## Introduction

“Stationary Waves are the combination of two Waves which move in opposite directions having the same amplitude as well as frequency.” It is also known as standing Waves. It is the phenomenon that is the outcome of interference that means when the Waves are superimposed; their energies are added at the same time or canceled. When Waves move in the same direction, then their interference generates traveling Waves. When Waves move in opposite directions, then their interference generates an oscillating Wave which is fixed in space.

### Cases in Stationary Waves

The rope which is vibrating tied at one end generates a standing Wave. The Wave train will get reflected and superimposed on oneself same as the other Wave train in the same plane after reaching the fixed side of the rope. Due to interference between the Waves, the final amplitude is the sum of their particular respective amplitudes.

Case I: The Wave train coincides so that Stationary Waves R have a value twice to amplitude.

Case II: In this case, 1/8 period later, the Wave train gets shifted by 1/8 of Wavelength.

Case III: In this case, another 1/8 period later, the amplitudes of Wave trains get in the opposite direction.

Every time, there is a position along the rope which is known as nodes (N) at which there is no movement anyway, and the Wave trains are always opposite in direction. On the other side of the node is the vibrating node known as antinode (A). The antinodes are alternatively in the direction of the distance so that the rope at any instant produces a graph of the mathematical function which is known as sine symbolized by R.

### Stationary Waves Definition

When two progressive Waves of similar amplitude, as well as Wavelength, travel with a straight line and in the opposite direction which gets superimposed on each other, it leads to the creation of Stationary Waves.

### Analytical Equation for Stationary Waves

Suppose the progressive Waves of amplitude and Wavelength λ travel in the X-axis direction.

\[ y_{1} = a sin 2π \frac{t}{T}– \frac{x}{λ} \] ----------- (1)

This Wave gets reflected from the free end and it passes the X-axis but in the negative direction, then the equation formed is:

\[ y_{2} = a sin 2π \frac{t}{T} + \frac{x}{λ} \] ---------- (2)

As per the objective of superposition, the final displacement is:

\[ y = y_{1} + y_{2} \]

\[ = a sin2π(\frac{t}{T}–\frac{x}{λ}) + sin2π(\frac{t}{T} + \frac{x}{λ}) \]

\[ = a 2sin(2π \frac{t}{T})cos(2π \frac{x}{λ}) \]

So, \[ y = 2a cos (2π \frac{x}{λ}) sin (2π \frac{t}{T}) \] ---------- (3)

Thus, this is the equation of Stationary Waves. There are some observed points described below.

At the points where x= 0, λ/2, λ, 3λ/2, then the values of cos2πx/λ = ±1. Thus, A = +2a.

At these points, the final resultant amplitude is the maximum which is known as antinodes.

At points of x= λ/4, 3λ/4, 5λ/4, then the values of cos2πx/λ = 0. Thus, A = 0.

The final resultant amplitude is 0 at these points which are known as nodes.

The difference between the distance of any two successive nodes or antinodes is equal to λ/2 and the difference between the distance of one node and one antinode is λ/4.

When the value of t = 0, T/2, T, 3T/2, 2T, then the value of sin2πt/T = 0, and the displacement is 0.

When the value of t = T/4, 3T/4, 5T/4, then the value of sin2πt/T = ±1, and the displacement is minimum.

### Features of Stationary Waves

The form of the Wave remains Stationary.

Nodes and antinodes are produced alternately.

The points where the value of displacement is zero are known as nodes and the points where the value of displacement is maximum are now known as antinodes.

The changes in pressure are maximum at nodes and minimum at antinodes.

The particles, except for the position of the node, implement the simple harmonic movements of the same period.

The amplitude of each and every particle is not equal, and it is maximum at antinodes which reduces drastically and the value is zero at the nodes.

The speed and velocity of all the particles at the nodes are 0 which increases rapidly and is the maximum at the antinodes.

The difference between the distance of any two consecutive nodes or antinodes is equal to λ/2, while the difference between the distance of any node and its adjacent antinode is equal to λ/4.

There is no exchange and transfer of energy where all the particles of the medium travel through their mean position at the same time twice while each vibration.

Particles in the same division vibrate in the same phase and between the neighboring segments, the particles vibrate in the opposite phase.

### Stationary Waves in The Strings

In some musical instruments such as sitar, violin, guitar, etc., a sound produced with it is due to the vibration of the stretched strings. Here, we will see the different modules of vibration of a string which is fixed at both ends.

When the string is under teinsion, it is set into vibration, where transverse Waves move to the end of the wire and then reflect. Therefore, Stationary Waves are produced.

### Stationary Waves or Standing Waves

The two progressive Waves of the same Wavelength and amplitude travel with the same speed through a medium in opposite directions and superpose upon each other, they give rise to what are called Stationary or standing Waves.

While in Stationary Waves, there are certain points, where the amplitude is zero and strain is maximum. These points are known as nodes. Midway between these nodes, there is another category of points, where the amplitude is maximum and this strain is zero. These points are known as antinodes. The amplitude is zero and maximum between the nodes and the antinodes. There is no flow of energy so they are called as Stationary Waves because there is no flow of energy along with the Wave. Like the progressive Waves, Stationary Waves may be transverse or longitudinal in nature.

### Equations of Stationary Waves or Standing Waves

Analytical Method

Let us consider a progressive Wave of amplitude a and Wavelength λ traveling in the direction of X axis.

\[ y_{1} = a sin 2π (\frac{t}{T} – \frac{x}{λ}) \] …... (1)

This Wave is reflected from a free end and it travels in the negative direction of X axis, then

\[ y_{2} = a sin 2π (\frac{t}{T} + \frac{x}{λ})\] …... (2)

According to the principle of superposition, the resultant displacement is,

\[ y = y_{1} + y_{2} \]

\[ = a sin 2π (\frac{t}{T} – \frac{x}{λ}) + sin 2π (\frac{t}{T} + \frac{x}{λ}) \]

\[ = a 2 sin (2π \frac{t}{T}) cos (2π \frac{x}{λ}) \]

So, \[ y = 2a cos (2π \frac{x}{λ}) sin (2π \frac{t}{T}) \] …... (3)

This is the equation of a Stationary Wave.

