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 [170,263]. 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 with wave impedance changing from $R_0$ to $R_1$ to $R_2$ along the horizontal axis, resulting in signal scattering (i.e., power-conserving transmission and reflection). b) Digital simulation diagram for the same situation. The section propagation delay is denoted as $z^{-T}$ . At an impedance discontinuity, such as from $R_0$ to $R_1$ , a propagating wave splits losslessly into transmitted and reflected components. The force-wave reflection coefficients are denoted by $r_i=(R_i-R_{i-1})/(R_i+R_{i-1})$ to the right, with corresponding transmission coefficients $1+r_i$ to the right. To the left, the impedance step negates, so the reflection coefficients negate for waves propagating to the left.

A single waveguide section between two partial sections is shown in Fig.C.19. 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 (C.46)), the traveling longitudinal plane waves in each section satisfy [170,263]

$$\begin{array}{rcrl} f^{{+}}_i(t)&=&&R_iv^{+}_i(t) \\ f^{{-}}_i(t)&=&-&R_iv^{-}_i(t) \end{array}$$ (C.57)

where the wave impedance is now $R_i=\sqrt{E\rho}$ , with $\rho$ being the mass density, and $E$ being the Young's modulus of the medium (defined as the stress over the strain, where strain means relative displacement--see §B.5.1) [170,263]. As before, velocity $v_i=v^{+}_i+v^{-}_i$ is defined as positive to the right, and $f^{{+}}_i$ is the right-going traveling-wave component of the stress, and it is positive when the rod is locally compressed.

If the wave impedance $R_i$ is constant, the shape of a traveling wave is not altered as it propagates from one end of a rod-section to the other. In this case we need only consider $f^{{+}}_i$ and $f^{{-}}_i$ at one end of each section as a function of time. As shown in Fig.C.19, we define $f^\pm _i(t)$ as the force-wave component at the extreme left of section $i$ . Therefore, at the extreme right of section $i$ , we have the traveling waves $f^{{+}}_i(t-T)$ and $f^{{-}}_i(t+T)$ , where $T$ is the travel time from one end of a section to the other.

For generality, we may allow the wave impedances $R_i$ to vary with time. A number of possibilities exist which satisfy (C.57) in the time-varying case. For the moment, we will assume the traveling waves at the extreme right of section $i$ are still given by $f^{{+}}_i(t-T)$ and $f^{{-}}_i(t+T)$ . This definition, however, implies the velocity varies inversely with the wave impedance. As a result, signal energy, being the product of force times velocity, is ``pumped'' into or out of the waveguide by a changing wave impedance. Use of normalized waves $\tilde{f}^\pm _i$ avoids this. However, normalization increases the required number of multiplications, as we will see in §C.8.6 below.

As before, the physical force density (stress) and velocity at the left end of section $i$ are obtained by summing the left- and right-going traveling wave components:

$$f_i = f^{{+}}_i+ f^{{-}}_i$$ (C.58)

$$v_i = v^{+}_i+ v^{-}_i$$

Let $f_i(t,x_i)$ denote the force at position $x_i$ and time $t$ in section $i$ , where $x_i\in[0,cT]$ is measured from the extreme left of section $i$ along its axis. Then we have, for example, $f_i(t,0)\isdef f^{{+}}_i(t)+f^{{-}}_i(t)$ and $f_i(t,cT)\isdef f^{{+}}_i(t-T)+f^{{-}}_i(t+T)$ at the boundaries of section $i$ . More generally, within section $i$ , the physical stress may be expressed in terms of its traveling-wave components by

$$f_i(t,x_i) \eqsp f^{{+}}_i\left(t-\frac{x_i}{c}\right)+f^{{-}}_i\left(t+\frac{x_i}{c}\right), \quad 0\leq x_i\leq cT.$$