What is Superposition of Waves?

Superposition of Waves is a phenomenon that occurs when two or more waves interact and the result is a combination of the individual waves. The total displacement at any point in space is the sum of the individual displacements caused by each wave.

According to the principle of superposition, the displacement at a specific point in a medium caused by a combination of waves is the vector sum of the individual displacements caused by each wave at the same point.

Principle of Superposition of Waves

We can observe that the net displacement of any element of the string at a given time is the result of the algebraic sum of the displacements caused by two waves travelling simultaneously along the same stretched string in opposite directions, as illustrated in the figure above. The waveforms of these waves can be seen in the string at each instant of time.

The resultant displacement when two waves travelling alone overlap can be represented by y(x,t), where y1(x, t) and y2(x, t) are the displacements of any element of the two waves.

Mathematically, $y(x, t) = y_1(x, t) + y_2(x, t)$

We can use the principle of superposition to add the overlapped waves algebraically and obtain the resultant wave. Let us assume that the wave functions of the moving waves are…

y1 = f1(x - v*t)

y2 = f2(x - v*t)

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The dog barked loudly.

y_n = f_n(x - v_t)

Then the wave function describing the disturbance in the medium can be expressed as

y = $f_1(x - vt) + f_2(x - vt) + \dots + f_n(x - vt)$

y = $\sum_{i=1}^{n} f_i(x-v_t)$

Let us consider two waves travelling along a stretched string, given by:

• y1(x, t) = A sin (kx – ωt)
• y2(x, t) = A sin (kx – ωt + φ), shifted from the first by a phase φ

We can conclude from the equations that both the waves have the same angular frequency, angular wave number k, wavelength and the same amplitude A.

The resultant wave has displacement $y(x, t) = A \sin (kx - \omega t) + A \sin (kx - \omega t + \phi)$, which is obtained by applying the superposition principle.

The above equation can be rewritten as:

y(x, t) = 2A cos(kx - ωt + ϕ/2). sin(ϕ/2)

The resultant wave is a sinusoidal wave, travelling in the positive X direction, with the phase angle being half of the phase difference of the individual waves, and the amplitude being [2cos ϕ/2] times the amplitudes of the original waves.

Superposition of Waves - Video Lesson

Interference of Light is the phenomenon in which two waves superimpose to form a resultant wave of greater, lower, or the same amplitude. It typically occurs when two waves of the same frequency are traveling in opposite directions and overlapping.

The phenomena of formation of maximum intensity at some points and minimum intensity at some other point when two (or) more waves of equal frequency having constant phase difference arrive at a point simultaneously, superimpose with each other is known as interference.

Types of Superposition of Waves

According to the phase difference in superimposing waves, interference can be categorized into two types:

Constructive Interference

If two waves superimpose with each other in the same phase, the amplitude of the resultant is equal to the sum of the amplitudes of individual waves resulting in the maximum intensity of light, this is known as constructive interference.

Destructive Interference

If two waves superimpose with each other in opposite phase, the amplitude of the resultant is equal to the difference in amplitude of individual waves, resulting in the minimum intensity of light, this is known as destructive interference.

Resultant Intensity in Interference of Two Waves

The resultant displacement at point p when two waves of vertical displacements $y_1$ and $y_2$ superimpose is given by:

y = y1 + y2

Waves are meeting at some point p at the same time, with the only difference being in their phases. The displacements of individual waves are given by.

y1 = a \sin \omega t

y2 = b \sin (\omega t + \varphi)

Where $a$ and $b$ are their respective amplitudes and $\Phi$ is the constant phase difference between the two waves.

Applying the superposition principle as stated above, we obtain:

y = a sin ωt + b sin (ωt + θ) ~~~~~ (1)

Representing equation 1 in the phasor diagram

The resultant has an amplitude A and a phase angle relative to wave -1.

y = A * sin(ω * t + θ)

A sin(ωt + θ) = a sin ωt + b sin (ωt + θ)

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For destructive interference, the minimum intensity, Imin, occurs when the phase difference, φ, is equal to -1 (cos φ = -1).

When $\cos \phi = -1$

φ = π, 3π, 5π, …

⇒ φ = (2n - 1)π, where n = 1, 2, 3, …

Of Δx is the path difference between the waves at point p.

