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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 [318].

Perhaps the apparently simplest modification to Eq.$ \,$(C.21) yielding frequency-dependent damping is to add a third-order time-derivative term [392]:

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

While this model has been successful in practice [77], 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.$ \,$(C.28) is given by

$\displaystyle Ky''= \epsilon {\ddot y}+ \mu{\dot y}+ \mu_2{\dot y''} \protect$ (C.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in which the third-order partial derivative with respect to time, $ {\dddot y}$, has been replaced by a third-order mixed partial derivative--twice with respect to $ x$ and once with respect to $ t$.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.C.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 [392], 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 §C.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 (C.21) is modified by adding terms proportional to $ \mu_3{\dddot y}$ and $ \mu_5{\partial^5 y/\partial t^5}$, for instance, then the per-sample propagation gain $ G(\omega)$ has the form

$\displaystyle G(\omega) = g_0 + g_2 \omega^2 + g_4 \omega^4

where the $ g_i$ are constants depending on the constants $ \mu_3$ and $ \mu_5$ in the wave equation. Since these per-sample loss filters are linear and time-invariant [449], they may also be consolidated at a minimum number of points in the waveguide without introducing any approximation error, just like the constant gains $ g$ in Fig.C.5. This result does not extend precisely to the waveguide meshC.14). In view of the above, we see that we can add odd-order time derivatives to the wave equation to approximate experimentally observed frequency-dependent damping characteristics in vibrating strings [73]. However, we then have the problem that such wave equations are ill posed, leading to possible stability failures at high sampling rates. As a result, it is generally preferable to use mixed derivatives, as in Eq.$ \,$(C.29), and try to achieve realistic damping using higher order spatial derivatives instead.
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Well Posed PDEs for Modeling Damped Strings
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Loss Consolidation