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Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          Longitudinal Waves in Rods

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Longitudinal Waves in Rods

In this section, elementary scattering relations will be derived for the case of longitudinal force and velocity waves in an ideal string or rod. In solids, force-density waves are referred to as stress waves [172,268]. Longitudinal stress waves in strings and rods have units of (compressive) force per unit area and are analogous to longitudinal pressure waves in acoustic tubes.

Figure: A waveguide section between two partial sections. a) Physical picture indicating traveling waves in a continuous medium whose wave impedance changes from $ R_0$ to $ R_1$ to $ R_2$. b) Digital simulation diagram for the same situation. The section propagation delay is denoted as $ z^{-T}$. The behavior at an impedance discontinuity is characterized by a lossless splitting of an incoming wave into transmitted and reflected components.
\includegraphics[width=\twidth]{eps/Fwgfs}

A single waveguide section between two partial sections is shown in Fig.H.20. The sections are numbered 0 through $ 2$ from left to right, and their wave impedances are $ R_0$, $ R_1$, and $ R_2$, respectively. Such a rod might be constructed, for example, using three different materials having three different densities. In the $ i$th section, there are two stress traveling waves: $ f^{{+}}_i$ traveling to the right at speed $ c$, and $ f^{{-}}_i$ traveling to the left at speed $ c$. To minimize the numerical dynamic range, velocity waves may be chosen instead when $ R_i>1$.

As in the case of transverse waves (see the derivation of (H.46)), the traveling longitudinal plane waves in each section satisfy [172,268]

\begin{displaymath}\begin{array}{rcrl} f^{{+}}_i(t)&=&&R_iv^{+}_i(t) \\ f^{{-}}_i(t)&=&-&R_iv^{-}_i(t) \end{array}\end{displaymath} (H.59)

where the wave impedance is now $ R_i=\sqrt{E\rho}$, with $ \rho$ being the mass density, and