The interference of two identical waves moving in opposite directions produces *standing waves. *For a string with fixed ends, the standing wave is given by

Standing waves are characterized by fixed locations of zero displacement called *nodes *and fixed locations of maximum displacements called **antinodes***. *The separation between two consecutive nodes or anti-nodes is λ/2.

For a stretched string of length L, fixed at both ends, the two ends of the string have to be nodes. If one of the ends is chosen as position *x *= 0, then the other end is *x *= *L*. In order that this end is a node; the length *L *must satisfy the condition

This condition shows that standing waves on a string of length *L *have restricted wavelength given by

The frequencies corresponding to these wavelengths

where *v *is the speed of traveling waves on the string.

The set of frequencies given by the above relation are called **the normal modes of oscillation of the system**. The oscillation mode with the lowest frequency

is called the **fundamental mode or the first harmonic**.

The *second harmonic *is the oscillation mode with *n *= 2 and so on.

Put in above equation,

The set of frequencies represented by the above relation are the normal modes of oscillation of such a system. The lowest frequency given by *v/*4*L *is the fundamental mode or the first harmonic.

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