A straightforward multivariable generalization of the telegrapher's
equations (2) and (3) gives the following -variable
generalization of the wave equation (5):

(6) |

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
eigenfunction

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

Substituting (9) into (7), the eigensolutions of (5) are found to be of the form

Having established that (10) is a solution of
(5) when condition (8) holds for the matrices
and
, 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 [132]. As we will see later, the bridge acts like a junction of several multivariable waveguides.

When the matrices
and
are
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 [60]. However, it turns out [122]
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 [125].

Download wgj.pdf

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

[Automatic-links disclaimer]