1. If a string (or a wire) stretched between two rigid supports is plucked at some point, the disturbance produced travels along the string in the form of transverse waves. If T is the tension applied to the string and m is the mass per unit length (i.e., linear density) of the string, the speed of the transverse waves is
v = \(\sqrt{\frac Tm}\)
2. The transverse waves moving along the string are reflected from the supports. The reflected waves interfere and under certain conditions set up stationary waves in the string. At each support, a node is formed.
3. The possible or allowed stationary waves are subject to the two boundary conditions that there must be a node at each fixed end of the string. The different ways in which the string can then vibrate are called its modes of vibration.
4. In the simplest mode of vibration, there are only two nodes (N), one at each end and an antinode (A) is formed midway between them, as shown in
A string vibrating fundamental mode
In this case, the distance between successive nodes is equal to the length of the string (L) and is equal to λ/2, where λ is the wavelength.
∴ L = \(\frac λ2\) or λ = 2L
The frequency of vibrations is n = \(\frac vλ\) Substituting v = \(\sqrt{\frac Tm}\) and λ = 2L in this relation, we get,
n = \(\frac 1{2L}\)\(\sqrt{\frac Tm}\)
This is the lowest frequency of the stationary waves on a stretched string and is called the fundamental frequency.
