Water, sound, light, and even invisible fields like radio waves are all examples of waves. In actual life, multiple waves frequently pass through the same area simultaneously. Thus, the question that naturally arises is:
When two or more waves traverse in the same medium, the displacement of any element of the medium is the algebraic sum of the displacement due to each wave, this known as the principle of superposition of waves.
Applying superposition principle, the resultant wave will be the algebraic sum of the two waves, y = y1 + y2 y (x, t) = a sin(kx – ωt) + a sin(kx + ωt + ϕ)
On solving we get,
y (x, t) = 2a cos ϕ / 2 sin (kx – ωt + ϕ / 2)
For constructive interference ϕ = 0, 2π, 4π, …..
For deconstructive interference ϕ = π, 3π, 5π, ….
The amplitude of the resultant wave depends on the phase difference ϕ.
What happens when two or more waves meet each other?
The answer lies in a very important and simple rule called the Principle of Superposition of Waves.
What is the Principle of Superposition?
In simple words, the Principle of Superposition states:
This implies that waves just add up when they overlap. The effect you observe at any one time is simply the total of the actions that each wave would have taken on its own; the waves do not permanently disturb one another; rather, they flow through one another.
A Closer Look with an Example
Suppose ourself standing next to a quiet pond. Two little stones are dropped into the water at two separate locations. Every stone produces waves that radiate outward in a circular pattern. The two sets of waves begin to overlap after a few seconds. Where do they meet?
The displacement, or water height, at each location where two waves overlap, is equal to the sum of the wave heights. The water rises higher if both waves push it upward at the same spot. Two waves partially cancel each other out if one pushes up and the other pulls down.
The principle of superposition precisely captures this accumulation of wave effects.
Mathematically
Two waves are traveling through the same medium.The displacement of the first wave at a point is y1 and the displacement due to the second wave is y2.
According to the principle of superposition, the total displacement y at that point is:
y = y1+y2
If there are more than two waves, you just keep adding their displacements:
y = y1+y2+y3+…
Depending on the direction (e.g., upward or downhill, forward or backward), displacement in this case can be either positive or negative.
Important Points to Remember
Only when waves are of the same kind, such as all electromagnetic waves (like light) or all mechanical waves (like sound), does superposition occur.
The medium through which the waves are traveling must be the same.
The waves don’t permanently alter or destroy one another. They move on as if nothing had happened after passing through one another.
Interference (where waves combine to make larger waves or cancel each other out) and beats (a regular variation in loudness when two sounds of slightly different frequencies are played simultaneously) are two intriguing effects that can be produced by superposition.
Types of Superposition
Depending on how the waves interact, superposition can create two special types of effects:
1.Constructive Superposition:Two waves reinforce one another and form a larger wave if they collide in a way that causes both of their displacements to be in the same direction, either upward or downward.
2. Destructive Superposition:
Two waves can partially or totally cancel each other out, producing a smaller wave or perhaps no wave at all, if they collide and their displacements are in opposing directions (one upward, one downward).
Real-Life Applications
There are applications for the superposition principle everywhere; it is not only a theoretical concept.
Destructive interference is used by noise-canceling headphones to block out undesired sounds.
Rich, complex tones are produced in musical instruments by combining various sound waves.
Electromagnetic waves are superposed in radio signals and Wi-Fi.
Light interference patterns are utilised in physics labs to detect very small distances with remarkable accuracy.
Summary,
By simply combining the effects of each wave at each location, the Principle of Superposition of Waves explains how numerous waves interact. It is a straightforward effective concept that aids in our understanding of wave behaviour in practically every branch of science and technology.
According to the superposition principle, the displacement that results from the intersection of two or more waves at a certain location is equal to the algebraic total of the displacements that each wave causes.
In the early 19th century, while researching the behaviour of light and other wave phenomena, scientists like Thomas Young and Augustin-Jean Fresnel developed and clarified the theory of superposition.
Yes, as long as the waves are of the same kind and moving through the same medium, the principle of superposition holds true for both mechanical waves (such as sound and water waves) and electromagnetic waves (such as light and radio waves).
The principle of superposition states that two waves moving in opposite directions will superpose when they collide. They can produce constructive interference, which results in a bigger displacement, or destructive interference, which results in a smaller or zero displacement, depending on their displacements.
When waves accumulate because their displacements are in the same direction, a larger final wave is produced. This phenomenon is known as constructive superposition.
When waves have opposite displacements, they can partially or totally cancel each other out, a phenomenon known as destructive superposition.
No, the effect of waves superposing is transient. Following interaction, each wave keeps moving in the same direction while maintaining its initial characteristics.
Examples are:
Beats produced when two sound waves of slightly different frequencies are played together.
Light interference patterns (such as the vibrant patterns on soap bubbles).
Noise-canceling headphones that use destructive interference to cut down on unwanted sound.