Definitions and approximations

By ideal string, in this context, we mean a one-dimensional, perfectly flexible object.

The parameters describing an ideal string are summarised in the following table:

When studying the physics of real objects, we must find conditions that approximate the ideal ones.

• An object much longer than its diameter approximates the condition of one-dimensionality, provided that the motion of the string about its axis (torsion) can be neglected.
• The condition that the string is perfectly flexible can reasonably be assumed if the bending stiffness and all external forces are negligible compared with the tension.
• Moreover, the tension is assumed not to change much during the motion of the string, that is, not to depend on the amplitude of the oscillation. This condition is generally met only for small displacements from equilibrium.
• The bending stiffness, negligible at low frequencies, matters more and more as the frequency increases: for frequencies that are "not too high" the string therefore behaves as an ideal string (see the page wave equation for the string).

When a real string departs from these conditions, the results given here of course do not hold, and more complex models are needed (for example, stiffness makes the higher frequencies slightly sharper than integer multiples of the fundamental: this is the inharmonicity of strings).

Transverse vibrations of the ideal string fixed at both ends

The fundamental frequency of an ideal string fixed at both ends is given by

${\displaystyle f_0=\frac{1}{2L}\sqrt{\frac{T}{\mu }}}$.

The higher frequencies are integer multiples of its fundamental frequency f₀

${\displaystyle f_n=n{f}_{0}n=1,2,\dots }$.

How are these values calculated?

We can follow two equivalent routes:

1. solve the wave equation with the boundary conditions describing the string fixed at both ends, namely that the displacement of the string is zero at the ends (see wave equation)
2. study the normal modes of the string (see normal modes and normal modes of a string)

Natural frequencies of the string in the presence of friction

In the section wave equation we briefly described the wave equation in the presence of friction, and its general travelling solution. Here we are interested in the stationary solutions, whose frequencies correspond to the natural frequencies of the string. Calling γ the friction coefficient, we get

${\displaystyle f_n=\frac{1}{2\pi }\sqrt{\frac{{\pi }^{2}T}{L^2\mu }{n}^2-\frac{{\gamma }^2}{4}}}$

Note that, for the same mode (the integer n), the frequency with friction is slightly lower than that of the string without friction.

String instruments

physics of string instruments

Further reading and links

• Related questions and answers:
  • Why are some guitar strings made from nylon while others are covered with metal?

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