Fasten one end of a string to a firm support and hold its other free end by your hand. The up and down motion of the hand will produce waves in the string. If we keep on increasing the rate of up and down motion of hand, then at a particularly frequency, the string will start vibrating making a shape, like the one shown in fig. 12.18.

Such waves are called stationary waves. Due to such waves, the amplitude of the string on different sections of the string is different. At some portions of the string the amplitude is zero while it is maximum on the others. The former points like N1, N2…where amplitude is zero are called nodes while others like A1,A2,……where amplitude is maximum are called antinodes. The wavelength of the stationary wave is twice the distance between two successive nodes or antinodes.

## The number of Loops

The number of loops on the string depend upon the frequency of up and down motion of the hand. The string makes a single loop at the lowest frequency.

This frequency is called fundamental frequency or first harmonic. If the string makes two loops, the frequency at which it does so is called second harmonic or first overtone and this frequency is found to be two times the fundamental frequency. Even at higher frequencies, the string can be seen making three, four loops representing third, fourth harmonics, in fig. 12.19, third harmonic of a string has been shown.

To generate stationary waves, two exactly identical waves in all respects travelling in the opposite direction along the same line are necessarily required. In above mentioned experiments stationary waves are produced due to the interference of a wave being generated by the up and down motion of the hand and the one (exactly identical) being reflected by the support and travelling in the opposite direction on the string.

## Example 12.4:

A stationary wave of fundamental frequency 250 Hz has been generated in a medium. Calculate three higher frequencies by using formula Fn = nF1 at which stationary waves could be produced.

## Solution:

Fundamental frequency = F1 = 250hz

Therefore, three higher frequencies at which stationary waves could be generated are 2f1, 3f1, 4f1. So,

F2=2F1 = 500 Hz

F3 = 3F1 = 750 Hz

F4 = 4F1 = 1000 Hz