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Coen Offline OP
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Ladies and Gentlemen,

If anyone could spare the time, I am looking for an elaboration on the history and definition of a "Bessel Function" and its applications in mathematically calculating radiation from a piston. It's not in my Audio Dictionary by Glen White or any other book I have. I'm pretty sure I know what it is, but I want to know how it was derived. Also is "Rayls" a standard SI unit of measurement now or is it still Newton s/m^3. Any info or reference to a more inclusive reference dictionary would be appreciated. Take it light.

Coen Leeland

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I can give you the definition,
Bessel function J of order n :

Jn (x) = (x/2)^n * sum ( ((-1)^i)/(i!*(n+i)!) * (x/2)^(2*i), for i=0 to infinite )

I don't know about history, I just had to use it to calculate the skin effect in a speaker cable.

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A rayl is a unit of acoustical impedance—how much a given medium resists the transmission of sound. The unit is equivalent to kg/(m^2 s). For air, the acoustic impedance is normally taken to be 405 rayls (at standard pressure and temperature).

As for Bessel Functions, they come about as linearly independent solutions to a second order differential equation that naturally arises when looking at wave propagation in a cylindrical coordinate system as you would find in a (round) drum head, or in your case, a piston!

In Newton’s notation, the differential equation would read:

x^2 y” + x y’ + (x^2 – n^2) y = 0

with y as a function of x, y” being the second derivative and y’ being the first.

Euler and Daniel Bernoulli were involved in early informal, but otherwise complete solutions using series ca. 1739-1745, although James Bernoulli did offer a specialized solution in a letter to Liebniz in 1703. Friedrich Wilhelm Bessel solved the above equation which now bears his name (Bessel’s Equation) in the 1820s, and provided the resulting functions that are solutions also bear his name. There's an awful lot of literature on Bessel functions, because lots of folks have worked on them.

Just about any elementary differential equations text should give you what you need. The history lesson is from “Mathematical Thought form Ancient to Modern Times, Volume 2”.

I have a better reference for your particular piston problem lying around somewhere, and I'll post if I find it.

-Dennis

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Coen Offline OP
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Thanks Pio and Retreading.


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