In systems with feedback, impedance (or admittance) provides information about resonances or anti-resonances.
Consider a string of length and tension rigidly fixed at position .
Let us assume the string is being driven at position 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:
(2)
where and 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 (), the boundary condition implies
At , the driving force must be compensated by a string force of
, resulting in the expression
These two equations can be used to solve for and , 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 ():
where 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
, and when
Note that this analysis does not account for any losses in the string or end support.
If the driving point () 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
, for