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Multivariable Formulation
The variable lossy wave equation is

(25) 
where
is a matrix that represents a viscous resistance.
If we plug the eigensolution (10) into (26), we get, in the Laplace domain

(26) 
or, by letting

(27) 
we get

(28) 
By restricting the Laplace analysis to the imaginary (frequency) axis
, decomposing the (diagonal) spatial frequency matrix into its real and imaginary parts
, and equating the real and imaginary parts of equation (29), we get the equations
The term
can be interpreted as attenuation per unit length, while
keeps the role of spatial frequency, so that the traveling wave solution is

(31) 
Defining
as the ratio^{} between the real and imaginary parts of
(
), the equations (30) and (31) become
Following steps analogous to those of eq. (18),
the admittance matrix turns out to be

(34) 
which, for
, collapses to the reciprocal of (19).
For the discretetime case, we may map
from the plane to
the plane via the bilinear transformation [43], or we may
sample the inverse Laplace transform of
and take its transform
to obtain
.
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