqn. (7) as


Here tex2html_wrap_inline161 is the angle the tex2html_wrap_inline155 is pointing relative to the x-axis. Letting tex2html_wrap_inline165 and noticing that the integrand is periodic, we can rewrite this as


This is the general solution to the two dimensional Helmoltz equation.

Now how do we relate this to eqn. (4) above? This was obtained by saying we wanted a special solution that looked like


So we look for solutions to eqn. (9) which are of this form. That is we have to hunt for the appropriate tex2html_wrap_inline167 . Its not impossible to see that tex2html_wrap_inline169 does the trick! This gives


Well this does indeed seem to have separated out the r and tex2html_wrap_inline173 components into the desired form. So comparing with eqn. (10) we see that


This integral will be defined to be equal to a special function. We'll call it tex2html_wrap_inline175 . The tex2html_wrap_inline177 is just a pesty normalization factor that we must include but is quite uninteresting. The big news is the thing tex2html_wrap_inline179 . This is called a ``Bessel function of the first kind and order n''. The above integral is an integral representation of that function. And this by construction is a solution to eqn. (4). There is a closely related form to the above integral. Let tex2html_wrap_inline183 . Then


By noting that tex2html_wrap_inline185 we have


In summary, Bessel functions can be thought of as the sum of two dimensional plane waves going in all possible directions.

next up previous
Next: About this document Up: No Title Previous: No Title

Joshua Deutsch
Wed Oct 22 13:04:23 PDT 1997