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DWNs and Numerical Integration

To answer this question, let us consider the two-dimensional waveguide mesh at a junction with coordinates $ x=i\Delta$ and $ y=j\Delta$, for integer $ i$ and $ j$. The discrete-time junction pressure $ p_{J, i, j}(n)$ at time $ t = nT$, for integer $ n$ (recall that our digital waveguide network operates with a sampling period of $ T$), can be written in terms of the four incident wave variables at the same location, from (1.14), as

$\displaystyle p_{J,i,j}(n) = \frac{1}{2}\left(p_{N,i,j}^{+}(n)+p_{E,i,j}^{+}(n)+p_{S,i,j}^{+}(n)+p_{W,i,j}^{+}(n)\right)$    

By tracing the propagation of the wave variables through the network backwards in time through two time steps, it is in fact possible to write a recursion in terms of the junction pressures alone,

$\displaystyle p_{J,i,j}(n)+p_{J,i,j}(n-2) = \frac{1}{2}\Big(p_{J,i-1,j}(n-1)+p_{J,i+1,j}(n-1)+p_{J,i,j-1}(n-1)+p_{J,i,j+1}(n-1)\Big)$ (1.17)

Assume, for the moment, that these discrete time and space junction pressure signals are in fact samples of a continuous function $ p(x,y,t)$ of $ x$, $ y$ and $ t$. Expanding the terms in the recursion above in Taylor series about the location with coordinates $ x=i\Delta$ and $ y=j\Delta$, at time $ t = (n-1)T$ gives

$\displaystyle T^{2}\left.\frac{\partial^{2} p}{\partial t^{2}}\right\vert _{x,y...
...frac{\partial^{2} p}{\partial y^{2}}\right)\right\vert _{x,y,t-T}+O(\Delta^{4})$    

Recalling that $ \Delta = \gamma T$, where $ \gamma$ is the speed of wave propagation in the one-dimensional tubes, and discarding higher-order terms in $ T$ and $ \Delta$ (they are assumed to be small), we get

$\displaystyle \frac{\partial^{2} p}{\partial t^{2}} = \tilde{\gamma}^{2}\left(\frac{\partial^{2} p}{\partial x^{2}}+\frac{\partial^{2} p}{\partial y^{2}}\right)$ (1.18)

This is simply the two-dimensional wave equation, with the wave speed $ \tilde{\gamma}$ defined by

$\displaystyle \tilde{\gamma} = \gamma/\sqrt{2}$ (1.19)

This equation describes wave propagation in a lossless two-dimensional acoustic medium, and the DWN of Figure 1.8(b) can thus be considered to be a numerical integrator of this equation, assuming the wave speeds in the tubes are set according to (1.19); the discrete Huygens' principle interpretation of the behavior of the mesh is justified, at least in the limit as $ T$ and $ \Delta$ become small% latex2html id marker 78450
\setcounter{footnote}{2}\fnsymbol{footnote}. The recursion (1.17) in the junction pressures, however, can be seen as a simple finite difference scheme which could have been derived directly from (1.18) by replacing the partial derivatives by differences between values of a grid function $ p_{i,j}(n)$ on a numerical grid. Because the DWN operates using wave variables, we can see that the DWN is simply a different organization of the same calculation; in particular, it has been put into a form for which all operations (scattering, and shifting) rigidly enforce conservation of energy, in a discrete sense.

We can also reconsider the Kelly-Lochbaum model in this light; forgetting, for the moment, about the approximation of the tube by a series of concatenated uniform tubes, it is possible to write the equations of motion for the gas in the tube directly [145] as

$\displaystyle \begin{eqnarray}\frac{\rho}{A(x)}\frac{\partial u}{\partial t} + ...{\partial p}{\partial t} + \frac{\partial u}{\partial x} &=& 0 \end{eqnarray}$ (1.20a)

subject to initial conditions and boundary conditions at the glottis and lips. This system is identical in form to (1.1) for a uniform tube, except for the variation in $ x$ of the cross-sectional area. It can be condensed to a single second-order equation in the pressure alone,

$\displaystyle \frac{\partial^{2}p}{\partial t^{2}} = \frac{\gamma^{2}}{A(x)}\frac{\partial}{\partial x}\left(A(x)\frac{\partial p}{\partial x}\right)$ (1.21)

which is sometimes called Webster's horn equation [15,30,66]. Due to the variation in the cross-sectional area, it is not equivalent to the one-dimensional wave equation (1.3), and does not possess a simple solution in terms of traveling waves (which is why we needed a concatenated uniform tube model in the first place). Returning now to the DWN of Figure 1.4, it can be shown that the junction pressures $ p_{J,i}(n)$ (at spatial locations $ x=i\Delta$ and at time $ t = nT$ for $ i$ and $ n$ integer) satisfy a recursion of the form

$\displaystyle p_{J,i}(n)+p_{J,i}(n-2) = \frac{2}{Y_{i}+Y_{i+1}}\Big(Y_{i}p_{i-1}(n-1)+Y_{i+1}p_{i+1}(n-1)\Big)$ (1.22)

where $ Y_{i}$ is the admittance of the $ i$th acoustic tube, running from $ x=(i-1)\Delta$ to $ x=i\Delta$. With the $ Y_{i}$ set according to (1.7), it is again possible to show that (1.22) is a finite difference scheme for (1.21), with $ \Delta = \gamma T$.

next up previous
Next: A General Approach: Multidimensional Up: Digital Waveguide Networks Previous: Waveguide Meshes and the
Stefan Bilbao 2002-01-22