Spatial Derivatives

Slope waves are simply related to velocity waves.
By the chain rule,

$$\begin{eqnarray*} y'(t,x) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial x}y(t,x) \\ [5pt] &=& y'_r(t-x/c) + y'_l(t+x/c) \\ [5pt] &=& -\frac{1}{c} {\dot y}_r(t-x/c) + \frac{1}{c}{\dot y}_l(t+x/c) \\ [5pt] &\rightarrow& -\frac{1}{c} v^{+}(n-m) + \frac{1}{c}v^{-}(n+m) \end{eqnarray*}$$

$\Rightarrow$

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

or

$$\begin{array}{rcrl} v^{+}&=&-&cy'^{+}\\ [5pt] v^{-}&=&&cy'^{-} \end{array}$$

• Physical string slope = (lower rail - upper rail)/c
  in a velocity-wave simulation
• $\Rightarrow v^{-}(0+m) = v^{+}(0-m) \, \forall m$ on a struck string