Principle of Superposition of Waves

When two waves propagating in the same medium interfere with each other the amplitude of the resultant of the two waves is the vector sum of the amplitude of the two waves, this is called the Principle of Superposition of Waves. Waves are disturbances that transfer energy between two points without there being actual contact between the two points. We are completely surrounded by waves and these are used for performing a variety of tasks in our daily life. We encounter different types of such as Radio waves, Light waves, Microwaves and others on a regular basis. The superposition of waves is the process of adding different waves together and finding their results. 

In this article, we will learn about the Superposition of Waves and others in detail.

What is Superposition of Waves?

The concept of superposition of wave state that, “The resultant displacement of the waves in a medium formed by the number of individual waves is the vector sum of all the individual displacement formed by the individual waves at that point.”

Principle of Superposition

When two or more waves travel in the same medium, they are bound to interact with each other. They retain their wave nature after combining with each other, but usually, the resultant wave is different from both of the individual waves. The superposition principle helps us describe the resulting wave or motion that is produced when two or more waves combine with each other. 

The figure below shows two waves that are producing some displacement in the particles of the given medium.

In the figure given above, two waves with individual displacements y₁ and y₂ are given. Notice that the resultant wave from the superposition of these two waves has a greater displacement than the two individual waves. 

The principles of superposition can be applied to any type of wave provided that: 

• The waves that are superimposed are of the same type.
• The medium in which the waves are propagating behaves linearly, this means that the particles of the medium which are displaced with twice the displacement experience twice the amount of restoring force on them.

In the case of waves, the figure above shows two waves that are travelling in opposite directions. These waves produce equal displacements in the rope.

In mathematical terms, the superposition principle can be described as given below. Let’s say y₁(x, t) and y₂(x, t) is the displacements produced by two waves in the medium. Let P be the point where these two ways come and meet. Now using the principle of superposition to find the resultant displacement (y). 

y = y₁(x, t) + y₂(x, t) 

If two or more waves are travelling and meeting at one point in a medium and the wave functions for the individual waves are given by, 

y = f₁(x – vt)

y = f₂(x – vt)

… 

y = f_n(x – vt)

The resultant wave after displacement is given by, 

y = f₁(x – vt) + f₂(x – vt) +  f₃(x – vt) + …. f_n(x – vt)

Types of Superposition of Waves

Generally, the superposition of waves is studied under two headings,

• Constructive Interference
• Destructive Interference

Constructive Interference

When two waves in the same phase superimpose their resultant amplitude is equal to the sum of the individual amplitude of the wave. This phenomenon is called Constructive Interference. In this case, their amplitude increases the most.

Let us consider two waves that are travelling at the same velocity. Since these two waves travel at the same velocity and meet at a particular point. Let’s analyze the resultant amplitude of the wave which originates after the superposition of these waves. Assuming that the equations for displacement produced by both the waves are the same and are given by, 

y₁ = acos(ωt) 

y₂ = acos(ωt) 

Now using the principle of superposition to find the resultant displacement (y). 

y = y₁ + y₂ 

y = acos(ωt) + acos(ωt) 

y = 2acos(ωt) 

It’s known that the intensity is proportional to the square of the amplitude. Since amplitude, in this case, becomes twice the original amplitude. The resulting intensity is given by

I = 4I₀

where 
I₀ is the intensity of the original wave

Condition for Constructive Interference

We know that the Intensity of the resultant wave is given using the formula,

I = I₁ + I₂ +2√(I₁I₂)cosθ

For constructive Interference, I is the maximum which is possible if 

cosθ = 1 , then θ = 0, 2π, 4π,…., 2nπ i.e.

Phase Difference is even multiple of π

Also, 

△x = λ/2π (θ)

△x = λ/2π (2nπ)

△x = nλ

Thus, for Constructive Interference

• Phase Difference: 2nπ
• Path Difference: nλ

Destructive Interference

When two waves in the opposite phase superimpose their resultant amplitude is equal to the sum of the individual amplitude of the wave. This phenomenon is called Destructive Interference. In this case, their amplitude decreases the most.

Let us consider two waves that are travelling at the same velocity. Since these two waves travel at the same velocity and meet at a particular point. Let’s analyze the resultant amplitude of the wave which originates after the superposition of these waves. Assuming that the equations for displacement produced by both the waves are the opposite this time and are given by, 

y₁ = acos(ωt) 

y₂ = acos(ωt + 3π) 

y = y₁ + y₂

y = acos(ωt)  + acos(ωt + 3π)

y = acos(ωt) – acos(ωt) 

y = 0

So, this time two displacements in opposite directions resulted in zero amplitude and zero intensity. This is called destructive interference. 

