From 3fd9b3225f3dd82c4eb85c3a239c91360fe50b7e Mon Sep 17 00:00:00 2001
From: Jake Peterson <jdpeterson072@gmail.com>
Date: Fri, 23 Feb 2024 15:01:24 -0500
Subject: [PATCH] add absorption to docs/theory

---
 docs/Theory/Absorption_Theory.md | 68 ++++++++++++++++++++++++++++++++
 docs/Theory/Reflection_Theory.md |  4 +-
 2 files changed, 71 insertions(+), 1 deletion(-)

diff --git a/docs/Theory/Absorption_Theory.md b/docs/Theory/Absorption_Theory.md
index e69de29..217d7be 100644
--- a/docs/Theory/Absorption_Theory.md
+++ b/docs/Theory/Absorption_Theory.md
@@ -0,0 +1,68 @@
+The frequency-dependent absorption coefficients $(A)$ are calculated directly from the total transfer matrix $(T_{t})$ of a multilayered structure.  These coefficients are a measure of the amount of sound energy absorbed by the structure.
+
+The coefficients can be calculated under both normal and diffuse sound field conditions.  Under a normal incidence sound field, the sound impinges on the surface from a single, perpendicular angle.  In the diffuse field case, the incident sound theoretically strikes the surface of the material from all possible angles -- though the acoustipy implementation defaults to angles between 0 and 79, as seen in literature on the topic.
+
+The acoustipy implementation for both cases can be found [here](https://jakep72.github.io/acoustipy/AcousticTMM/#src.acoustipy.TMM.AcousticTMM.absorption).
+
+### Normal Incidence
+
+
+
+Starting from the total transfer matrix:
+
+\[
+T_{t} = 
+\begin{bmatrix}
+T_{11} & T_{12}\\
+T_{21} & T_{22} \\
+\end{bmatrix}
+\]
+
+First, the surface impedence $(Z_{s})$ is calculated:
+
+\[
+Z_{s} = \frac{T_{11}}{T_{21}}  
+\]
+
+Then the reflection coeffients are:
+
+\[
+R = \frac{Z_{s}-Z_{0}}{Z_{s}-Z_{0}}  
+\]
+
+where $Z_{0}$ is the characteristic impedence of air:
+
+\[
+Z_{0} = \rho_{0} c_{0}
+\]
+
+and $\rho_{0}$ is the density of air and $c_{0}$ is the speed of sound in air.
+
+Finally, the absorption coefficients are detemined via:
+
+\[
+A = 1-|R|^2  
+\]
+
+### Diffuse Incidence
+
+Under the diffuse sound field condition, the calculation of surface impedence $(Z_{s})$ is the same as the normal incidence condition.
+
+The reflection coefficients at each angle are then:
+
+\[
+r = \frac{Z_{s}\cos(\theta)-Z_{0}}{Z_{s}\cos(\theta)+Z_{0}}
+\]
+
+Similar to the normal field condition, the absorption coefficients at each angle are then:
+
+\[
+a = 1-|r|^2  
+\]
+
+which yields a vector of shape $[f, \theta]$.  To collapse this vector to shape $[f,1]$, Paris' formula is used as shown below.
+
+
+\[
+A = \frac{\sum a\cos(\theta)\sin(\theta)}{\sum \cos(\theta)\sin(\theta)}
+\]
\ No newline at end of file
diff --git a/docs/Theory/Reflection_Theory.md b/docs/Theory/Reflection_Theory.md
index 40f2fb2..a694ccb 100644
--- a/docs/Theory/Reflection_Theory.md
+++ b/docs/Theory/Reflection_Theory.md
@@ -1,7 +1,9 @@
-The frequency-dependent reflection coefficients $(R)$ are calculated directly from the total transfer matrix $(T_{t})$ of a multilayered structure.
+The frequency-dependent reflection coefficients $(R)$ are calculated directly from the total transfer matrix $(T_{t})$ of a multilayered structure.  These coefficients are a measure of how much sound is reflected off the surface of a structure.
 
 The coefficients can be calculated under both normal and diffuse sound field conditions.  Under a normal incidence sound field, the sound impinges on the surface from a single, perpendicular angle.  In the diffuse field case, the incident sound theoretically strikes the surface of the material from all possible angles -- though the acoustipy implementation defaults to angles between 0 and 79, as seen in literature on the topic.
 
+The acoustipy implementation for both cases can be found [here](https://jakep72.github.io/acoustipy/AcousticTMM/#src.acoustipy.TMM.AcousticTMM.reflection).
+
 ### Normal Incidence