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 , we obtain the modified wave equation

Thus, the wave equation has been extended by a ``first-order'' term,

Setting
in the wave equation to find the relationship
between temporal and spatial frequencies in the eigensolution, the wave
equation becomes

where is the wave velocity in the lossless case. At high frequencies (large ), or when the friction coefficient is small relative to the mass density at the lowest frequency of interest, we have the approximation

(C.22) |

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

(C.23) |

The eigensolution is then

(C.24) |

The right-going part of the eigensolution is

(C.25) |

which

(C.26) |

which also decays exponentially in the direction of travel.

Setting and using superposition to build up arbitrary traveling wave shapes, we obtain the general class of solutions

(C.27) |

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.

- Loss Consolidation
- Frequency-Dependent Losses
- Well Posed PDEs for Modeling Damped Strings
- Digital Filter Models of Damped Strings
- Lossy Finite Difference Recursion

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University