When two or more waves pass through the same region of space at the same moment in time, interference takes place. The principle of superposition provides a way to combine the waves. The combined wave, also called the composite wave, is obtained by taking the algebraic sum of displacements at every single point. The two main types of wave interference are Constructive interference and Destructive interference.

Figure 1: Superposition of waves

If two waves are in phase, they interfere constructively. This is called constructive interference. That is, the peaks of the waves will be in phase. When two waves are completely out of phase with each other, destructive interference occurs. These two waves are out of phase by 180 degrees or π radians and hence they interfere destructively and cancel each other out.

Principle of Superposition

When two waves of the same frequency and amplitude combine, they interfere with each other. If the waves have the same frequency and move in the same direction in a medium, their combined intensity at any given point can be different from the sum of their individual intensities. Sometimes, the combined wave is stronger at certain points and weaker at others.

In the case of waves, the crests represent positive amplitude, and the troughs represent negative amplitude. The Principle of Superposition describes the amplitude of waves before and after the collision. It states that at any given point in a medium, the resulting displacement is equal to the sum of the individual displacements caused by each wave passing through that point. The superposition of waves is shown in figure 1.

During constructive interference, the crests of two waves combine. For example, if one wave has an amplitude of +1 V and the other has an amplitude of +2 V, the resulting wave will have an amplitude of 3 V. In contrast, when a crest and trough combine, they can cancel each other out or reduce the amplitude of the resulting wave. For example, if one wave has a crest that measures +1 V in amplitude and another wave has a trough with a measurement of -2 V, the resultant wave will have an amplitude of -1 V at that particular point. 

Consider two waves having the same frequency traveling in the same direction:

Case 1: Constructive Interference

Figure 2: Constructive Interference

The two waves must travel through the same medium before constructive interference occurs. Also, the waves must have the same properties. i.e.; they should have the same amplitude and wavelength. The height of the crest or trough gives the amplitude of the wave. In order to determine the wavelength, the distance between two neighbouring crests or troughs of a wave is measured. When two waves intersect, their crests or troughs merge, leading to the formation of a new wave that has a higher amplitude. 

When the two waves (shown in figure 2) are combined by adding them point-by-point, the crests and troughs of the interfering waves meet. The resulting wave appears similar to the original waves, but with a higher amplitude. This occurrence is referred to as constructive interference, where the resultant wave's magnitude is greater than either of the two initial waves. Hence, a bigger wave is formed by adding the waves together. 

There is a certain condition for constructive interference to occur. It is given by the path difference and phase difference between the waves that interfere.

Condition for Constructive interference:

The path difference between the waves is equal to an integer multiple of the wavelength

Path difference between the waves is given by,

Where n is an integer representing the number of wavelengths difference between the waves (n=0,1,2,3) and λ is the wavelength of the waves

The phase difference between the two waves is an even multiple of π.

Phase difference between the waves,

Case 2: Destructive Interference

Figure 3: Destructive Interference

Destructive interference is a phenomenon that occurs when the crests and troughs of two waves that interfere meet. This interference causes their amplitudes to cancel out and a flat line is created instead of a larger wave. This phenomenon happens when the crest of one wave coincides with the trough of another wave. The crest will cancel out the trough, causing the medium to remain undisturbed. For two waves to exhibit destructive interference, they must have equal amplitudes that are opposite in direction.

When one wave is at its peak, the other wave is at its lowest point, resulting in the sum of the two waves being zero (as shown in figure 3). Similarly, when one wave is at its lowest point, the other wave is at its peak, resulting in a sum of zero. Hence, they cancel each other and there will be no wave left. This phenomenon is known as destructive interference, where the sum of two waves can be zero.

Condition for Destructive interference:

The path difference between the waves is equal to a half-integer multiple of the wavelength.

Path difference between the waves is given by,

Where n is an integer representing the number of half-wavelengths difference between the waves (n=0,1,2,3) and λ is the wavelength of the wave

The phase difference between the two waves is an odd multiple of π.

Phase difference between the waves, 

Applications of Interference

Constructive interference is used to enhance the sound quality of recordings in music production. For example, multiple microphones can be positioned around a musical instrument to capture the sound waves generated from different directions, and the signals can be combined to create a more detailed and natural sound. Destructive interference can also be used to reduce unwanted noise or echoes by canceling out specific frequencies.

In optics, constructive interference is used to create interference patterns that reveal information about the properties of light, such as its wavelength and polarization. This technique is commonly used in interferometry, which is the measurement of small distances, angles, or changes in refractive index.

Constructive interference is also used to improve signal strength by focusing the waves in a specific direction. This is achieved by using antenna arrays that are designed to produce constructive interference in the desired direction. Destructive interference is used to reduce interference from other radio sources operating on the same frequency.

In quantum mechanics, constructive and destructive interference play a crucial role in the behavior of subatomic particles, such as electrons and photons. These phenomena are exploited in many applications, including quantum computing, where the interference of quantum states is used to perform complex calculations.