Other factors which affect the frequency of a vibrating string
(a) Tension. The effect of tension can be investigated if the length of the string is kept constant while the tension is altered to bring the wire into unison with a number of standard forks in turn.
The results show that the frequency is proportional to the square root of the tension T. (b) Mass per unit length. The thicker and heavier a string is, the lower is its frequency for a given length and tension. In musical instruments, however, merely using thicker strings of the same material is accompanied by stiffness leading to rapid decay in amplitude. To prevent this, the necessary loading without loss of flexibility is achieved by binding the strings with wire. This can be seen by looking inside a piano.
Experiment shows that the frequency is inversely proportional to the square root of the mass per unit length, m.