Perhaps the most straightforward multivariable generalization of
(2) and (3) is

in the spatial coordinates at time , where and are non-singular matrices playing the respective roles of multidimensional mass and tension. Differentiating (5) with respect to and (6) with respect to , and eliminating the term yields the -variable generalization of the wave equation

The second spatial derivative is defined here as

(8) |

Similarly, differentiating (5) with respect to and
(6) with respect to
, and eliminating
yields

For digital waveguide modeling, we desire solutions of the
multivariable wave equation involving only sums of traveling waves.
Consider the eigenfunction

where is the diagonal matrix of sound-speeds along the coordinate axes. Since , we have

Substituting (12) into (10), the eigensolutions of (7) are found to be of the form

Similarly, applying (10) to (9) yields

The generalized sound-speed matrices and are the same whenever and

Having established that (13) is a solution of
(7) when condition (11) holds on the matrices
and
, we can express the general traveling-wave
solution to (7) in both pressure and velocity as

When the mass and tension matrices and are diagonal, our analysis corresponds to considering separate waveguides as a whole. For example, the two transversal planes of vibration in a string can be described by (7) with . In a musical instrument such as the piano [29], the coupling among the strings and between different vibration modalities within a single string, occurs primarily at the bridge [30]. Indeed, the bridge acts like a junction of several multivariable waveguides (see section IV).

When the matrices
and
are
non-diagonal, the physical interpretation can be of the form

Besides the existence of physical systems that support multivariable traveling wave solutions, there are other practical reasons for considering a multivariable formulation of wave propagation. For instance, modal analysis considers the vector (whose dimension is infinite in general) of coefficients of the normal mode expansion of the system response. For spaces in perfectly reflecting enclosures, can be compacted so that each element accounts for all the modes sharing the same spatial dimension [32]. admits a wave decomposition as in (16), and is diagonal. Having walls with finite impedance, there is a damping term proportional to that functions as a coupling term among the ideal modes [33]. Coupling among the modes can also be exerted by diffusive properties of the enclosure [32,9].

Note that the multivariable wave equation (7)
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 [27]. However, it turns out that a good
simulation of wave propagation in a multidimensional medium may in
fact be obtained by forming a *mesh* of unidirectional
waveguides as considered here, each described by (7);
such a mesh of 1D waveguides can be shown to solve numerically a
discretized wave equation for multidimensional media
[34,35,13,14].

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