Diffuse Sound Reflection at a Locally Reacting Plane

Generally, the absorption coefficient $\alpha _{{dif}}$ follows from the absorption coefficient $\alpha(\Theta)$ for oblique incidence by integration over the polar angle $\Theta$. The integrals are

in 2-dimensional space: $\displaystyle\alpha _{{2-dif}}=\int\limits _{0}^{{\pi/2}}{\alpha(\Theta)\cdot\cos\Theta\; d\Theta}$, (1)

in 3-dimensional space: $\displaystyle\alpha _{{3-dif}}=2\int\limits _{0}^{{\pi/2}}{\alpha(\Theta)\cdot\cos\Theta\cdot\sin\Theta\; d\Theta}$. (2)

The integral in three dimensions has an analytical solution for a locally reacting plane with normalised input admittance $Z_{{0}}G=g^{{\prime}}+j\cdot g^{{\prime\prime}}$:

$$\begin{array}[]{@{}l@{\,}c@{\,}l}\alpha _{{3-dif}}&=&\displaystyle 8\,{g}^{{\prime}}\left[{1+\frac{{g}^{{\prime 2}}-{g}^{{\prime\prime 2}}}{{g}^{{\prime\prime}}}\cdot\arctan\,\frac{{g}^{{\prime\prime}}}{{g}^{{\prime}}+{g}^{{\prime 2}}+{g}^{{\prime\prime...