### 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 [169,261]. 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.

A single waveguide section between two partial sections is shown in Fig.C.19. The sections are numbered 0 through from left to right, and their wave impedances are , , and , respectively. Such a rod might be constructed, for example, using three different materials having three different densities. In the th section, there are two stress traveling waves: traveling to the right at speed , and traveling to the left at speed . To minimize the numerical dynamic range, velocity waves may be chosen instead when .

As in the case of transverse waves (see the derivation of (C.46)), the traveling longitudinal plane waves in each section satisfy [169,261]

where the wave impedance is now , with being the mass density, and being the

*Young's modulus*of the medium (defined as the stress over the strain, where

*strain*means relative displacement--see §B.5.1) [169,261]. As before, velocity is defined as positive to the right, and is the right-going traveling-wave component of the stress, and it is positive when the rod is locally

*compressed*.

If the wave impedance 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 and at one
end of each section as a function of time. As shown in Fig.C.19,
we define
as the force-wave component at the *extreme
left* of section . Therefore, at the extreme right of section ,
we have the traveling waves
and
, where is
the travel time from one end of a section to the other.

For generality, we may allow the wave impedances 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 are still given by and . 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 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 are obtained by summing the left- and
right-going traveling wave components:

Let denote the force at position and time in section , where is measured from the extreme left of section along its axis. Then we have, for example, and at the boundaries of section . More generally, within section , the physical stress may be expressed in terms of its traveling-wave components by

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