A Lossy 1D Wave Equation

In any real vibrating string, there are energy losses due to yielding terminations, drag by the surrounding air, and internal friction within the string. While losses in solids generally vary in a complicated way with frequency, they can usually be well approximated by a small number of odd-order terms added to the wave equation. In the simplest case, force is directly proportional to transverse string velocity, independent of frequency. If this proportionality constant is $\mu$ , we obtain the modified wave equation

$$Ky''= \epsilon {\ddot y}+ \mu{\dot y}.$$ (C.21)

Thus, the wave equation has been extended by a ``first-order'' term, i.e., a term proportional to the first derivative of $y$ with respect to time. More realistic loss approximations would append terms proportional to ${\dddot{y}}$ , ${\partial^5 y/\partial t^5}$ , and so on, giving frequency-dependent losses.

Setting $y(t,x) = e^{st+vx}$ in the wave equation to find the relationship between temporal and spatial frequencies in the eigensolution, the wave equation becomes

$$K\left(v^2 y\right) = \epsilon \left(s^2 y\right)+ \mu\left(s y\right) \,\,\Rightarrow\,\,Kv^2 = \epsilon s^2 + \mu s$$

$$\,\,\Rightarrow\,\,v^2 = \frac{\epsilon }{K} s^2 + \frac{\mu}{K} s = \frac{\epsilon }{K} s^2\left(1 + {\frac{\mu}{\epsilon s}} \right) \isdef \frac{s^2}{c^2}\left(1 + {\frac{\mu}{\epsilon s}} \right)$$

$$\,\,\Rightarrow\,\,v = \pm \frac{s}{c} \sqrt{1 + {\frac{\mu}{\epsilon s}}}$$

where $c \isdef \sqrt{K/\epsilon }$ is the wave velocity in the lossless case. At high frequencies (large $\vert s\vert$ ), or when the friction coefficient $\mu$ is small relative to the mass density $\epsilon$ at the lowest frequency of interest, we have the approximation

$$\left(1 + {\frac{\mu}{\epsilon s}}\right)^\frac{1}{2} \approx 1 + \frac{1}{2}{\frac{\mu}{\epsilon s}}$$ (C.22)

by the binomial theorem. For this small-loss approximation, we obtain the following relationship between temporal and spatial frequency:

$$v \approx \pm \frac{1}{c}\left({s + \frac{\mu}{2\epsilon }} \right)$$ (C.23)

The eigensolution is then

$$e^{st+vx} = \exp{\left[st\pm \left({s + \frac{\mu}{2\epsilon }}\right)\frac{x}{c}\right]} = \exp{\left[s\left(t\pm \frac{x}{c}\right)\right]} \exp{\left(\pm\frac{\mu}{2\epsilon }\frac{x}{c}\right)}.$$ (C.24)

The right-going part of the eigensolution is

$$e^{-{\left(\mu/2\epsilon \right)}{x/c}} e^{s \left(t - {x/c}\right)}$$ (C.25)

which decays exponentially in the direction of propagation, and the left-going solution is

$$e^{{\left(\mu/2\epsilon \right)}{x/c}} e^{s \left(t + {x/c}\right)}$$ (C.26)

which also decays exponentially in the direction of travel.

Setting $s=j\omega$ and using superposition to build up arbitrary traveling wave shapes, we obtain the general class of solutions

$$y(t,x) = e^{-{\left(\mu/2\epsilon \right)}{x/c}} y_r\left(t-{x/c}\right) + e^{{\left(\mu/2\epsilon \right)}{x/c}} y_l\left(t+{x/c}\right).$$ (C.27)

Sampling these exponentially decaying traveling waves at intervals of $T$ seconds (or $X=cT$ meters) gives

$$\begin{eqnarray*} y(t_n,x_m) &=& e^{-{\left(\mu/2\epsilon \right)}{x_m/c}} y_r\left(t_n-{x_m/c}\right) + e^{{\left(\mu/2\epsilon \right)}{x_m/c}} y_l\left(t_n+{x_m/c}\right)\\ &=& e^{-{\left(\mu/2\epsilon \right)}{mX/c}} y_r\left(nT-{mX/c}\right) + e^{{\left(\mu/2\epsilon \right)}{mX/c}} y_l\left(nT+{mX/c}\right)\\ &=& e^{-{\mu mT/2\epsilon }} y_r\left[(n-m)T\right] + e^{ {\mu mT/2\epsilon }} y_l\left[(n+m)T\right]\\ &=& \left(e^{-{\mu T/2\epsilon }}\right)^m y^{+}(n-m) + \left(e^{ {\mu T/2\epsilon }}\right)^m y^{-}(n+m) \\ &\isdef & g^{m} y^{+}(n-m) + g^{-m} y^{-}(n+m). \end{eqnarray*}$$

The simulation diagram for the lossy digital waveguide is shown in Fig.C.5.

Figure C.5: Discrete simulation of the ideal, lossy waveguide.

Again the discrete-time simulation of the decaying traveling-wave solution is an exact implementation of the continuous-time solution at the sampling positions and instants, even though losses are admitted in the wave equation. Note also that the losses which are distributed in the continuous solution have been consolidated, or lumped, at discrete intervals of $cT$ meters in the simulation. The loss factor $g = e^{-{\mu T/2\epsilon }}$ summarizes the distributed loss incurred in one sampling interval. The lumping of distributed losses does not introduce an approximation error at the sampling points. Furthermore, bandlimited interpolation can yield arbitrarily accurate reconstruction between samples. The only restriction is again that all initial conditions and excitations be bandlimited to below half the sampling rate.