When stating the principle of superposition of waves, we learn that the resultant displacement of multiple waves in a medium at a specific point is the vector summation of different displacements. These individual displacements relate to every wave at a specific point. Read on to learn more about the principle of superposition of waves.
Principle of Superposition of Waves
Suppose we have two waves traveling simultaneously along the exact stretched string but in directions opposite to one another. We will notice the production of waveforms at every instant of time in the string.
Further, we will observe that any string element’s net displacement at a given time is equal to the algebraic summation of displacement due to every single wave.
Let us consider that two waves are traveling separately, then we express the displacement of any element regarding these two waves as:
y1 = f1(x–vt),
y2= f2(x–vt)
…
yn = fn (x–vt)
The expression for wave function that describes the disturbance within the medium is:
y = f1(x – vt)+ f2(x – vt)+ …+ fn(x – vt)
or, y = ∑ fi (x−vt)
At a time when there is an overlap between these two waves, we express the resultant displacement as:
y(x,t)
Mathematically, y (x, t) = y1(x, t) and y2(x, t)
According to the principle of superposition of waves, we are allowed to conduct an algebraic addition of the overlapping waves to find a resultant wave.
Let us consider these expressions as wave functions of the traveling waves:
Expression for a wave that has been traveling along a stretched string:
y1(x, t) and y2(x, t)(x, t) = A sin (kx – ωt) and another wave, shifted from the first by a phase φ, given as y2(x, t) and y2(x, t)(x, t) = A sin (kx – ωt + φ)
The above equation depicts that both waves maintain the similar angular wave number ‘k’ and similar angular frequency. Hence their wavelength and amplitude are also the same.
Here when we apply the principle of superposition of waves, we find that the resultant wave is some of the two consecutive waves algebraically. It maintains displacement:
y(x, t) = A sin (kx – ωt) + A sin (kx – ωt + φ)
We can write the above equation as:
y(x, t) = 2A cos (ϕ/2) sin (kx − ωt + ϕ/2)
Here the resultant wave is known as a sinusoidal wave which travels towards the positive X direction. Here the angle of the face is half of the phase difference of different waves. And the amplitude value is [2cos (ϕ/2)] times the amplitude values of the original waves.
Explain the interference of light
When there is the formation of highest intensity at specific points and lowest intensity at other specific points when two or more waves with similar frequency and constant phase difference meet at a single point at the same time, they get superimposed with one another, and this phenomenon is known as interference of light.
What are the different types of superposition of waves?
As per the phase difference in superimposing waves, there are two classifications in which interference is divided:
Constructive Interference
When two different waves superimpose with one another within a similar phase, then the resultant amplitude becomes equal to the amplitude summation of each wave. This results in the highest intensity of light, and this occurrence is termed constructive interference.
Destructive Interference
When two different waves superimpose with one another in the opposite phase, then the resultant amplitude becomes equal to the amplitude differences of each wave. This results in the lowest intensity of light, and this occurrence is termed destructive interference.
Conclusion
The principle of superposition of waves contains important parts regarding different types of waves and their functions. Here we covered the definition of the principle of superposition of waves in detail; we explored the calculation of superposition of waves and learned about constructive and destructive interference.
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