Impedance

*Impedance* is defined for mechanical systems as
force divided by velocity, while the inverse (velocity/force) is
called an *admittance*. For dynamic systems, the impedance of a
``driving point'' is defined for each frequency
, so that the
``force'' in the definition of impedance is best thought of as the
peak amplitude of a sinusoidal applied force, and similarly for the
velocity. Thus, if
denotes the Fourier transform of the
applied force at a driving point, and
is the Fourier
transform of the resulting velocity of the driving point, then the
*driving-point impedance* is given by

In the lossless case (no dashpots, only masses and springs), all driving-point impedances are purely imaginary, and a purely imaginary impedance is called a

In acoustics [320,321], force takes the form of
*pressure*
(*e.g.*, in physical units of newtons per meter squared),
and velocity may be either *particle velocity* in open air
(meters per second) or *volume velocity* in acoustic tubes
(meters cubed per second) (see §B.7.1 for
definitions).
The *wave impedance* (also called the *characteristic
impedance*) in open air is the ratio of pressure to particle velocity
in a sound wave traveling through air, and it is given by
, where
is the density (mass
per unit volume) of air,
is the speed of sound propagation,
is ambient pressure, and
is the ratio of the specific
heat of air at constant pressure to that at constant volume. In a
vibrating string, the wave impedance is given by
, where
is string density (mass per unit length) and
is
the tension of the string (stretching force), as discussed further in
§C.1 and §B.5.2.

In circuit theory [110], force takes the form of electric potential in volts, and velocity manifests as electric current in amperes (coulombs per second). In an electric transmission line, the characteristic impedance is given by , where and are the inductance and capacitance, respectively, per unit length along the transmission line. In free space, the wave impedance for light is , where denotes the speed of light, and and are the permeability and permittivity, respectively, of free space. One might be led from this to believe that there must exist a medium, or ``ether'', which sustains wave propagation in free space; however, this is one instance in which ``obvious'' predictions from theory turn out to be wrong (see the special theory of relativity for details).

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