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Physical Audio Signal Processing
Digital Waveguide Theory
The Lossy 1D Wave Equation

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The 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 $\mu$ , we obtain the modified wave equation

$\displaystyle Ky''= \epsilon {\ddot y}+ \mu{\dot y} \protect$

(H.21)

Thus, the wave equation has been extended by a ``first-order'' term, i.e., a term proportional to the first derivative of

with respect to time. More realistic loss approximations would append terms proportional to ${\dddot y}$ , ${\partial^5 y/\partial t^5}$ , and so on, giving frequency-dependent losses.

Setting $y(t,x) = e^{st+vx}$ in the wave equation to find the relationship between temporal and spatial frequencies in the eigensolution, the wave equation becomes

$\displaystyle K\left(v^2 y\right)$	$\displaystyle =$	$\displaystyle \epsilon \left(s^2 y\right)+ \mu\left(s y\right) \,\,\Rightarrow\,\,Kv^2 = \epsilon s^2 + \mu s$
$\displaystyle \,\,\Rightarrow\,\,v^2$	$\displaystyle =$	$\displaystyle \frac{\epsilon }{K} s^2 + \frac{\mu}{K} s = \frac{\epsilon }{K} s... ...on s}} \right) \isdef \frac{s^2}{c^2}\left(1 + {\frac{\mu}{\epsilon s}} \right)$
$\displaystyle \,\,\Rightarrow\,\,v$	$\displaystyle =$	$\displaystyle \pm \frac{s}{c} \sqrt{1 + {\frac{\mu}{\epsilon s}}}$

where $c \isdef \sqrt{K/\epsilon }$ is the wave velocity in the lossless case. At high frequencies (large $\vert s\vert$ ), or when the friction coefficient $\mu$ is small relative to the mass density $\epsilon$ at the lowest frequency of interest, we have the approximation

$\displaystyle \left(1 + {\frac{\mu}{\epsilon s}}\right)^\frac{1}{2} \approx 1 + \frac{1}{2}{\frac{\mu}{\epsilon s}}$

(H.22)

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