Damped Traveling Waves

Figure 1.3: Damped traveling-wave simulator. In addition to propagation delay by $M$ samples, there is attenuation by $g^M<1$.

The delay line shown in Fig. 1.1 on page  can be used to simulate any traveling wave, where the traveling wave must propagate in one direction with a fixed waveshape. If a traveling wave attenuates as it propagates, with the same attenuation factor at each frequency, the attenuation can be simulated by a simple scaling of the delay line output (or input), as shown in Fig. 1.3. This is perhaps the simplest example of the important principle of lumping distributed losses at discrete points. That is, it is not necessary to implement a small attenuation $g$ for each time-step of wave propagation; the same result is obtained at the delay-line output if propagation is ``lossless'' within the delay line, and the total cumulative attenuation $g^M$ is applied at the output. The input-output simulation is exact, while the signal samples inside the delay line are simulated with a slight gain error. If the internal signals are needed later, they can be tapped out using correcting gains. For example, the signal half way along the delay line can be tapped using a coefficient of $g^{M/2}$ in order to make it an exact second output. In summary, computational efficiency can often be greatly increased at no cost to accuracy by lumping losses only at the outputs and points of interaction with other simulations.

Modeling traveling-wave attenuation by a scale factor is only exact physically when all frequency components decay at the same rate. For accurate acoustic modeling, it is usually necessary to replace the constant scale factor $g$ by a digital filter $G(z)$ which implements frequency-dependent attenuation, as depicted in Fig. 1.4. In principle, a linear time-invariant (LTI) filter can provide an independent attenuation factor at each frequency. Section 1.3 addresses this case in more detail.

Figure 1.4: Damped traveling-wave simulator. In principle, the digital filter $G(z)$ provides an arbitrary attenuation at each frequency. Physically, we must have $\left \vert G(e^{j\omega T})\right \vert\leq 1$ for all $\omega$.