Equation of destructive interference

Introduction

Destructive interference is a concept based on superposition of waves. Wave is the disturbance in a medium to transfer energy without the transfer of mass. Waves are usually in a periodic motion, like a simple harmonic oscillator. They have crests and troughs. The amplitude of the wave is the height of the wave. The distance covered between two consecutive crests is called the wavelength of the wave. The number of waves passed per second is called the frequency of the wave. Phase is not the property of waves. But it gives the relation between two signals with the same frequency. A particle is a localized object that possesses physical and chemical properties such as volume, mass, density, etc. They may be in different sizes like electrons, atoms, and powder which are subatomic, microscopic, and macroscopic particles.

What is wave interference?

Interference is a phenomenon of superposition of waves. When two waves travel in the same medium, they interfere with each other, known as wave interference. When the two waves are combined, their amplitudes are added at every single point, forming a resultant wave. The resultant wave may have a low, high, or the same amplitude. The interference effect is applied to different waves such as light, radio, acoustic, and water waves. The interference of waves is of two types. They are constructive interference and destructive interference.

What is destructive interference?

When the maxima of the two waves that are going to interact are 180 degrees out of phase with each other, the displacement of one wave is positive, and another is negative. So during the interaction, the displacement of each wave cancels each other. This happens when the amplitude of the first wave rises, the amplitude of the second wave descends, and vice versa. As the amplitude of the two waves cancels each other at each point, the resultant wave is no wave.

The most important criteria for interference are at least the appearance of two waves. The waves having the same amplitude and frequency can superimpose and undergo interference. If the two waves of comparable frequency superimpose on each other, the intensity of the resultant wave is different from the total intensity. The strength of the resultant wave is different at different points.

Equation of destructive interference

The interference between the two waves is represented mathematically as given below. Let us consider two waves which are having the same amplitude in the same medium. The equation of the wave which is traveling in a positive X-direction is given by,

$$\mathrm{Y_1=Acos(kx-\omega t)}$$

Here A denotes the amplitude

k denotes the wave number and $\mathrm{k=\frac{2π}{λ}}$

λ denotes the wavelength of the wave.

ω denotes the angular frequency of the wave and ω=2πf

Let us take the second wave with some phase difference and have the same amplitude and frequency. The equation of the second wave is given by

$$\mathrm{Y_2=Acos(kx-\omega t+\varphi)}$$

While superimposing, the amplitudes are added together.

$$\mathrm{Y_1+Y_2=Acos(kx-\omega t)+Acos(kx-\omega t+\varphi)}$$

$$\mathrm{ Y_1+Y_2=A[cos(kx-\omega t)+cos(kx-\omega t+\varphi)]}$$

$$\mathrm{cos A +cos B=2cos(\frac{A-B}{2})cos(\frac{A+B}{2})}$$

So,

$$\mathrm{Y_1+Y_2=2A[cos(\frac{kx-\omega t-kx+\omega t+\varphi}{2}) cos(\frac{kx-\omega t+kx-\omega t+\varphi}{2})]}$$

$$\mathrm{Y_1+Y_2=2A[cos(\frac{φ}{2}) cos(kx-ωt-\frac{φ}{2})]}$$

Constructive interference

For constructive interference, the phase difference is an even multiple of π. φ=0, 2π, 4π etc. So $\mathrm{cos(\frac{φ}{2})=1}$. The amplitude of the resultant wave is doubled.

$$\mathrm{Y_1+Y_2=2A cos(kx-\omega t)}$$

Destructive interference

For destructive interference, the phase difference is an odd multiple of π. φ=0, 2π, 4π etc. So $\mathrm{cos(\frac{φ}{2})=0}$. The amplitude of the resultant wave is zero.

Examples of destructive interference

Some examples of destructive interference are given below.

• Moving electrons and radio waves are also responsible for destructive interference. Destructive interference is easy for moving electrons and can follow destructive interference rules.

• In noise-cancelling headphones, destructive interference is an important factor. As the amplitude of the sound wave while playing a sound, is in opposite directions, destructive interference occurs, and they cancel out each other and the noise is reduced.

• One example of destructive interference is the gravity wave. As the gravity waves are travelling at different speeds at different places, frame finding is hard. That is why destructive interference occurs there.

• Light beams are also examples of destructive interference.

• Automobile mufflers are also examples of destructive interference. Generally, it is also denoted as a silencer. So it is fixed in any vehicle to cancel the noise.

• Most of musical instruments are examples of destructive interference.

• Speaker waves also undergo destructive interference.

Conclusion

The disturbance in the medium for transferring energy is called a wave. Phenomenon such as diffraction, interference, and polarization indicates that the light behaves like a wave. When two waves are combined, they superimpose and form interference waves. According to the combination, wave interference is classified into two. They are destructive interference and constructive interference. When two waves of the same amplitude are 180 out of phase with each other, their opposite amplitudes cancel each other and the amplitude of the resultant wave is zero. The equation of destructive interference and some of their examples were also discussed in detail in this tutorial.

FAQs

1.What are the properties of waves?

Properties of waves are given below.

• Amplitude − The maximum displacement done by the particle in the wave from the mean position.

• Time period − The time taken for one complete to and fro motion of the wave.

• Wavelength − How far away the two consecutive crests or troughs are given by the wavelength.

• Frequency − The number of to and fro motions in one second.

• Velocity − The distance covered by the wave in one second.

2.What is constructive interference?

If the maxima of two waves are in phase with each other, the displacement of two waves is positive or both are negative. So during the interaction of two such waves, the amplitude of the two waves is added together and the amplitude of the resultant wave is the sum of the amplitudes. That is known as constructive interference.

3.What is known as wave number?

In spectroscopy, the wave number is defined as the number of waves per unit distance. It is denoted as the spatial wave number, Its measurement is done by using cm^-1. Wave number is defined in theoretical physics as the number of radians that are present in the unit distance. It is denoted as the angular wave number. It is represented as,

$$\mathrm{ k=\frac{2π}{λ}}$$

4.What happens during destructive interference, and what is their phase difference and path difference?

During destructive interference two waves overlap in such a way that they cancel one another out. Two waves have a phase difference that is an odd multiple of π.

$$\mathrm{ Phase\: difference\: between\: waves \:= \phi= (2n-1) \pi}$$

Two waves' paths difference is an odd multiple of λ/2.

$$\mathrm{Path\: difference\: between\: the\: waves\: =\Delta= (2n-1) λ/2.}$$

5.What is the principle of the superposition of waves?

The principle of superposition is also denoted as superposition property. When two waves travel in the same plane they undergo superimpose and the resultant wave has the amplitude, which is the addition of the amplitude of the two individual waves.