Stationary Waves for IIT JEE

Stationary Waves - Definition and Equations

Definition:


“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 which is the outcome of interference that means when the waves are superimposed; their energies are added at the same time or cancelled. When waves move in the same direction, then their interference generates travelling waves. When waves move in opposite directions, then their interference generates an oscillating wave which is fixed in space. 

The rope which is vibrating tied at one end generates a standing wave. The wave train will get reflected back 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 get shifted by 1/8 of wavelength.
Case III: In this case, another 1/8 period later, the amplitudes of wave train 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


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


Suppose the progressive waves of amplitude and wavelength λ travel in the X-axis direction.

y1 = a sin 2π [t/T – 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:
y2 = a sin 2π [t/T + x/λ] ---------- (2)
As per the objective of superposition, the final displacement is:
y = y1+y2
= a [sin 2π (t/T – x/λ) + sin 2π (t/T + x/λ)]
= a [2sin (2πt/T) cos (2πx/λ)]
So, y = 2a cos (2πx/λ) sin (2πt/T) ---------- (3)

Thus, this is the equation of stationary waves. There are some observed points described below.
a. 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.
b. 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.

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

d. 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


1. The form of wave remains stationary.
2. Nodes and antinodes are produced alternately.
3. The points where the value of displacement is zero are known as nodes and the points where the value of displacement is maximum is now known as antinodes.
4. The changes in pressure are maximum at nodes and minimum at antinodes.
5. The particles except for the position of the node, implement the simple harmonic movements of the same period.
6. 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.
7. The speed and velocity of all the particles at the nodes are 0 which increases rapidly and is the maximum at the antinodes.
8. 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.
9. 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.
10. Particles in the same division vibrate in the same phase and between the neighbouring 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 tension, it is set into vibration, where transverse waves move to the end of the wire and then reflect back. Therefore, stationary waves are produced.

Sonometer


The sonometer includes a hollow sounding box which is a meter long in length 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 is 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


1. 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 are 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 fundamental frequency.
n1 = v/λ1
Hence, n1 = (1/2l) √T/m …... (3)
2. 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 favourites is its use in the musical instruments.

In the string musical instruments, a bowed or plucked string creates the note as certain frequencies of standing or stationary waves have the ability to 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 cancelled fast and thus, it is the stationary wave frequencies which we hear.

In the same way, in the instruments of woodwind family, we create the notes as stationary waves have the ability to produce in the tube of air inside the musical instruments.

Strings and woodwind instruments sound or tones are different from each other as they provide different types of combination 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 opposite of the progressive waves. As the name suggests, they are stationary waves which do not travel through space in unit time.