Impedance in Other Physical Systems

• For electrical systems, impedance is voltage divided by current: $R = V/I$

• For transverse traveling waves on a vibrating string, the wave impedance (or characteristic impedance) is given by
  $$R = \frac{F(j\omega)}{V(j\omega)} = \sqrt{K\epsilon }=\epsilon c$$
  where
  • $F =$ transverse force
  • $V =$ transverse velocity
  • $\epsilon =$ string density (mass per unit length)
  • $K=$ string tension (stretching force)

• For longitudinal waves in air, the wave impedance is given by pressure $p$ divided by particle velocity $u$:
  $$R = \frac{P(j\omega)}{U(j\omega)} = \sqrt{\gamma P_0\rho } = \rho c$$
  where $\rho$ is the density (mass per unit volume) of air, $c$ is the speed of sound propagation, $P_0$ is ambient air pressure, and $\gamma_c = 1.4$ is the adiabatic gas constant for air (ratio of the specific heat of air at constant pressure to that at constant volume)

• For longitudinal plane-wave sections in an acoustic tube, the wave impedance is given by pressure $p$ divided by volume velocity $v$:
  $$R = \frac{P(j\omega)}{V(j\omega)} = \frac{\rho c}{A}$$
  where $A$ is the cross-sectional area of the tube section. Note that volume velocity is in units of meters cubed per second.

• Typical physical units used in practice are the Standard International (SI) units:
  • force in Newtons (kilograms times meters per second squared)
  • pressure in Newtons per meter squared
  • velocity in meters per second
  • mass in kilograms

• Related terms:
  • admittance = $$\frac{1}{\mbox{impedance}}$$
  • reactance = purely imaginary impedance
  • susceptance = purely imaginary admittance
  • immittance = either impedance or admittance

• For transverse electromagnetic (TEM) waves in a transmission line, the characteristic impedance is given by electric potential $V$ in volts divided by electric current $I$ in amperes (coulombs per second):
  $$R=\frac{V}{I} = \sqrt{\frac{L}{C}}=Lc$$
  where $L$ and $C$ are the inductance and capacitance, respectively, per unit length along the transmission line

• In a vacuum, the wave impedance for light (also a TEM wave) is
  $$R=\sqrt{\frac{\mu_0}{\epsilon_0}} =\mu_0 c$$
  where $\mu_0$ and $\epsilon_0$ are the permeability and permittivity, respectively, of the vacuum

  It is odd and interesting that waves in the vacuum are subject to the special theory of relativity (speed of light always measured to be the same, irrespective of one's velocity)