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

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

Introduction  to stationary waves 

  1. Principle of superposition of waves is one such outstanding principle related to wave motion. Formation of beats, studied in the previous article, due to the superposition of sound waves having slightly diterent frequencies is an interesting outcome of this principle. 
  2. The superposition of incident and reflected waves gives rise to the fomation of another kind of waves which are 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 strihged instruments such as veena, violin, Guitar, sitar, harmonium etc.
  4. The formation, properies, analytical treatment, modes of vibration of air columns, alongwith free and forced vibration, resonance phenomenon and various applications of staionary waves etc.
  5. A medium which has fixed boundary and whose boundaries are separated from other medium of distinct surface is called boundaries or finite medium. 
  6. When two identical waves of the same amplitude same speed travelling through a medium along the same path in exactly opposite directions super imposed with each other their resultant wave is called stationary waves.
  7. Stationary waves are localised 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 extemal periodic force is applied to it are called forced vibrations.
  9. If a body is made to vibrate, by an external periodic force with a frequency which is 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 tuning fork and to verify the laws of vibrating string.

Defination of stationary wave:

Stationary waves The resultant waves formed due to superposition of two progressive waves having the same amplitude, wavelength and speed but travelling 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 as stationary wave.

What are stationary wave :

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.

Characteristics of Stationary waves

  1. Stationary waves are formed due to superposition of two exactly identical waves travelling 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 the and periodic in space.
  7. Stationary waves do not transier energy through the medium
  8. In stationary waves all the paricles 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 are 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 wave 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.

Types of stationary wave

There are two types of stationary wave.

Transverse stationary wave

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

Longitudinal stationary wave

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

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.

How stationary waves are formed on string

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

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

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

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

Stationary waves

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 superposition of two exactly identical waves traveiling through medium in opposite direcions.
  3. Stationary waves d0 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 particle 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 unifom amplitude.
  5. Every particle lags behind the previous particle in phase.
  6. The particle of the medium begins to vibrate as the vwave approaches it.