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
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.
Table of Content
- What is Superposition of Waves?
- Principle of Superposition
- Types of Superposition of Waves
- Constructive Interference
- Destructive Interference
- Resultant Intensity in Interference of Two Waves
- What is Interference of Light?
- Solved Examples on Superposition of Waves