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.
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.
A short section of a stretched, vibrating string.
The vertical velocity of the wave component
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.
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 (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