Multivariable Wave Impedance

From (5), we have, using (13),

$$\begin{array}{lrcl} & \partial {\mbox{\boldmath$p$}}({\mbox{... ...ox{\boldmath$R$}}{\mbox{\boldmath$u$}}\nonumber \,, \end{array}$$

where `$+$' is for right-going and `$-$' is for left-going. Thus, following the classical definition for the scalar case, the $m \times m$ wave impedance is defined by

$${\mbox{\boldmath$R$}}\stackrel{\triangle}{=}{\mbox{\boldmath... ...ldmath$M$}} = {\mbox{\boldmath$K$}}{\mbox{\boldmath$C_u$}} \,,$$ (18)

and we have

$$\begin{array}{rcrl} % Result looks like rccl - don't know wh... ...=&-&{\mbox{\boldmath$R$}}{\mbox{\boldmath$u$}}^-\,. \end{array}$$ (19)

Thus, the wave impedance ${\mbox{\boldmath$R$}}$ is the factor of proportionality between pressure and velocity in a traveling wave. In the cases governed by the ideal wave equation (7), ${\mbox{\boldmath$R$}}$ is diagonal if and only if the mass matrix ${\mbox{\boldmath$M$}}$ is diagonal (since ${\mbox{\boldmath$C_p$}}$ 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$}}^{-1}$.

A linear propagation medium in the discrete-time case is completely determined by its wave impedance ${\mbox{\boldmath$R$}}(z,{\mbox{\boldmath$x$}})$ which, in a generalized formulation, is frequency dependent and spatially varying. Examples of such general cases will be given in the sections that follow. 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 ${\mbox{\boldmath$x$}}$, or else it varies smoothly with ${\mbox{\boldmath$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$}}(z)$, which is intended to be an $m \times m$ function of the complex variable $z$, analytic for $\vert z\vert > 1$.

The generalized version of (20) is

$$\begin{array}{rcrl} {\mbox{\boldmath$p$}}^+&=& & {\mbox{\bol... ...box{\boldmath$R$}}^*(1/z^*) {\mbox{\boldmath$u$}}^- \end{array}$$ (20)

where ${\mbox{\boldmath$R$}}^*(1/z^*)$ is the paraconjugate of ${\mbox{\boldmath$R$}}(z)$, i.e., the unique analytic continuation (when it exists) from the unit circle to the complex plane of the conjugate transposed of ${\mbox{\boldmath$R$}}(z)$ [39].