## Frequency-Dependent Damping

In real vibrating strings, damping typically increases with frequency for a variety of physical reasons [73,77]. A simple modification [392] to Eq.(6.14) yielding frequency-dependent damping is

The result of adding such damping terms to the wave equation is that
traveling waves on the string decay at *frequency-dependent
rates*. This means the loss factors of the previous section
should really be *digital filters* having gains which decrease
with frequency (and never exceed for stability of the loop).
These filters *commute* with delay elements because they are
*linear* and *time invariant* (LTI) [449]. Thus,
following the reasoning of the previous section, they can be lumped at
a single point in the digital waveguide. Let
denote the
resulting *string loop filter* (replacing in
Fig.6.12^{7.7}). We have the stability (passivity)
constraint
, and making the filter
*linear phase* (constant delay at all frequencies) will restrict
consideration to *symmetric* FIR filters only.

Restriction to FIR filters yields the important advantage of keeping
the approximation problem *convex* in the weighted least-squares
norm. Convexity of a norm means that gradient-based search techniques
can be used to find a global miminizer of the error norm without
exhaustive search [64],[428, Appendix A].

The linear-phase requirement halves the degrees of freedom in the filter coefficients. That is, given for , the coefficients are also determined. The loss-filter frequency response can be written in terms of its (impulse response) coefficients as

A further degree of freedom is eliminated from the loss-filter approximation by assuming all losses are insignificant at 0 Hz so that (taking the approximation error to be zero at ). This means the coefficients of the FIR filter must sum to . If the length of the filter is , we have

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The Stiff String

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The Damped Plucked String