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Wave Equation Interpretation

The second-order PDE describing the voltage distribution $ u(x,t)$ along an electrical transmission line with constant inductance and capacitance $ l$ and $ c$ per unit length and which runs parallel to the $ x$-axis is

$\displaystyle \frac{\partial^{2} u}{\partial t^{2}} = \gamma^{2}\frac{\partial^{2} u}{\partial x^{2}}$ (4.7)

where the wave speed $ \gamma$ is $ 1/\sqrt{lc}$. As we saw in §1.1.1, the solution to this equation, if we set aside boundary conditions for the moment, can be written in terms of traveling waves:

$\displaystyle u(x,t) = u^{l}(x+\gamma t)+u^{r}(x-\gamma t)$ (4.8)

That is, the solution at any time $ t\geq 0$ is made up of a sum of two shifted copies of the initializing functions $ u^{l}(x)$ and $ u^{r}(x)$, which have traveled to the left and right respectively with velocity $ \gamma$ over a distance $ \gamma t$. For any $ \Delta$ we have, for the leftward-traveling wave, the identity

$\displaystyle u^{l}(x+\gamma t) = u^{l}\big((x+\Delta) + \gamma(t - \Delta/\gamma)\big)$ (4.9)

If we set $ \gamma = \Delta/T$, then at time $ t = nT$,

$\displaystyle u^{l}(x+\gamma nT) = u^{l}\big((x+\Delta) + \gamma(n-1)T)\big)$ (4.10)

Associate now with a particular waveguide a delay $ T$ and a physical length $ \Delta$, so that in Figure 4.1 $ U_{1}^{+}$ represents an outgoing voltage wave quantity at position $ x$, and $ U_{2}^{-}$ an incoming wave at position $ x+\Delta$. It is then clear that if we have $ \Delta/T=\gamma$, then (4.10) is equivalent to the second equation of (4.1), with $ m=1$, and with $ u^{l}(x + \gamma nT) = U_{1}^{+}(n)$ and $ u^{l}(x+\Delta+\gamma(n-1)T) = U_{2}^{-}(n-1)$. A similar correspondence holds for the right-going traveling wave component $ u^{r}$ and the wave variables at either end of the rightward waveguide, $ U_{1}^{-}$ and $ U_{2}^{+}$. A chain of bidirectional delay lines, connected in cascade will then implement an exact traveling wave solution to the wave equation. The physical voltage $ u$ may be obtained (as should be clear from (4.8)) by summing the leftward and rightward traveling components at any particular location in the cascade, as per equation (4.3). Note that because $ \gamma = \Delta/T$, the delay period and the waveguide length cannot be chosen independently, if the discrete wave quantities are to behave as traveling wave solutions to (4.7).

next up previous
Next: Note on the Different Up: Digital Waveguides Previous: Impedance
Stefan Bilbao 2002-01-22