Note that the case of zero amplitude is only when the two waves are of the same amplitude but in opposite directions. However, this is not mandatory case to have waves of the same amplitude. Hence, the below figure represents a general concept of destructive interference for waves of different amplitude and opposite directions.

Condition for Destructive Interference

We know that the Intensity of the resultant wave is given using the formula,

I = I₁ + I₂ +2√(I₁I₂)cosθ

For constructive Interference, I is the minimum which is possible if 

cosθ = -1, then θ = π, 3π,…., (2n-1)π i.e.

Phase Difference is the odd multiple of π

Also, 

△x = λ/2π (θ)

△x = λ/2π ((2n-1)π)

△x = (n – 1/2)λ

Thus, for Destructive Interference

• Phase Difference: (2n-1)π
• Path Difference: (n – 1/2)λ

Resultant Intensity in Interference of Two Waves

When two waves superimpose their resultant intensity can be easily calculated and to calculate their intensity let’s take an example.

Take two waves of displacements y₁ and y₂ that superimpose at point O in 3-D space and the phase difference between these two waves be “φ”. Now we can write the equation of wave y₁ and y₂ as,

y₁ = a sin ωt

y₂ = b sin ( ωt + φ)

where a and b are the amplitude of y₁ and y₂

The resultant displacement of the wave is given using the,

y = y₁ + y₂

y = a sin ωt + b sin (ωt + φ) . . . . . . . . . . (1)

y = a sin ωt + b sin ωt cos φ + b cos ωt sin φ

y = (a + b cos φ)sin ωt + b sin φcos ωt

Taking 

(a + b cos φ) = A cosθ and b sin φ = A sinθ

y = A cosθ sin ωt + A sinθ cos ωt

y = A (cosθ sin ωt + sinθ cos ωt)

y = A sin(ωt + θ)

Now as,

(a + b cos φ) = A cosθ…(a)

b sin φ = A sinθ…(b)

Now adding and squaring eq. (a) and eq. (b)

 (A cosθ)² + (A sinθ)² = (a + b cos φ)² + (b sin φ)²

A² (cos²θ + sin²θ) = a² + b² + 2absinφ

A = √(a² + b² + 2absinφ)…(c)

This is the formula for the amplitude of the resultant wave.

Also, Dividing eq.(b) by eq.(a)

tan θ = (b sin φ)/(a + b cos φ)…(d)

We know that,

I = KA²

where
I is the intensity of the wave
A is the amplitude of the wave

From (c) we get,

I = I₁ + I₂ +2√(I₁I₂)cosθ

What is Interference of Light?

When two or more light rays interact at some point the resultant wave so formed is called the superimposed wave. These superimposed waves have maximum intensity and maximum amplitude at some points and minimum intensity and minimum amplitude at some other points. This phenomenon is called the Interference of Light

Read More,

• Radio Waves
• Electromagnetic Waves
• Properties of Waves

Solved Examples on Superposition of Waves

Example 1: Two waves travelling in a medium are given by the following equations, 

y₁ = 2acos(ωt) 

y₂ = 2acos(ωt + π) 

Find the resulting amplitude after their superposition. 

Solution: 

y = 2acos(ωt) + 2acos(ωt + π)

y = 2acos(ωt) – 2acos(ωt)

y = 0 

The resulting amplitude becomes zero. 

Example 2: Two waves travelling in a medium are given by the following equations, 

y₁ = acos(0.5ωt) 

y₂ = acos(0.5ωt + 2π) 

Find the resulting amplitude after their superposition. 

Solution: 

y = acos(0.5ωt) + acos(0.5ωt + 2π)

y = acos(0.5ωt) + acos(0.5ωt)

y = 2acos(0.5ωt)

The resulting amplitude becomes “2a”. 

FAQs on Superposition of Waves

Q1: What is Superposition of waves?

Answer:

When two waves travelling in a similar medium interact their resultant displacement is the vector sum of their individual displacement at that point.

Q2: What is Superposition of Waves Formula?

Answer:

The superposition of wave formula state that for two waves y₁ and y₂ when they interact their resultant wave is given using the formula,

y₁ = a sin ωt
y₂ = b sin ( ωt + φ)

Resultant (y) = A sin(ωt + θ)

where
A = √(a² + b² + 2absinφ)

Q3: What are Nodes and Antinodes?

Answer:

The highs made by the waves while propagating are called the Nodes whereas the lows made by the waves while propagating are called the Antinodes

Q4: What is Constructive Interference?

Answer:

If the two waves in the same phase interact the interference pattern formed by them is called constructive interference. In constructive interference, the amplitude and the intensity of the superimposed wave is the highest.

Q5: What is Destructive Interference?

Answer:

If the two waves in the opposite phase interact the interference pattern formed by them is called destructive interference. In destructive interference, the amplitude and the intensity of the superimposed wave is the lowest.