What are stationary waves | Characteristics | Equation | Types  | Formation 

Introduction  to stationary waves 

1. The principle of superposition of waves is one such outstanding principle related to wave motion. The formation of beats, studied in the previous article, due to the superposition of sound waves having slightly different frequencies is an interesting outcome of this principle. 
2. The superposition of incident and reflected waves gives rise to the formation of another kind of wave which is called stationary waves. 
3. Such waves produced in air columns in pipes, open at both ends or closed at one end are used in resonating tubes and air columns, the wind, and stringed instruments such as the veena, violin, Guitar, sitar, harmonium, etc.
4. The formation, properties, analytical treatment, modes of vibration of air columns, along with free and forced vibration, resonance phenomenon and various applications of stationary waves, etc.
5. A medium that has fixed boundaries and whose boundaries are separated from other mediums of the distinct surfaces is called boundaries or finite medium. 
6. When two identical waves of the same amplitude same speed travel through a medium along the same path in exactly opposite directions superimposed with each other their resultant wave is called stationary waves.
7. Stationary waves are localized and do not transfer energy through the medium the points which are permanently at rest along stationary waves are called nodes, while the points which are having maximum displacement are called antinodes.
8. The vibrations made by a body under the action of its own restoring force are called free vibrations. The vibration made by the body when an external periodic force is applied to it is called forced vibrations.
9. If a body is made to vibrate, by an external periodic force with a frequency that is the same as the natural frequency of a body, the body begins to vibrate with maximum amplitude. This phenomenon is called resonance.
10. A sonometer can be used to determine the frequency of the tuning fork and to verify the laws of the vibrating string.

Stationary waves The resultant waves formed due to the superposition of two progressive waves having the same amplitude, wavelength, and speed but traveling in opposite directions are called stationary waves. These waves do not travel in any direction, and they do not transfer energy through the medium that’s why it is called stationary waves.

Standing waves are periodic oscillations produced by interference between waves of equal frequency and propagating along the same direction but in opposite directions. When an incident wave meets a wave reflected by a fixed end of a string, standing waves are formed, also known as harmonics. Standing waves are so called because they do not propagate through space. Its oscillations occur exclusively in the direction perpendicular (90º) to the direction of the waves that produced them.

1. Stationary waves are formed due to the superposition of two exactly identical waves traveling through a medium in opposite directions.
2. When stationary waves are set up in a uniform, the particles at some points are permanently at rest (amplitude is zero). Such points are called nodes. The distance between two successive nodes is λ/2.
3. When stationary waves are set up in a medium, the particles at some points vibrate with maximum amplitude; such points are called antinodes. The distance between two successive antinodes is λ/2.
4. Nodes and antinodes are alternately situated. The distance between any node and its adjacent antinodes is λ/4.
5. In stationary waves, loops are formed between two successive nodes. All the particles in one loop are in the same phase while particles in successive loops are out of phase by π.
6. Stationary waves are periodic in wavelength and periodic in space.
7. Stationary waves do not transfer energy through the medium
8. In stationary waves, all the particles except at the nodes vibrate with the same period as that of the interfering waves.
9. The amplitudes of vibrations are different for different particles, and they increase from node to antinodes.
10. A loop is formed between the two nodes for two reasons. Firstly, the amplitude of the particles gradually increases from its value at the node (which is zero) to a maximum at the antinodes. This amplitude then decreases from the antinodes to the node. Secondly, all the particles reach their maximum displacements at the same time.
11. Stationary waves can be produced by the interference of longitudinal as well as transverse waves.

What are the Conditions for the formation of stationary waves?

For the formation of a stationary wave, the two simple harmonic progressive waves must

1. have equal wavelength 
2. have equal period
3.  have equal amplitude,
4. have equal frequency
5. have equal speed and Travel in the same medium along the same path but in opposite direction.

There are two types of stationary waves.

1. Transverse stationary wave : 

A stationary wave formed due to the superposition of two identical transverse progressive waves traveling at the same speeds in opposite directions along a straight line is called a transverse stationary wave.

Ex.:The stationary waves formed on the vibrating string of i) sonometer ii) Melde’s experiment. iii) musical chord instruments. Transverse stationary waves are set up on a stretched string.

2. Longitudinal stationary wave: 

A stationary wave formed due to the superposition of two identical longitudinal progressive waves traveling with the same speed in the opposite direction is called a longitudinal stationary wave.

Ex. : The stationary waves formed in the vibrating air column of i) resonance tube ii) closed organ pipe iii) open organ pipe, iv) musical vocal instruments.

When two identical progressive waves (transverse or longitudinal) traveling along  Add in opposite directions Interfere with each other by superposition of various resultants were obtained in the form of loops called stationary waves.

Consider two simple harmonic progressive waves of equal amplitude and frequency propagating on a long uniform string in opposite direction.

If wave (frequency n and wavelength travelling along positive direction of X-axis  then
$$
Y_{1}=a sin 2 pileft(n t-frac{x}{lambda}right)
$$
If wave (frequency ‘n’ and wavelength ‘ $lambda^{prime}$ ) is travelling along negative direction of $X$ -axis then
$$
mathrm{Y}_{2}=mathrm{a} sin 2 pileft(mathrm{nt}+frac{x}{lambda}right)
$$
These waves interfere to produce stationary wave. The resultant displacement of the stationary waves is given by the principle of superposition of waves.
$$
begin{array}{l}
Y=Y_{2}+Y_{1} \
Y=a sin 2 pileft(n t+frac{x}{lambda}right)+a sin 2 pileft(n t-frac{x}{lambda}right)
end{array}
$$
Using $sin C+sin D=2 sin left(frac{C+D}{2}right) cos left(frac{C-D}{2}right)$
nd $cos (-theta)=cos theta)$
$$
begin{array}{l}
mathrm{Y}=2 mathrm{a}left[sin (2 pi mathrm{nt}) cos left(frac{2 pi x}{lambda}right)right] \
mathrm{Y}=left(2 mathrm{a} cos frac{2 pi mathrm{x}}{lambda}right) sin (2 pi mathrm{nt}) \
mathrm{Y}=mathrm{A} sin (2 pi mathrm{nt})
end{array}
$$
But $omega=2 pi mathrm{n}$
$$
Y=A sin omega t quadleft(text { where } A=2 a cos frac{2 pi x}{lambda}right)
$$

 A is the amplitude of the resultant stationary wave. i.e amplitude is periodic in space hence we can see loops in the case of transverse waves forming stationary waves on a string. Thus in stationary waves, frequency is the same as that of progressive waves but amplitude varies with the position of particles. The above equation represents the resultant displacement of two simple harmonic progressive waves which are not moving because of the absence of the term x in the equation, therefore, it is called a stationary wave.

In the above equation time, t, and distance X are not present in the same harmonic function so it is not a progressive wave it is a stationary wave.

Difference between stationary waves and Progressive waves:

Stationary waves

1. Stationary waves are not propagated in any direction
2. Stationary waves are formed due to the superposition of two exactly identical waves traveling through a medium in opposite directions.
3. Stationary waves do not transfer energy through a medium.
4. Amplitude increases from node to antinode.
5. All the particles in one loop are in the same phase, while particles in two successive loops are out of phase by π.
6. Particles at nodes are permanently at rest

Progressive waves : 

1. Progressive waves are propagated in the forward direction. 
2. Progressive waves are produced due to a disturbance created in the medium.
3. Progressive waves transfer energy through a medium from particle to particle.
4. Every particle vibrates with uniform amplitude.
5. Every particle lags behind the previous particle in phase.
6. The particle of the medium begins to vibrate as the wave approaches it.