Generalized Wave Impedance

From the multivariable generalization of (2), we have, using (10), $\partial {\mbox{\boldmath$p$}}({\mbox{\boldmath$x$}},t) / \partial {\mbox{\bold... ...dmath$u$}}\stackrel{\triangle}{=}\pm {\mbox{\boldmath$R$}}{\mbox{\boldmath$u$}}$, where `$+$' is for right-going and `$-$' is for left-going. Thus, following the classical definition for the scalar case, the wave impedance is defined by

$${\mbox{\boldmath$R$}}\stackrel{\triangle}{=}{\mbox{\boldmath... ...ldmath$M$}}^{1/2} = {\mbox{\boldmath$C$}}{\mbox{\boldmath$M$}}$$

and we have

$$\begin{array}{rcr} {\mbox{\boldmath$p$}}^+&=& {\mbox{\boldma... ...-&=& - {\mbox{\boldmath$R$}}{\mbox{\boldmath$u$}}^- \end{array}$$ (13)

Thus, the wave impedance ${\mbox{\boldmath$R$}}$ is the factor of proportionality between pressure and velocity in a traveling wave. It is diagonal if and only if the mass matrix ${\mbox{\boldmath$M$}}$ is diagonal (since ${\mbox{\boldmath$C$}}$ is assumed diagonal). The minus sign for the left-going wave ${\mbox{\boldmath$p$}}^-$ accounts for the fact that velocities must move to the left to generate pressure to the left. The wave admittance is defined as ${\mbox{\boldmath$\Gamma$}}= {\mbox{\boldmath$R$}}^{-1}$.

More generally, when there is a loss represented by a diagonal matrix ${\mbox{\boldmath$G$}}$, we have, in the continuous-time case,

$${\mbox{\boldmath$p$}}= e^{{{\mbox{\boldmath$G$}}}{\mbox{\bol... ...^{-1}{\mbox{\boldmath$X$}}\right)} \cdot {\mbox{\boldmath$1$}}$$ (14)

where ${\mbox{\boldmath$X$}}\stackrel{\triangle}{=}\hbox{diag}({\mbox{\boldmath$x$}})$ as before, leading to the admittance matrix

$${\mbox{\boldmath$\Gamma$}}= {\mbox{\boldmath$M$}}^{-1} {\mbo... ...\frac{1}{s} {\mbox{\boldmath$M$}}^{-1} {{\mbox{\boldmath$G$}}}$$ (15)

For the discrete-time case, we may map ${\mbox{\boldmath$\Gamma$}}(s,{\mbox{\boldmath$x$}})$ from the $s$ plane to the $z$ plane via the bilinear transform [65], or we may sample the inverse Laplace transform of ${\mbox{\boldmath$\Gamma$}}(s,{\mbox{\boldmath$x$}})$ and take its $z$ transform to obtain $\hat{{\mbox{\boldmath$\Gamma$}}}(z,{\mbox{\boldmath$x$}})$.

A linear propagation medium in the discrete-time case is completely determined by its wave impedance ${\mbox{\boldmath$R$}}(z,{\mbox{\boldmath$x$}})$ (generalized here to permit frequency-dependent and spatially varying wave impedances). A waveguide is defined for purposes of this paper as a length of medium in which the wave impedance is either constant with respect to spatial position $x$, or else it varies smoothly with $x$ in such a way that there is no scattering (as in the conical acoustic tube). For simplicity, we will suppress the possible spatial dependence and write only ${\mbox{\boldmath$R$}}(z)$.³