(\Delta x = \frac{\lambda}{2\pi} \phi)

(\Delta x=\frac{\lambda}{2\pi}\left(2n-1\right)\pi)

Therefore, $\Delta x = \frac{2n-1}{2}\lambda$

Condition for Destructive Interference

‘Phase Difference = $(2n - 1)\pi$’

Path Difference = (2n - 1)λ/2

I = I_{min}

(\begin{array}{l}{I_{\min}} = I_1 + I_2 - 2\sqrt{I_1I_2}\end{array})

\(\left( \sqrt{{I_1}} - \sqrt{{I_2}} \right)^2\)

Conditions for Interference of Light

Sources must be consistent

Coherent sources must have the same frequency (mono chromatic light source).

The amplitudes of the waves from the coherent sources should be equal.

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#Frequently Asked Questions on Superposition of Waves

What is the result of wave superposition?

Constructive Interference and Destructive Interference.

Nodes and Antinodes

A node is a point along a standing wave where the wave has minimal amplitude. An antinode is a point along a standing wave where the wave has the maximum amplitude.

Nodes are points of zero amplitude, and antinodes are points of maximum amplitude.

What is Constructive Interference?

Constructive interference occurs when two waves meet, resulting in a wave with a greater amplitude than either of the individual waves. This can happen when two waves are in phase, meaning they have the same frequency and amplitude, and their peaks and troughs line up.

The resultant wave from constructive interference is bigger than either of the two original waves when two waves are added together by superposition. This wave has a larger amplitude but looks similar to the original waves.

What is Destructive Interference?

Destructive interference is a phenomenon that occurs when two waves of the same frequency combine to form a resultant wave with a smaller amplitude than the original waves. This occurs when the two waves are out of phase with each other, meaning that the peaks of one wave line up with the troughs of the other.

When the two waves superimpose, the sum of two waves can be less than either wave and can also be zero. This is called destructive interference.— title: “Superposition Of Waves” link: “/superposition-of-waves” draft: false

What is Superposition of Waves?

Superposition of waves is a principle in physics which states that when two or more waves come together, the resulting wave is the sum of the individual waves. This means that the amplitude of the resulting wave is equal to the sum of the amplitudes of the individual waves.

According to the principle of superposition, the displacement of a medium at a certain point due to a combination of waves is the sum of the individual displacements of each wave at that point.

Principle of Superposition of Waves

We can observe that the net displacement of any element of the stretched string at a given time is the algebraic sum of the displacements caused by the two waves travelling in opposite directions, as shown in the waveforms.

The resultant displacement of two waves travelling alone, when they overlap, can be represented by $y(x,t)$, where $y1(x,t)$ and $y2(x,t)$ are the displacements of any element of these two waves.

Mathematically, $$y(x, t) = y_1(x, t) + y_2(x, t)$$

As per the principle of superposition, we can add the overlapped waves algebraically to produce a resultant wave. Let us assume that the wave functions of the moving waves are $\psi_1$ and $\psi_2$. Then, the resultant wave can be calculated as $\psi_{resultant} = \psi_1 + \psi_2$.

y1 = f1(x - v*t)

y2 = f2(x - v * t)

It’s a beautiful day!

Yes, it sure is a beautiful day!

y_n = f_n(x - v_t)

Then the wave function describing the disturbance in the medium can be expressed as

y = $f_1(x - vt) + f_2(x - vt) + \dots + f_n(x - vt)$

$y = \sum_{i=1}^{n} f_i(x - v_t)$

Let us consider two waves travelling along a stretched string, given by:

1. y1(x, t) = A sin (kx - ωt)
2. y2(x, t) = A sin (kx - ωt + φ), shifted from the first wave by a phase φ

From the equations, we can see that both the waves have the same angular frequency, same angular wave number k, hence the same wavelength and the equal amplitude A.

The resultant wave has displacement $y(x, t) = 2A \sin(kx - \omega t + \phi)$

The above equation can be written as:

y(x, t) = 2A cos(kx - ωt + ϕ/2). sin(ϕ/2)

The resultant wave is a sinusoidal wave, travelling in the positive X direction, with a phase angle of $\frac{1}{2}\phi$, where $\phi$ is the phase difference of the individual waves, and an amplitude of $2\cos\frac{\phi}{2}$ times the amplitudes of the original waves.

