Definition and result

• The mechanical impedance of a system is the ratio between the force applied at a point and the velocity of that point. In the mks system it is measured in mechanical ohms (kg/s). For the string we refer to the propagation of transverse waves in an ideal string (see wave equation in a string and natural frequencies of a string), as a model for the motion of the strings in all string instruments.

We define

The mechanical impedance of the string is the ratio between the force exerted on the string to keep it oscillating and the transverse velocity

${\displaystyle Z=\frac{F}{v}}$

We are usually interested in the linear frequency response of the system, which means that we study the motion of the system when we apply to it, as "input", a force of small amplitude oscillating periodically in time (in particular sinusoidally). In this case we find that

the impedance of the string is a function only of characteristics of the string itself (density ${\displaystyle \rho }$ and tension T), and is

${\displaystyle Z=\frac{T}{c}=\rho c=\sqrt{\rho T}}$.

How the result is obtained

The force F applied from outside at a given point x is balanced by the transverse component of the tension (the force of gravity is usually not considered, because the work it does is negligible compared with that of the tension). On a small element dx of the string it is

${\displaystyle F(x,t)=-T\frac{\partial y}{\partial x}}$

When a harmonic wave travels along the string, its shape is

${\displaystyle y(x,t)=A\sin (\omega t-kx)}$,

so, differentiating with respect to x, we find that the force is

${\displaystyle F(x,t)=kTA\cos (\omega t-kx)}$,

while differentiating with respect to t we find the velocity

${\displaystyle v(x,t)=\omega A\cos (\omega t-kx)}$.

The characteristic mechanical impedance of the string is therefore

${\displaystyle Z=\frac{F}{v}=\frac{kT}{\omega }}$.

Now recall that, for harmonic waves (see fundamental quantities of waves and harmonic waves),

${\displaystyle c=\frac{\omega }{k}}$,

where c is the propagation speed of the wave, and moreover in an ideal string k and ${\displaystyle \omega }$ are related by (see wave equation in a string)

${\displaystyle c^2=\frac{T}{\rho }}$,

where ${\displaystyle \rho }$ is the linear density of the string (its mass per unit length, in kg/m). We can therefore write the impedance of the string as a function only of characteristics of the string itself, density and tension, as

${\displaystyle Z=\frac{T}{c}=\rho c=\sqrt{\rho T}}$.

In the section refraction in one dimension (in Italian) you will find an example, illustrated with animations, of an application of the concept of string impedance introduced here.

Remarks

1. We have found that the impedance of the string is a constant, depending only on its density and tension, and related to the speed c of waves in the string.
2. Note that the impedance depends neither on the position of the point on the string, nor on the particular instant of time, nor on the frequency of the applied force. It is therefore not a function of x and t, but a true numerical constant.
3. The ideal string presents the same impedance at all frequencies, just as waves of any frequency travel at the same speed ${\displaystyle c=\sqrt{T/\rho }}$: impedance and speed both depend only on the tension and the density. This is no longer exactly true for real strings, in which bending stiffness makes high-frequency waves slightly faster (see the discussion of dispersion, with particular reference to the consequences for string instruments).
4. The result holds for a wave travelling in one direction only, as on an infinitely long string or on a string whose end does not reflect waves. On a string of finite length, fixed at its ends, the reflected waves come back and add to the one produced by the force: the impedance measured at the point where the force is applied then depends on frequency and becomes very small at the natural frequencies of the string, that is at the resonances. The impedance ${\displaystyle \sqrt{\rho T}}$ is therefore called the characteristic impedance of the string.

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