Wave Impedance

We just showed

$$\begin{array}{rcrl} y'^{+}&=&-&\frac{1}{c}v^{+}\\ [5pt] y'^{-}&=&&\frac{1}{c}v^{-} \end{array}$$

Define new wave variables in terms of slope waves as

$$\begin{eqnarray*} f^{{+}}&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& - Ky'^{+}\\ f^{{-}}&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& - Ky'^{-} \end{eqnarray*}$$

Note that $f^\pm$ are in physical units of force.
We have

$$\begin{array}{rcrl} f^{{+}}&=&&\frac{K}{c}v^{+}\\ [5pt] f^{{-}}&=&-&\frac{K}{c}v^{-} \end{array}$$

Recall

$$\begin{eqnarray*} c &=& \sqrt{\frac{K}{\epsilon }}\\ [10pt] \Rightarrow\qquad ... ... \sqrt{K\epsilon } \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\; R \end{eqnarray*}$$

which is the wave impedance of the ideal string (force/velocity for traveling waves). Thus,

$$\zbox{ \begin{array}{rcrl} f^{{+}}&=&&R\,v^{+}\\ f^{{-}}&=&-&R\,v^{-} \end{array}}$$