Superposition of Waves - Video Lesson

What is Interference of Light?

Interference of Light is the phenomenon in which two or more light waves interact with each other, resulting in the reinforcement or cancellation of their combined amplitudes. This phenomenon is most commonly observed in the form of an interference pattern created by overlapping waves.

The phenomena of formation of maximum intensity at some points and minimum intensity at some other point when two (or) more waves of equal frequency having constant phase difference arrive at a point simultaneously, superimpose with each other is known as interference.

Types of Superposition of Waves

According to the phase difference in superimposing waves, interference is divided into two categories: constructive interference and destructive interference.

Constructive Interference

If two waves superimpose with each other in the same phase, the amplitude of the resultant is equal to the sum of the amplitudes of individual waves resulting in the maximum intensity of light, this is known as constructive interference.

Destructive Interference

If two waves superimpose with each other in opposite phase, the amplitude of the resultant is equal to the difference in amplitude of individual waves, resulting in the minimum intensity of light, this is known as destructive interference.

Resultant Intensity in Interference of Two Waves

The resultant displacement at point p when two waves of vertical displacements, y1 and y2, superimpose is given by:

y = y1 + y2

Waves are meeting at some point p at the same time, with the only difference being in their phases. The displacements of individual waves are given by.

y1 = a \sin(\omega t)

y2 = b \sin (\omega t + \phi)

Where $a$ and $b$ are their respective amplitudes and $\Phi$ is the constant phase difference between the two waves.

Applying the superposition principle as stated above, we obtain:

y = a sin ωt + b sin (ωt + θ) …………………………………. (1)

Representing equation 1 in a phasor diagram

The resultant has an amplitude A and a phase angle of -1 with respect to the wave.

y = A * sin(ω * t + θ)

A sin($\omega t + \theta$) = a sin $\omega t$ + b sin ($\omega t + \theta$)

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For destructive interference, the minimum Intensity, I = Imin, is achieved when the value of cosφ is equal to -1.

When $\cos{\phi} = -1$

φ = π, 3π, 5π, …

⇒ φ = $(2n - 1)\pi$, when $n = 1, 2, 3, \ldots$

Of Δx is the path difference between the waves at point p.

\(\Delta x = \frac{\lambda \phi}{2\pi}\)

(\Delta x=\frac{\lambda}{2\pi}\left(2n-1\right)\pi)

Therefore, $\Delta x = \frac{(2n - 1)}{2}\lambda$

Destructive Interference Conditions

Phase difference = $(2n - 1)\pi$

Path Difference = (2n - 1)λ/2

I$_{min}$ = I

(\begin{array}{l}{I_{\min }}=I_1+I_2-2\sqrt{I_1I_2}\end{array})

\(\left( \sqrt{{I_1}} - \sqrt{{I_2}} \right)^2\)

Interference of Light Conditions

Sources must be consistent

Coherent sources must have the same frequency (mono chromatic light source).

The amplitudes of the waves from the coherent sources should be equal.

Related Articles:

HC Verma Solutions for Physics

HC Verma Solutions Vol 1

HC Verma Solutions Vol 2

#Frequently Asked Questions on Superposition of Waves

What is the result of superposition of waves?

Constructive Interference and Destructive Interference.

Nodes and Antinodes

Nodes are points along a standing wave where the wave has minimal amplitude. Antinodes are points along a standing wave where the wave has maximum amplitude.

Nodes are points of zero amplitude, and antinodes are points of maximum amplitude.

What is Constructive Interference?

Constructive interference is the phenomenon that occurs when two waves of the same frequency combine to form a wave of greater amplitude. The two waves are said to be “in-phase” and the resulting wave is of greater amplitude than either of the two original waves.

The resultant wave from constructive interference is larger in amplitude than either of the two original waves when they are added together by superposition.

What is Destructive Interference?

Destructive interference is a phenomenon in wave theory where two waves of the same frequency combine to form a wave with a smaller amplitude. This occurs when the two waves are out of phase with each other, meaning that the crest of one wave coincides with the trough of the other wave.

When the two waves superimpose, the sum of two waves can be less than either wave and can also be zero. This is called destructive interference.