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

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The Dispersive 1D Wave Equation

In the ideal vibrating string, the only restoring force for transverse displacement comes from the string tension (§H.1 above); specifically, the transverse restoring force is equal the net transverse component of the axial string tension. Consider in place of the ideal string a bundle of ideal strings, such as a stranded cable. When the cable is bent, there is now a new restoring force arising from some of the fibers being compressed and others being stretched by the bending. This force sums with that due to string tension. Thus, stiffness in a vibrating string introduces a new restoring force proportional to bending angle. It is important to note that string stiffness is a linear phenomenon resulting from the finite diameter of the string.

In typical treatments,^H.3bending stiffness adds a new term to the wave equation that is proportional to the fourth spatial derivative of string displacement:

$\displaystyle \epsilon {\ddot y}= Ky''- \kappa y'''' \protect$

(H.32)

where the moment constant $\kappa = YI$ is the product of Young's modulus

(the ``relative-displacement spring constant per unit cross-sectional area,'' discussed in §F.4.1) and the area moment of inertia

(§F.4.4); as derived in §F.4.5, a cylindrical string of radius

has area moment of inertia equal to $\pi a^2 \cdot (a/2)^2 = \pi a^4/4$ . This wave equation works well enough for small amounts of bending stiffness, but it is clearly missing some terms because it predicts that deforming the string into a parabolic shape will incur no restoring force due to stiffness. More accurate wave equations for stiff strings are given [172,268].

To solve the stiff wave equation, we may set $y(t,x) = e^{st+vx}$ to get

$\displaystyle \epsilon s^2 = Kv^2 - \kappa v^4.$

At very low frequencies, or when stiffness is negligible in comparison with

, we obtain again the non-stiff string: $\epsilon s^2\approx Kv^2 \,\,\Rightarrow\,\,v=\pm s/c$ .

At very high frequencies, or when the tension is negligible relative to $\kappa v^2$ , we obtain the ideal bar approximation

$\displaystyle \epsilon s^2 \approx -\kappa v^4 \,\,\Rightarrow\,\,v \approx \pm e^{\pm j\frac{\pi}{4}} \left(\frac{\epsilon }{\kappa} \right)^{1/4}\sqrt{s}$

In an ideal bar, the only restoring force is due to bending stiffness. Setting $s=j\omega$ gives solutions