A straightforward multivariable generalization of the telegrapher's
equations (2) and (3) gives the following -variable
generalization of the wave equation (5):
For digital waveguide modeling, we desire solutions of the multivariable
wave equation which involve only sums of traveling waves, because traveling
wave propagation can be efficiently simulated digitally using only delay
lines, digital filters, and scattering junctions. Consider the
Having established that (10) is a solution of
(5) when condition (8) holds for the matrices
, we can express the general traveling-wave
solution to (5) in both pressure and velocity as
When the mass and stiffness matrices and are diagonal, our analysis corresponds to considering separate waveguides as a whole. For example, the three directions of vibration (one longitudinal and two transverse) in a single terminated string can be described by (5) with . The coupling among the strings occurs primarily at the bridge in a piano . As we will see later, the bridge acts like a junction of several multivariable waveguides.
When the matrices
non-diagonal, the physical interpretation can be of the form
Note that the multivariable wave equation (5) considered here does not include wave equations governing propagation in multidimensional media (such as membranes, spaces, and solids). In higher dimensions, the solution in the ideal linear lossless case is a superposition of waves traveling in all directions in the -dimensional space . However, it turns out  that a good simulation of wave propagation in a multidimensional medium may be in fact be obtained by forming a mesh of unidirectional waveguides as considered here, each described by (5). Such a mesh of 1D waveguides can be shown to solve numerically a discretized wave equation for multidimensional media .