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Dealing With a Summation

  1. Nov 14, 2009 #1

    FredGarvin

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    Hey guys and gals,

    While this technically isn't homework, I figured this is the place to post.

    I am working over a problem and I am at a point in the solution that has me a bit stumped. Perhaps someone may provide some guidance.

    In acoustics, we run into the problem of a radiating body in cylindrical coordinates. Essentially, after a bit of work, I have come to the point where I am stuck in equating a body's radial velocity and the radial derivative of the velocity potential. What I have is:

    [tex]C cos(\theta) = \sum_{n=0}^\infty A_n \left(\frac{\omega}{a_o} \right) H_{n+1}^{(1)} \left(\frac{\omega}{a_o} R\right) cos(n \theta)[/tex]

    Where C is a constant, An is what I am trying to solve for, and H is the Hankel function of the first kind.

    Now, normally I don't have both sides as a function of theta and the solving for An is pretty straight forward. However, this time it is not the case. Is there a way to somehow come up with a general solution to An that does not include the summation? I'm thinking no, but I figured I'd ask.

    Thanks!
     
  2. jcsd
  3. Nov 14, 2009 #2

    lanedance

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    could you do the fourier type approach & mutiply both side by [itex] cos(m\theta) [/itex] then integrate over theta?
     
  4. Nov 15, 2009 #3

    FredGarvin

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    I am not seeing what that will do if I multiply by [tex]cos (m \theta)[/tex]. Can you elaborate a bit more on that? Thanks!
     
  5. Nov 15, 2009 #4

    lanedance

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    could be missing something/oversimplifying, but here's what i was thinking... directly along the lines of how you determine the fourier co-efficients

    so if i'm getting it correctly, for the purpose of evaluating the An constants, the hankel functions are effectively just a constant evaluated at the boundary r = R, so write the total coefficient as Bn for now:
    [tex]C cos(\theta) = \sum_{n=0}^\infty B_n cos(n \theta)[/tex]

    now, if the terms of the sum were something more like [itex] cos(n\pi \theta) [/itex], then multiplying by [itex] cos(m\pi \theta)[/itex], and integrating you get:
    [tex]C \int cos(m\pi \theta) cos(\pi \theta) = \sum_{n=0}^\infty B_n \int cos(\pi n \theta)cos(m\pi \theta) [/tex]

    which will cancel out pretty simply due to the orthogonal nature of the cos functions...

    so maybe you can find a similar approach...

    though its been a while since i done any of these, & that was only really in SL type problems & so some issues i could see:
    - if the n aren't nice integers or have a clear integer difference type relationship (no longer othogonal..? is there another orthogonal set?)
    - and if any of the hankel functions had zeroes at the boundary
     
    Last edited: Nov 16, 2009
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