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Physical Audio Signal Processing
    Digital Waveguide Theory
       Digital Waveguide Mesh
          2D Mesh and the Wave Equation

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2D Mesh and the Wave Equation

Figure C.33: Region of nodes in a rectilinear waveguide mesh.
\includegraphics{eps/mesh}

Figure C.34: Zoom-in about node $ (l,m)$ at time $ n$ in a rectilinear waveguide mesh, showing traveling-wave components entering and leaving the node. (All variables are at time $ n$,)
\includegraphics{eps/meshzoom}

Consider the 2D rectilinear mesh, with nodes at positions $ x=lX$ and $ y=mY$, where $ l$ and $ m$ are integers, and $ X$ and $ Y$ denote the spatial sampling intervals along $ x$ and $ y$, respectively (see Fig.C.33). Then from Eq.$ \,$(C.105) the junction velocity $ v_{lm}$ at time $ n$ is given by

$\displaystyle v_{lm}(n) = \frac{1}{2}\left[ v_{lm}^{+\textsc{n}}(n) + v_{lm}^{+\textsc{e}}(n) + v_{lm}^{+\textsc{s}}(n) + v_{lm}^{+\textsc{w}}(n)\right] $

where $ v_{lm}^{+\textsc{n}}(n)$ is the ``incoming wave from the north'' to node $ (l,m)$, and similarly for the waves coming from east, west, and south (see Fig.C.34).

These incoming traveling-wave components arrive from the four neighboring nodes after a one-sample propagation delay. For example, $ v_{lm}^{+\textsc{n}}(n)$, arriving from the north, departed from node $ (l,m+1)$ at time $ n-1$, as $ v_{l,m+1}^{-\textsc{s}}(n-1)$. Furthermore, the outgoing components at time $ n$ will arrive at the neighboring nodes one sample in the future at time $ n+1$. For example, $ v_{lm}^{-\textsc{n}}(n)$ will become $ v_{l,m+1}^{+\textsc{s}}(n+1)$. Using these relations, we can write $ v_{lm}(n+1)$ in terms of the four outgoing waves from its neighbors at time $ n$:

$\displaystyle v_{lm}(n+1) = \frac{1}{2}\left[ v_{l,m+1}^{-\textsc{s}}(n) + v_{l... ...}}(n) + v_{l,m-1}^{-\textsc{n}}(n) + v_{l-1,m}^{-\textsc{e}}(n)\right] \protect$ (C.116)

where, for instance, $ v_{lm}^{-\textsc{n}}(n)$ is the ``outgoing wave to the north'' from node $ (l,m)$. Similarly, the outgoing waves leaving $ v_{lm}(n-1)$ become the incoming traveling-wave components of its neighbors at time $ n$:

$\displaystyle v_{lm}(n-1) = \frac{1}{2}\left[ v_{l,m+1}^{+\textsc{s}}(n) + v_{l... ...}}(n) + v_{l,m-1}^{+\textsc{n}}(n) + v_{l-1,m}^{+\textsc{e}}(n)\right] \protect$ (C.117)

This may be shown in detail by writing

\begin{eqnarray*} v_{lm}(n-1) &=& \frac{1}{2}[v_{lm}^{+\textsc{n}}(n-1) + \cdot... ...}^{-\textsc{n}}(n-1) + \cdots + v_{lm}^{-\textsc{w}}(n-1)\right] \end{eqnarray*}

so that

\begin{eqnarray*} v_{lm}(n-1) &=& \frac{1}{2}[v_{lm}^{-\textsc{n}}(n-1) + \cdot... ... v_{l,m-1}^{+\textsc{n}}(n) + v_{l-1,m}^{+\textsc{e}}(n)\right]. \end{eqnarray*}

Adding Equations (C.116-C.116), replacing terms such as $ v_{l,m+1}^{+\textsc{s}}(n) + v_{l,m+1}^{-\textsc{s}}(n)$ with $ v_{l,m+1}(n)$, yields a computation in terms of physical node velocities:

\begin{eqnarray*} \lefteqn{v_{lm}(n+1) + v_{lm}(n-1) = } \\ & & \frac{1}{2}\left[ v_{l,m+1}(n) + v_{l+1,m}(n) + v_{l,m-1}(n) + v_{l-1,m}(n)\right] \end{eqnarray*}

Thus, the rectangular waveguide mesh satisfies this equation giving a formula for the velocity at node $ (l,m)$, in terms of the velocity at its neighboring nodes one sample earlier, and itself two samples earlier. Subtracting $ 2v_{lm}(n)$ from both sides yields

\begin{eqnarray*} \lefteqn{v_{lm}(n+1) - 2 v_{lm}(n) + v_{lm}(n-1)} \\ &=& \fra... .... \left[v_{l+1,m}(n) - 2 v_{lm}(n) + v_{l-1,m}(n)\right]\right\} \end{eqnarray*}

Dividing by the respective sampling intervals, and assuming $ X=Y$ (square mesh-holes), we obtain

\begin{eqnarray*} \lefteqn{\frac{v_{lm}(n+1) - 2 v_{lm}(n) + v_{lm}(n-1)}{T^2}} ... ...ft.\frac{v_{l+1,m}(n) - 2 v_{lm}(n) + v_{l-1,m}(n)}{X^2}\right]. \end{eqnarray*}

In the limit, as the sampling intervals $ X,Y,T$ approach zero such that $ X/T = Y/T$ remains constant, we recognize these expressions as the definitions of the partial derivatives with respect to $ t$, $ x$, and $ y$, respectively, yielding

$\displaystyle \frac{\partial^2 v(t,x,y)}{\partial t^2} = \frac{X^2}{2T^2} \left... ...^2 v(t,x,y)}{\partial x^2} + \frac{\partial^2 v(t,x,y)}{\partial y^2} \right]. $

This final result is the ideal 2D wave equation $ \ddot v = c^2 \nabla^2 v$, i.e.,

$\displaystyle \frac{\partial^2 v}{\partial t^2} = c^2 \left[ \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right] $

with

$\displaystyle c = \frac{1}{\sqrt{2}}\frac{X}{T} = \frac{\sqrt{2}}{2}\frac{X}{T}. \protect$ (C.118)

Discussion regarding solving the 2D wave equation subject to boundary conditions appears in §B.8.3. Interpreting this value for the wave propagation speed $ c$, we see that every two time steps of $ 2T$ seconds corresponds to a spatial step of $ \sqrt{2}X$ meters. This is the distance from one diagonal to the next in the square-hole mesh. We will show later that diagonal directions on the mesh support exact propagation (of plane waves traveling at 45-degree angles with respect to the $ x$ or $ y$ axes). In the $ x$ and $ y$ directions, propagation is highly dispersive, meaning that different frequencies travel at different speeds. The exactness of 45-degree angles can be appreciated by considering Huygens' principle on the mesh.

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Next: The Lossy 2D Mesh

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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