The Problem
I have the following information from an article. I am not seeking help with the engineering side of this, but am more looking for information on how to write out / solve a portion of a given equation:
My Understanding
In the above article, it says that gamma (the surface of a Quadratic Residue Diffuser) is divided into a number of straight-line elements equal to n. n in my case is equal to the number of slots on a QRD (each with a length of 2.5 cm or 12 cm (the deepest wells with no child diffusers).
From what I understand the journal explains that in Equation 1 we must substitute rm where $m = 1, 2, 3, …, n$ in for $r$ in order to create n number of linear equations. These linear equations will allow us to find the sound pressures at each slot midpoint. Once we have the sound pressures at each slot midpoint we can then plug those values back into Equation 1’s summation portion (for $p(rn, r0$)) in order to solve for $p(r,r0)$ (the sound pressure at the receiver with the sound barrier in between. We then plug our known value into a logarithmic equation that compares the sound pressure with and without the sound barrier and gives us a value for insertion loss in dB.
My Questions
Where I am getting tripped up is the partial derivative part of equation 1. We are currently trying to figure out how to either simplify or compute the following part of the equation. Does anyone know how I can simplify this portion of the function or know how I would input this into Wolfram Mathematica?: