4) Its amplitude is this depends on position x.

5) Antinodes: At these points the amplitude of the vibrating particles is maximum

and the change in pressure and the density is minimum.

6) For the positions of antinodes:

7) Nodes: At these points the amplitude of particles is minimum and the change in pressure and density is maximum.

**13) for stationary waves: **

1) Waves produce nodes and antinodes at regular points in limited medium.

2) The distance between to consecutive nodes/ antinodes is .

3) Distance between a node and an adjacent antinode is .

4) Particles situated between two nodes execute simple harmonic motion whose amplitude is different but frequencies are same.

5) Amplitude depends on the position of the particle; maximum amplitude is obtained at antinodes and zero amplitude at the nodes.

6) The particles situated between two consecutive nodes vibrate in the phase with different amplitudes while the particles situated on either side of a node vibrate in the opposite phase.

** **

**14) vibration in a stretched string: **

1) Transverse progressive wave is produced in a stretched string.

2) Node is always formed at fixed end of string.

3) Velocity of the waves produced in a stretched string is

Where T = tension and m = mass per unit length of string.

Given density of material of string is d; radius is r, then;

4) For a string fixed at both the ends, nodes are formed at the ends.

5) If the length of a string is L and p loops are formed in it, then the frequency of string

(a) When p=1, then . In this condition string vibrates in one loop. is called the fundamental frequency. This is also called first harmonic.

(b) If p =2, then . In this condition is called the second harmonic or the first overtone.

(c) If p=3, then is called third harmonic or the second overtone.

(d) In strings all harmonics are produced and their ratio is ;

(e) Some harmonics are shown in the figure

Fig (4)

**15) laws of vibration of a stretched string: **

1) Law of length: where T and m are constants.

2) Law of tension: where L and m are constants.

3) Law of mass: where L and T are constants.

4) Law of radius: where d, L and T are constants.

5) Law of density: where r, T and L are constants.

On the basis of above Laws the formulae of frequency of vibration of string are:

M being the mass hanged on string.

**16) melde’s experiment: **

1) In a vibrating string of fixed length, the product of number of loops of loops in a vibrating string and square root of tension is a constant or

2)

3) In longitudinal vibration system the frequency of tuning fork is given by;

= 2 ´ (vibration frequency of string)

4) In this experiment vibrations of string are always transverse, but in longitudinal vibration system; the vibrations of arms of the tuning fork are along the direction of string. This experiment is also based on the stationary (transverse) waves.

**17) vibration of air columns in pipes: **

The pipe which contains air and in which sound vibrations are produced is called organ pipe.

1) CLOSED PIPE:

(a) One end of the pipe of this kind is closed and the other end is open.

(b) Node is formed at closed end and antinodes at the open end. In this pipe, number of antinodes and nodes are the same.

(c) Closed end of pipe reflects the compression as compression and rarefaction as rarefaction open end of the pipe reflects compression as rarefaction while rarefaction as compression.

(d) For a pipe of length L, the frequency of fundamental node of stationary waves produced. Fundamental frequency , is same as first harmonic while other are multiples of this frequency

wavelength . Frequency of third harmonic or the first overtone

wavelength

Frequency of the second overtone or the fifth harmonic

, wavelength