Wave Impedance

Using the above identities, we have that the force distribution along the string is given in terms of velocity waves by

$$f(t,x) = \frac{K}{c} \left[{\dot y}_r(t-x/c) - {\dot y}_l(t+x/c) \right], \protect$$ (C.44)

where $K/c \isdef K/\sqrt{K/\epsilon } = \sqrt{K\epsilon }$ . This is a fundamental quantity known as the wave impedance of the string (also called the characteristic impedance), denoted as

$$R\isdefs \sqrt{K\epsilon } \eqsp \frac{K}{c} \eqsp \epsilon c.$$ (C.45)

The wave impedance can be seen as the geometric mean of the two resistances to displacement: tension (spring force) and mass (inertial force).

The digitized traveling force-wave components become

$$\begin{array}{rcrl} f^{{+}}(n)&=&&R\,v^{+}(n) \\ f^{{-}}(n)&=&-&R\,v^{-}(n) \end{array} \protect$$ (C.46)

which gives us that the right-going force wave equals the wave impedance times the right-going velocity wave, and the left-going force wave equals minus the wave impedance times the left-going velocity wave.^C.4Thus, in a traveling wave, force is always in phase with velocity (considering the minus sign in the left-going case to be associated with the direction of travel rather than a $180$ degrees phase shift between force and velocity). Note also that if the left-going force wave were defined as the string force acting to the left, the minus sign would disappear. The fundamental relation $f^{{+}}= Rv^{+}$ is sometimes referred to as the mechanical counterpart of Ohm's law for traveling waves, and $R$ in c.g.s. units can be called acoustical ohms [263].

In the case of the acoustic tube [320,299], we have the analogous relations

$$\begin{array}{rcrl} p^+(n) &=& &R_{\hbox{\tiny T}}\, u^{+}(n) \\ p^-(n) &=& -&R_{\hbox{\tiny T}}\, u^{-}(n) \end{array} \protect$$ (C.47)

where $p^+(n)$ is the right-going traveling longitudinal pressure wave component, $p^-(n)$ is the left-going pressure wave, and $u^\pm (n)$ are the left and right-going volume velocity waves. In the acoustic tube context, the wave impedance is given by

$$R_{\hbox{\tiny T}}= \frac{\rho c}{A} \qquad\mbox{(Acoustic Tubes)}$$ (C.48)

where $\rho$ is the mass per unit volume of air, $c$ is sound speed in air, and $A$ is the cross-sectional area of the tube. Note that if we had chosen particle velocity rather than volume velocity, the wave impedance would be $R_0=\rho c$ instead, the wave impedance in open air. Particle velocity is appropriate in open air, while volume velocity is the conserved quantity in acoustic tubes or ``ducts'' of varying cross-sectional area.