How Can I Solve for Coefficients in Cylindrical Acoustic Modeling?

[FredGarvin]

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:

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

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!

 

[lanedance]

could you do the Fourier type approach & mutiply both side by $cos(m\theta)$ then integrate over theta?

 

[FredGarvin]

I am not seeing what that will do if I multiply by $$cos (m \theta)$$. Can you elaborate a bit more on that? Thanks!

 

[lanedance]

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 A_n constants, the hankel functions are effectively just a constant evaluated at the boundary r = R, so write the total coefficient as B_n for now:
$$C cos(\theta) = \sum_{n=0}^\infty B_n cos(n \theta)$$

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

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