Multivariable Formulation of the Acoustic Tube

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

$$\frac{\partial^2 {\mbox{\boldmath$p$}}({\mbox{\boldmath$x$}}... ...}}({\mbox{\boldmath$x$}},t)}{\partial {\mbox{\boldmath$x$}}^2}$$ (5)

in the spatial coordinates at time $t$, where ${\mbox{\boldmath$M$}}$ and ${\mbox{\boldmath$K$}}$ are $m \times m$ non-singular, non-negative matrices which play the respective roles of multidimensional mass and stiffness. The second spatial derivative is defined here as

$$\left[ \frac{\partial^2{\mbox{\boldmath$p$}}({\mbox{\boldmat... ...{\mbox{\boldmath$x$}},t)}{\partial x_m^2} \end{array}\right]$$ (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

$${\mbox{\boldmath$p$}}({\mbox{\boldmath$x$}},t) = \left[ \beg... ...oldmath$V$}}{\mbox{\boldmath$X$}}} \cdot {\mbox{\boldmath$1$}}$$ (7)

where $s$ is interpreted as a Laplace-transform variable $s=\sigma+j\omega$, ${\mbox{\boldmath$I$}}$ is the $m \times m$ identity matrix, ${\mbox{\boldmath$X$}}{\tiny\stackrel{\triangle}{=}} \hbox{diag}({\mbox{\boldmath$x$}})$, ${\mbox{\boldmath$V$}}{\tiny\stackrel{\triangle}{=}}\hbox{diag}([v_1, \ldots, v_m])$ is a diagonal matrix of spatial Laplace-transform variables (the imaginary part of $v_i$ being spatial frequency along the $i$th spatial coordinate), and ${\mbox{\boldmath$1$}}^T {\tiny\stackrel{\triangle}{=}}[1, \ldots, 1]$. Substituting the eigenfunction (7) into (5) gives the algebraic equation

$$s^2{\mbox{\boldmath$I$}}= {\mbox{\boldmath$K$}}{\mbox{\boldm... ...l{\triangle}{=}{\mbox{\boldmath$C$}}^2 {\mbox{\boldmath$V$}}^2$$ (8)

where ${\mbox{\boldmath$C$}}$ is the diagonal matrix of sound-speeds along the $m$ coordinate axes. Since ${\mbox{\boldmath$C$}}^2{\mbox{\boldmath$V$}}^2 = s^2{\mbox{\boldmath$I$}}$, we have

$${\mbox{\boldmath$V$}}=\pm s {\mbox{\boldmath$C$}}^{-1}.$$ (9)

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

$${\mbox{\boldmath$p$}}({\mbox{\boldmath$x$}}, t) = e^{s\left(... ...^{-1}{\mbox{\boldmath$X$}}\right)} \cdot {\mbox{\boldmath$1$}}$$ (10)

Having established that (10) is a solution of (5) when condition (8) holds for the matrices ${\mbox{\boldmath$M$}}$ and ${\mbox{\boldmath$K$}}$, we can express the general traveling-wave solution to (5) in both pressure and velocity as

$$\begin{array}{l} {\mbox{\boldmath$p$}}({\mbox{\boldmath$x$}... ...t)={\mbox{\boldmath$u$}}^++ {\mbox{\boldmath$u$}}^- \end{array}$$ (11)

where ${\mbox{\boldmath$p$}}^+{\tiny\stackrel{\triangle}{=}}f(t{\mbox{\boldmath$I$}}-{\mbox{\boldmath$C$}}^{-1}{\mbox{\boldmath$X$}})$, with $f$ being an arbitrary superposition of right-going components of the form (10) (i.e., taking the minus sign), and ${\mbox{\boldmath$p$}}^-{\tiny\stackrel{\triangle}{=}}g(t{\mbox{\boldmath$I$}}+{\mbox{\boldmath$C$}}^{-1}{\mbox{\boldmath$X$}})$ is similarly any linear combination of left-going eigensolutions from (10) (all having the plus sign). Similar definitions apply for ${\mbox{\boldmath$u$}}^+$ and ${\mbox{\boldmath$u$}}^-$. When the time and space arguments are dropped as in the right-hand side of (11), it is understood that all the quantities are written for time $t$ and position ${\mbox{\boldmath$x$}}$.

When the mass and stiffness matrices ${\mbox{\boldmath$M$}}$ and ${\mbox{\boldmath$K$}}$ are diagonal, our analysis corresponds to considering $m$ 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 $m=3$. 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 ${\mbox{\boldmath$M$}}$ and ${\mbox{\boldmath$K$}}$ are non-diagonal, the physical interpretation can be of the form

$${{\mbox{\boldmath$C$}}}^2\stackrel{\triangle}{=}{\mbox{\boldmath$K$}}{\mbox{\boldmath$M$}}^{-1}$$ (12)

where ${\mbox{\boldmath$K$}}$ is the stiffness matrix, ${\mbox{\boldmath$M$}}$ is the mass density matrix. ${\mbox{\boldmath$C$}}$ is diagonal if (8) holds, and in this case, the wave equation (5) is decoupled in the spatial dimensions. There are physical examples where the matrices ${\mbox{\boldmath$M$}}$ and ${\mbox{\boldmath$K$}}$ are not diagonal, even though ${{\mbox{\boldmath$C$}}}^2\stackrel{\triangle}{=}{\mbox{\boldmath$K$}}{\mbox{\boldmath$M$}}^{-1}$ is. One such example, in the domain of electrical variables, is given by $m$ conductors in a sheath or above a ground plane, where the sheath or the ground plane acts as a coupling element [63, pp. 67-68].

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 $m$-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].