Moving Termination: Ideal String

Uniformly moving rigid termination for an ideal string
(tension $K$ , mass density $\epsilon$ ) at time $0<t_0<L/c$ .

Driving-Point Impedance $F_0/V_0$ :

$$\begin{eqnarray*} y'(t_0,0) &=& -\frac{v_0 t_0}{c t_0} = -\frac{v_0}{c} = -\frac{v_0}{\sqrt{K/\epsilon }} \\ [10pt] \Rightarrow\quad f_0 &=& -K\sin(\theta)\approx -Ky'(t,0) = \sqrt{K\epsilon }\,v_0 \mathrel{\stackrel{\mathrm{\Delta}}{=}}R\, v_0 \end{eqnarray*}$$

• If the left endpoint moves with constant velocity $v_0$
  then the external applied force is $f_0 = Rv_0$
• $R\mathrel{\stackrel{\mathrm{\Delta}}{=}}\sqrt{K\epsilon } \mathrel{\stackrel{\mathrm{\Delta}}{=}}$ wave impedance (for transverse waves)
• Equivalent circuit is a resistor (dashpot) $R>0$
• We have the simple relation $f_0 = Rv_0$ only in the absence of return waves, i.e., until time $t_0 = 2L/c$ .
• Interactive Animation

• Successive snapshots of the ideal string with a uniformly moving rigid termination
• Each plot is offset slightly higher for clarity
• GIF89A animation at
  

  http://ccrma.stanford.edu/~jos/swgt/movet.html