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 , we obtain the modified wave equation
in the wave equation to find the relationship
between temporal and spatial frequencies in the eigensolution, the wave
Setting and using superposition to build up arbitrary traveling wave shapes, we obtain the general class of solutions
Sampling these exponentially decaying traveling waves at intervals of seconds (or meters) gives
The simulation diagram for the lossy digital waveguide is shown in Fig.C.5.
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 meters in the simulation. The loss factor 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.