reflection to docs/theory

docs/Theory/Reflection_Theory.md

@@ -0,0 +1,53 @@
+The frequency-dependent reflection coefficients $(R)$ are calculated directly from the total transfer matrix $(T_{t})$ of a multilayered 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.
+
+### 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.
+
+### 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:
+
+\[
+r = \frac{Z_{s}\cos(\theta)-Z_{0}}{Z_{s}\cos(\theta)+Z_{0}}
+\]
+
+which yields a vector of shape $[f, \theta]$.  To collapse this vector to shape $[f,1]$, Paris' formula is used as shown below.
+
+\[
+R = \frac{\sum r\cos(\theta)\sin(\theta)}{\sum \cos(\theta)\sin(\theta)}
+\]

mkdocs.yml

@@ -71,4 +71,8 @@ nav:
         - MAA Model: Theory/MAA_MPP_Model.md
         - Equivalent Fluid Model: Theory/EF_MPP_Model.md
       - Transfer Matrices: Theory/TMM_Theory.md
+      - Acoustic Indicators:
+        - Reflection: Theory/Reflection_Theory.md
+        - Absorption: Theory/Absorption_Theory.md
+        - Transmission Loss: Theory/Transmission_Theory.md