Plane Wave Reflection at a Locally Reacting Plane

An absorber is said to be locally reacting if there is no sound propagation inside the absorber parallel to the absorber surface.

A plane wave with amplitude A is incident in the plane (x,y) on an absorbent plane; the y axis is in the absorber surface; the x axis is normal to the absorber and directed into the absorber; the wave vector of the incident wave p${}_{{i}}$(x,y) forms a polar angle $\Theta$ with the normal to the surface.

The acoustic quality of the absorber is defined

by the wall admittance: $\displaystyle G=\frac{v_{x}(0,y)}{p(0,y)}=\frac{j}{k_{0}Z_{0}}\,\frac{\partial p(0,y)/\partial x}{p(0,y)}$?. (1)

Field formulation: $\begin{array}[]{l}\displaystyle p(x,y)=p_{i}(x,y)+p_{r}(x,y)\;,\\ p_{i}(x,y)=A\cdot e^{{-j\, k_{0}\left({x\cdot\cos\,\Theta+y\cdot sin\,\Theta}\right)}}\;,\\ p_{r}(x,y)=r\cdot A\cdot e^{{-j\, k_{0}\left({-x\cdot\cos\,\Theta+y\cdot sin\,\Theta}\right)}}\;.\\ \end{array}$ (2)

Reflection factor: \(\begin{array}[]{l}\displaystyle...