Wave Impedance

• Consider a sinusoidal traveling-wave component of displacement moving in the positive $x$ direction along an infinite string:
  $$\tilde{y}^{+}(x,t) = A e^{j (\omega t - k x)},$$
  where $A$ is a complex constant that describe the amplitude and phase of the traveling-wave component and $k = \omega / c = 2 \pi / \lambda$.

• At an arbitrary point and time along the string (Fig. 3), the vertical force exerted on the portion to the right by the left-side portion of the string is given by
  $$\tilde{f}^{+}(t,x) = -T \sin(\theta) \approx -T \tan(\theta) = -T \frac{\partial{\tilde{y}^{+}}}{\partial{x}} = T j k A e^{j (\omega t - k x)},$$
  where $T$ is the string tension and $\vert\partial{\tilde{y}^{+}}/\partial{x}\vert \ll 1$. This force is clearly balanced by an equal but opposite force exerted from the right-side portion of the string.

  Figure 3: A short section of a stretched, vibrating string.

  

• The vertical velocity of the wave component $\tilde{y}^{+}(t,x)$ is
  $$\tilde{v}^{+}(t,x) = \frac{\partial{\tilde{y}^{+}}}{\partial{t}} = j\omega A e^{j (\omega t - k x)}.$$

• The impedance of the wave component at this arbitrary point is the ratio of force to velocity, which is
  $$\frac{\tilde{f}^{+}(t,x)}{\tilde{v}^{+}(t,x)} = T / c,$$
  where $c$ is the speed of wave propagation. In deriving the wave equation for string motion, we determined $c = \sqrt{T/\epsilon}$ and thus $T/c$ is also equal to $\epsilon c$, where $\epsilon$ is the linear mass density of the string.

• This fundamental scalar quantity is referred to as the characteristic impedance or wave impedance of the string and is denoted
  $$R \ensuremath{\stackrel{\Delta}{=}}\sqrt{T \epsilon} = \frac{T}{c} = \epsilon c.$$
  It indicates that the force and velocity components of a traveling wave moving along a string are related by a constant, frequency-independent “resistance”.

• After similar consideration of a traveling-wave component moving to the left, a more general, frequency-independent expression for the relationship between vertical force and velocity on the string is:
  $$f(t,x) = R \left[ \frac{\partial}{\partial{t}} y^{+}(t - x/c) - \... ...l{t}} y^{-}(t + x/c)\right] = R \left[ v^{+}(t - x/c) - v^{-}(t + x/c)\right].$$

• Traveling-wave components of force are thus related to traveling-wave components of velocity by
  $$\begin{eqnarray*} f^{+} &=& R v^{+} \\ f^{-} &=& -R v^{-} \end{eqnarray*}$$