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Chapters

Physical Audio Signal Processing
    Digital Waveguide Theory
       The Lossy 1D Wave Equation
          Frequency-Dependent Losses

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Frequency-Dependent Losses

In nearly all natural wave phenomena, losses increase with frequency. Distributed losses due to air drag and internal bulk losses in the string tend to increase monotonically with frequency. Similarly, air absorption increases with frequency, adding loss for sound waves in acoustic tubes or open air [326].

Perhaps the apparently simplest modification to Eq.$ \,$(H.21) yielding frequency-dependent damping is to add a third-order time-derivative term [398]:

$\displaystyle Ky''= \epsilon {\ddot y}+ \mu{\dot y}+ \mu_3{\dddot y}. \protect$ (H.28)

While this model has been successful in practice [80], it turns out to go unstable at very high sampling rates. The technical term for this problem is that the PDE is ill posed [45].

A well posed replacement for Eq.$ \,$(H.28) is given by

$\displaystyle Ky''= \epsilon {\ddot y}+ \mu{\dot y}+ \mu_2{\dot y''} \protect$ (H.29)

in which the third-order partial derivative with respect to time, $ {\dddot y}$, has been replaced by a third-order mixed partial derivative--twice with respect to $ x$ and once with respect to $ t$.

The solution of a lossy wave equation containing higher odd-order derivatives with respect to time yields traveling waves which propagate with frequency-dependent attenuation. Instead of scalar factors $ g$ distributed throughout the diagram as in Fig.H.5, each $ g$ factor becomes a lowpass filter having some frequency-response per sample denoted by $ G(\omega)$. Because propagation is passive, we will always have $ \left\vert G(\omega)\right\vert\leq 1$.

More specically, As shown in [398], odd-order partial derivatives with respect to time in the wave equation of the form

$\displaystyle \frac{\partial^n}{\partial t^n} y(t,x), \quad n=1,3,5,\ldots, $

correspond to attenuation of traveling waves on the string. (The even-order time derivatives can be associated with variations in dispersion as a function of frequency, which is considered in §H.6 below.) For $ n>1$, the losses are frequency dependent, and the per-sample amplitude-response ``rolls off'' proportional to

$\displaystyle G(\omega) \propto \frac{1}{\omega^{n-1}},$   $\displaystyle \mbox{($n$\ odd)}$$\displaystyle . $

In particular, if the wave equation (H.21) is modified by adding terms proportional to $ \mu_3{\dddot y}$ and