- In AC electrical systems, impedance is defined as voltage divided by current.
- Electrical resistors have constant, frequency-independent impedances.
- Electrical capacitors and inductors, however, have current-to-voltage characteristics that change with respect to the frequency of an applied source.
- For mechanical systems, impedance is defined as the ratio of force to velocity.
- The inverse of impedance is called admittance. One can use the term immittance to refer to either an impedance or an admittance.
- In lossless systems, an immittance is purely imaginary and called a reactance.
- Immittances are steady state characterizations that imply zero initial conditions for elements with ``memory'' (masses and springs, capacitors and inductances).
- In acoustics, impedance is given by pressure divided by either particle or volume velocity.

- Consider a sinusoidal traveling-wave component of displacement moving in the positive
*x*direction along an infinite string:

where*A*is a complex constant that describe the amplitude and phase of the traveling-wave component and . - 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

where*T*is the string tension and . This force is clearly balanced by an equal but opposite force exerted from the right-side portion of the string. - The vertical velocity of the wave component
is

- Thus, the ratio of force to velocity for this wave component is

where*c*is the speed of wave propagation. In deriving the wave equation for string motion, we determined and thus*T*/c is also equal to , where 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

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:

- Traveling-wave components of force are thus related to traveling-wave components of velocity by

- 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: . - 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:

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 . - At the fixed end (
*x*=*L*), the boundary condition*y*(*L*,*t*) = 0 implies

- At
*x*=0, the driving force must be compensated by a string force of , resulting in the expression

- These two equations can be used to solve for
*C*^{+}and*C*^{-}, which when substituted back into Eq. 2 gives

- From this, the string velocity can be determined as:

- The driving-point, or input, impedance
is defined as the ratio of force to velocity at the driving point (
*x*= 0):

where*R*is the characteristic impedance of the string. This function is plotted below. - The impedance is zero for values of
, and when
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 , or when , from which we find the resonance frequencies*f*=*n c*/ (2*L*), for

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