# Impedance

In this section, we are interested in understanding the concept of impedance, which relates the frequency-dependent nature of force'' and motion'' in a system. In subsequent sections of this course, we will evaluate the impedance of various parts of musical instruments to understand how they vibrate.

## Impedance

• 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.

## Wave Impedance

• 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

## 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: .

• 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 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