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Wave Equation Interpretation
The second-order PDE describing the voltage distribution
along an electrical transmission line with constant inductance and capacitance
and
per unit length and which runs parallel to the
-axis is
![$\displaystyle \frac{\partial^{2} u}{\partial t^{2}} = \gamma^{2}\frac{\partial^{2} u}{\partial x^{2}}$](img1147.png) |
(4.7) |
where the wave speed
is
. 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:
![$\displaystyle u(x,t) = u^{l}(x+\gamma t)+u^{r}(x-\gamma t)$](img1148.png) |
(4.8) |
That is, the solution at any time
is made up of a sum of two shifted copies of the initializing functions
and
, which have traveled to the left and right respectively with velocity
over a distance
. For any
we have, for the leftward-traveling wave, the identity
![$\displaystyle u^{l}(x+\gamma t) = u^{l}\big((x+\Delta) + \gamma(t - \Delta/\gamma)\big)$](img1152.png) |
(4.9) |
If we set
, then at time
,
![$\displaystyle u^{l}(x+\gamma nT) = u^{l}\big((x+\Delta) + \gamma(n-1)T)\big)$](img1154.png) |
(4.10) |
Associate now with a particular waveguide a delay
and a physical length
, so that in Figure 4.1
represents an outgoing voltage wave quantity at position
, and
an incoming wave at position
. It is then clear that if we have
, then (4.10) is equivalent to the second equation of (4.1), with
, and with
and
. A similar correspondence holds for the right-going traveling wave component
and the wave variables at either end of the rightward waveguide,
and
. A chain of bidirectional delay lines, connected in cascade will then implement an exact traveling wave solution to the wave equation. The physical voltage
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
, 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: Note on the Different
Up: Digital Waveguides
Previous: Impedance
Stefan Bilbao
2002-01-22