Chain Circuit

→ See also Mechel, Vol. II, Ch. 2 (1995)

A chain circuit is a useful representation of multilayer absorbers (see → Sect.  D.4 ).

A chain network consists of longitudinal impedances Z${}_{{n}}$ and lateral admittance G${}_{{n}}$. Its sound pressures p${}_{{n}}$ in the nodes and velocities v${}_{{ln}}$ in the longitudinal elements, as well as v${}_{{qn}}$ in the transversal elements, can be evaluated by iteration.

If the network is open (as shown), i.e. v${}_{{l,N+1}}=0$, one begins with an assumed value p${}_{{N}}=1$.

The backward recursion is

$$\begin{array}[]{@{}l@{\,}c@{\,}l}v_{{q,n}}&=&p_{n}\cdot G_{n},\\ v_{{\ell,n}}&=&v_{{q,n}}+v_{{\ell,n+1}},\\ p_{{n-1}}&=&p_{n}+v_{{\ell,n}}\cdot Z_{n}.\\ \end{array}$$

(1)

One iterates over $n=N$, $N-1,{\ldots},1$. The last result is $p_{{0}}$. All field quantities are proportional to $p_{{N}}$. To replace this by $p_{{0}}$ as the reference pressure, divide all (saved) quantities by the value of $p_{{0}}$. If parameters Z