The String as a Waveguide

• The uniform stretched string forms a one-dimensional waveguide along which mechanical waves can travel with a constant speed of propagation (ignoring for the moment other dimensions of wave propagation).

• The time evolution of the string's shape is given by the linear superposition of traveling-wave components propagating in opposite directions along the string (Matlab example).

• The resonance frequencies (or normal modes) of the string are determined by its length, the wave speed of propagation, and the boundary conditions at each of its ends.

• For a string of length $L$, fixed at each of its ends, a discrete set of standing-wave patterns is possible with frequencies given by integer multiples of $f_{0} = c/(2 L)$, where $c$ is the speed of wave propagation on the string. These standing-wave frequencies correspond to the resonances of the system.

• The extent to which the resonance frequencies of the string are present in any particular string vibration pattern is determined by the way the system is driven or excited.

• The stiffness of the string produces a restoring force that was neglected in our derivation of the wave equation. When we account for this force, the string modes are “stretched” from perfect harmonic relationships (with greater stretching for higher modes).

• For thin metal strings, the decay time is determined mostly by air viscosity. For gut or nylon strings, internal damping is dominant for most modes.