The Driving Point Impedance of a Fixed String

• In systems with feedback, impedance (or admittance) provides information about resonances or anti-resonances.

• Consider a string of length $L$ and tension $T$ rigidly fixed at position $x = L$.

• Let us assume the string is being driven at position $x = 0$ by a transverse sinusoidal force in the form of a complex exponential: $\tilde{f}(t) = F e^{j \omega t}$.

• The motion of a string of finite length will be composed of both right- and left-going wave components. The resulting sinusoidal response of the string can then be represented as:
    $$\tilde{y}(x,t) = C^{+} e^{j (\omega t - k x)} + C^{-} e^{j (\omega t + k x)},$$ (6)
  where $C^{+}$ and $C^{-}$ are complex constants that describe the amplitude and phase of each traveling-wave component with respect to the driving force and $k = \omega / c = 2 \pi / \lambda$.

• At the fixed end ($x = L$), the boundary condition $y(L, t) = 0$ implies
  $$0 = [C^{+} e^{-j k L} + C^{-} e^{j k L}] e^{j \omega t}.$$

• At $x = 0$, the driving force must be compensated by a string force of $\tilde{F} \simeq -T (\partial{\tilde{y}} / \partial{x})$, resulting in the expression
  $$F e^{j \omega t} = T (j k C^{+} - j k C^{-}) e^{j \omega t}.$$

• These two equations can be used to solve for $C^{+}$ and $C^{-}$, which when substituted back into Eq. 6 gives
  $$\tilde{y}(x,t) = \frac{F}{k T} \frac{\sin k (L - x)}{\cos k L} e^{j \omega t}.$$

• From this, the string velocity can be determined as:
  $$\tilde{v}(x,t) = \frac{\partial{\tilde{y}}}{\partial{t}} = \frac{j \omega F}{k T} \frac{\sin k (L - x)}{\cos k L} e^{j \omega t}.$$

• The driving-point, or input, impedance $Z_{in}(\omega)$ is defined as the ratio of force to velocity at the driving point ($x = 0$):
  $$Z_{in}(\omega) = \frac{\tilde{f}(t)}{\tilde{v}(0, t)} = \frac{-j k T}{\omega} \cot k L = -j R \cot kL,$$
  where $R$ is the characteristic impedance of the string. This function is plotted below.

  Figure 4: Normalized driving-point impedance of a string.

  

• The impedance is zero for values of $k L = \pi/2, 3\pi/2, \ldots$, and $\pm j \infty$ when $k L = 0, \pi, \ldots$ Note that this analysis does not account for any losses in the string or end support.

• If the driving point ($x = 0$) is fixed, the velocity of the string at that point must equal zero. This would correspond to an infinite impedance. Thus, the resonance frequencies of a string rigidly fixed at both its ends correspond to the frequencies at which $Z_{in} = \infty$, or when $k L = 0, \pi, \ldots$, from which we find the resonance frequencies $f_n = n c / (2 L)$ for $n = 0, 1, 2, \ldots$ (which are the same as those found for the standing waves).

• On the other hand, if the point ($x = 0$) is perfectly free, the force of the string at that point must equal zero. This would correspond to an impedance of zero. Thus, the resonance frequencies of a string free on one end and rigidly fixed at the other correspond to the frequencies at which $Z_{in} = 0$, or when $k L = \pi/2, 3\pi/2, \ldots$, from which we find the resonance frequencies $f_n = (n+1) c / (4 L)$ for $n = 0, 1, 2, \ldots$ (which occur at odd integer multiples of the fundamental $f_0 = c / (4 L)$).