Multivariable Formulation

The $m$-variable lossy wave equation is

$$\frac{\partial^2 {\mbox{\boldmath$p$}}({\mbox{\boldmath$x$}}... ...\mbox{\boldmath$x$}},t)}{\partial {\mbox{\boldmath$x$}}^2} \;,$$ (25)

where ${\mbox{\boldmath$\Phi$}}$ is a $m \times m$ matrix that represents a viscous resistance. If we plug the eigensolution (10) into (26), we get, in the Laplace domain

$$s^2{\mbox{\boldmath$I$}}+ s {\mbox{\boldmath$K$}}{\mbox{\bol... ...iangle}{=}{\mbox{\boldmath$C_p$}}^2 {\mbox{\boldmath$V$}}^2 \,$$ (26)

or, by letting

$${\mbox{\boldmath$\Phi$}}{\mbox{\boldmath$K$}}^{-1} \stackrel{\triangle}{=}{\mbox{\boldmath$\Upsilon$}}\; ,$$ (27)

we get

$$s^2 {\mbox{\boldmath$I$}}+ s {\mbox{\boldmath$C_p$}}^2 {\mbo... ...lon$}}= {\mbox{\boldmath$C_p$}}^2 {\mbox{\boldmath$V$}}^2 \; .$$ (28)

By restricting the Laplace analysis to the imaginary (frequency) axis $s = j \omega_s$, decomposing the (diagonal) spatial frequency matrix into its real and imaginary parts ${\mbox{\boldmath$V$}}= {\mbox{\boldmath$V$}}_R + j {\mbox{\boldmath$V$}}_I$, and equating the real and imaginary parts of equation (29), we get the equations

$${{\mbox{\boldmath$V$}}_R}^2 - {{\mbox{\boldmath$V$}}_I}^2 = - {\mbox{\boldmath$C_p$}}^{-2} \omega_s$$ (29)

$$2 {\mbox{\boldmath$V$}}_R {\mbox{\boldmath$V$}}_I = \omega_s {\mbox{\boldmath$\Upsilon$}}\; .$$ (30)

The term ${\mbox{\boldmath$V$}}_R$ can be interpreted as attenuation per unit length, while ${\mbox{\boldmath$V$}}_I$ keeps the role of spatial frequency, so that the traveling wave solution is

$${\mbox{\boldmath$p$}}= e^{{{\mbox{\boldmath$V$}}_R}{\mbox{\b... ...I {\mbox{\boldmath$X$}}\right)} \cdot {\mbox{\boldmath$1$}}\,.$$ (31)

Defining ${\mbox{\boldmath$\Theta$}}$ as the ratio between the real and imaginary parts of ${\mbox{\boldmath$V$}}$ ( ${\mbox{\boldmath$\Theta$}}{\mbox{\boldmath$V$}}_I = {\mbox{\boldmath$V$}}_R$), the equations (30) and (31) become

$${\mbox{\boldmath$V$}}_I^2 = \left({\mbox{\boldmath$I$}}- {\mbox{\boldmath$\Theta$}}^2 \right)^{-1} {\omega_s}^2 {\mbox{\boldmath$C_p$}}^{-2}$$ (32)

$${\mbox{\boldmath$\Upsilon$}} = 2 {\mbox{\boldmath$\Theta$}}\left({\mbox{\boldmath$I$}}- {\mbox{\boldmath$\Theta$}}^2 \right)^{-1} \omega_s {\mbox{\boldmath$C_p$}}^{-2} \; .$$ (33)

Following steps analogous to those of eq. (18), the $m \times m$ admittance matrix turns out to be

$${\mbox{\boldmath$\Gamma$}}= {\mbox{\boldmath$M$}}^{-1} \left... ...}{s} {\mbox{\boldmath$M$}}^{-1} {{\mbox{\boldmath$V$}}_R} \, ,$$ (34)

which, for ${\mbox{\boldmath$\Phi$}}\rightarrow 0$, collapses to the reciprocal of (19). 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 transformation [43], 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$}})$.