Principles of Superposition

→ See also Ochmann/Donner  (1994); Mechel  (2000)

Some principles of superposition may help to reduce more general problems to a repetition of simpler standard tasks.

First principle of superposition (by Mechel): (unsymmetry superposition)

Two opposite walls, normal to the same co-ordinate, locally react with different admittances G${}_{{1}}$, G${}_{{2}}$. The sound fields at the walls have the corresponding indices 1,2.

The boundary conditions at these surfaces are (with normal particle velocity components v ${}_{{i}}$): $\begin{array}[]{rcl}v_{1}&=&G_{1}\cdot p_{1};\\     v_{2}&=&G_{2}\cdot p_{2}.\end{array}$

Set (with G${}_{{1}}$, G${}_{{2}}$ selected so that Re{G${}_{{a}}$}$\geq$0): G${}_{{s}}$ is the symmetrical and G${}_{{a}}$ the antisymmetrical part of the boundary conditions. $\begin{array}[]{@{}l}G_{s}=\displaystyle\frac{1}{2}\left({G_{1}+G_{2}}\right),\\ G_{a}=\displaystyle\frac{1}{2}\left({G_{1}-G_{2}}\right).\end{array}\hfill(1)$

Suppose the...