Rigidly Terminated Ideal String

• Reflection inverts for displacement, velocity, or acceleration waves (proof below)
• Reflection non-inverting for slope or force waves

Boundary conditions:

$$y(t,0) \equiv 0 \qquad y(t,L) \equiv 0 \qquad \hbox{($L = $\ string length)}$$

Expand into Traveling-Wave Components:

$$\begin{eqnarray*} y(t,0) &=& y_r(t) + y_l(t) = y^{+}(t/T) + y^{-}(t/T) \\ y(t,L) &=& y_r(t-L/c) + y_l(t+L/c) \end{eqnarray*}$$

Solving for outgoing waves gives

$$\begin{eqnarray*} y^{+}(n) &=& -y^{-}(n) \\ y^{-}(n+N/2) &=& -y^{+}(n-N/2) \end{eqnarray*}$$

$N\mathrel{\stackrel{\mathrm{\Delta}}{=}}2L/X=$ round-trip propagation time in samples