### Characteristics of Stationary Waves

The Waveform remains Stationary.

Nodes and antinodes are formed alternately.

The points where displacement is zero are called nodes and the points where the displacement is maximum are called antinodes.

The change in pressure is maximal at the nodes and minimal at the belly. All particles except nodes perform simple vibrations with the same period.

Not all particles have the same amplitude. It maximizes on the belly, gradually decreases, and reaches zero on node

The velocity of the particles at the node is zero. It gradually increases and reaches its maximum in the belly.

Distance between any two consecutive nodes or antinodes is equal to λ2, whereas the distance between a node and its adjacent antinode is equal to λ/4.

There is no transfer of energy. All particles of the medium pass through the intermediate position twice at the same time during each vibration.

Particles in the same segment oscillate in the same phase, and particles oscillate in opposite phases between adjacent segments.

### Sonometer

The sonometer includes a hollow-sounding box which is a meter long where one end of a thin metallic wire of cross-section is fixed to a hook and the other side is passed over a pulley attached to the weight hanger. Then the wire is stretched by two knife sides let’s say P and Q with the addition of weight on the hanger. The difference between the distances of two knife edges is in the adjustable form for changing the vibration length of the metallic wire.

Transverse Waves which are Stationary are generated in the wire. Though the ends are fixed, nodes are produced at P and Q and its antinode is generated in the middle.

The length of the vibrating generated l = λ/2.

Therefore, λ =2l. If n is considered as the frequency of the vibrating segment, then

n = v/λ = v/2l ------------- (1)

We all know that, v = √T/m,

where,

T is the tension,

m is the mass per unit length of the metallic wire.

Hence, n = 1/2l *√T/m ----- (2)

### Modes of Stretching String Vibration

Fundamental frequency

If the metallic wire is stretched between two points, then transverse Stationary Waves pass through the wire and get reflected at the fixed end. A transverse Wave is produced which is Stationary.

When the wire AB of length say l is made for creating vibration in one segment then,

l = λ1/2. Thus, λ1 = 2l which gives the lowest frequency known as the fundamental frequency.

n1 = v/λ1

Hence, n1 = (1/2l) √T/m …... (3)

Overtones in the stretched string

If the wire AB is made for vibration in two segments then, l = λ2/2 + λ2/2

Hence, λ2 = l and n2 = v/λ2

Thus, n2 = 1/l √T/m = 2n1 …... (4)

Where n2 is the frequency of the first overtone.

Though the frequency is the same as twice the fundamental which is also known as second harmonic. In the same way, higher overtones are generated, if the wire gets vibrated with more segments. If there are P number of segments, then the length of each segment will be

l/p = λp/2

that is, λp = 2l/P

Thus, the frequency np = (P/2l)√T/m = Pn1 ----- (5)

This means Pth harmonic is equal to (P-1)th overtone.

### Real-Time Applications of Stationary Waves

There are many such applications of generating Stationary Waves, but the easiest and one of the favorites is its use in musical instruments.

In string musical instruments, a bowed or plucked string creates the note as certain frequencies of standing or Stationary Waves can produce the note on that string under some conditions such as the finger holding the string at some position.

Any vibrations which are not at the right and appropriate frequencies for making Stationary Waves are canceled fast and thus, it is the Stationary Wave frequencies that we hear.

In the same way, in the instruments of the woodwind family, we create the notes as Stationary Waves can produce in the tube of air inside the musical instruments.

Strings and woodwind instruments’ sounds or tones are different from each other as they provide different types of combinations of overtones to be formed such as higher frequency Stationary or standing Waves.

The curve which gets produced is known as Stationary Waves when two Waves pass in the opposite directions where it meets and generate interference. And this is set to be an example of constructive interference.

Standing Waves is the term used for describing Stationary Waves which are the opposite of progressive Waves. As the name suggests, they are Stationary Waves that do not travel through space in unit time.

## FAQs on Stationary Waves for IIT JEE

**1. What is the difference between progressive and Stationary Waves?**

Stationary Waves are stilla and they do not travel in any medium. Progressive Waves travel in the medium which are in different phases. Stationary Waves even though they are at rest they do perform simple harmonic motion (SHM) in varying amplitude. Progressive Waves, on the other hand, perform simple harmonic motion in their mean position.

**2. Why are Stationary Waves called Stationary?**

The two identical Waves traveling along the same path in opposite directions interfere with each other, the resultant Wave is called a Stationary Wave. Stationary Waves does not travel in any direction and there is no transport of energy.

**3. Does Stationary Waves transfer energy?**

Unlike the traveling Waves, the standing Waves do not cause net transport of energy as the two Waves which make them up are carrying equal energy in opposite directions. The particles right at the edge of the standing Wave do not move.

**4. Are Stationary Waves in phase?**

Since adjacent points are in phase, no energy is transferred from one point to the next, unlike a traveling Wave. Standing Waves or Stationary Waves are formed by the superposition of two traveling Waves of the same frequency with the same polarization and the same amplitude travels in opposite directions.

**5. What are the uses of Stationary Waves?**

Standing Waves or Stationary Waves in the you directions have various applications in music. A circular drum head is a simple system on which standing Waves can be studied. Instead of having nodes at opposite ends as in guitar and piano strings, the entire rim of the drum is a node.