In this explainer, we will learn how to define wave interference and describe the constructive and destructive interference of waves that are mutually in or out of phase.

Suppose that two material objects, meaning objects made of matter, travel toward the same point and will reach the point at the same time. When the objects are sufficiently close to the point that they contact each other, they collide. The objects then exert forces on each other and so change their direction of motion. This is shown in the following figure.

Suppose now that two light beams travel toward the same point and will reach the point at the same time. When the light beams reach the point that they are both traveling toward, they both pass that point without any change of direction of motion. This is shown in the following figure in which successive wave fronts of the two beams are represented by the green and pink lines.

Suppose now that we replace each of the beams by a pulse that consists of one wavelength.

We can initially model the light waves as transverse waves that have values of displacement at different positions. The positions of these pulses and the displacements of the pulses at the points around their positions at two instants are shown in the following figure.

The two pulses will eventually reach a position where they overlap. At this position, the pulses will interfere with each other.

The interference between two pulses is simpler to model for two pulses that are traveling along the same line in opposite directions, as shown in the following figure.

Let us consider the situation where the leading wave fronts of the two pulses reach the same point at the same time, as shown in the following figure.

To understand what happens when light waves interfere, it is helpful to recall that light waves are electromagnetic waves. Light waves exert electric and magnetic forces. The displacement of the wave at a point indicates the direction and relative magnitude of the force.

We can represent the forces exerted at equally spaced points along a pulse with arrows that correspond to force vectors, as shown in the following figure.

Using this way of representing electromagnetic waves, the two pulses reaching the same point at the same time can be represented by the following figure.

The red cross shows the point at which the pulses meet. There is no force exerted by either pulse at that point.

We can now imagine a very short time later, when the pulses begin to overlap. This is shown in the following figure.

We can see that at the point where the pulses first met, the arrows from both pulses now have the same direction.

Two vectors can be added by connecting the heads of one vector to the tail of the other vector. If all the forces due to both pulses are now considered just as forces, not distinguishing between the two pulses, the forces will look as shown in the following figure.

We can see that the point at which the pulses start to interfere, they produce a force that is the sum of the forces due to either pulse at that point.

There are two ways that wave pulses can interfere that are of particular interest.

Let us consider two wave pulses with the same wavelength and amplitude.

The first case of particular interest is where the force vectors at every point on both pulses have the same direction. This is shown in the following figure.

In this case, the pulses are said to be in phase with each other, or to have a phase difference of zero. In this case, the resultant force vector at each point has a maximum magnitude. The interference between the pulses is called constructive interference.

The second case of particular interest is where the force vectors at every point on both pulses have opposite directions. This is shown in the following figure.

In this case, the pulses are said to be in antiphase with each other, or to have a phase difference of . In this case, the resultant force vector at each point has a minimum magnitude. The interference between the pulses is called destructive interference.

Phase difference can have any value. The following figure shows some of the possible phase differences between pairs of identical waves.

It is important to note that a phase difference of is equivalent to a phase difference of .

Let us now look at an example in which two waves produce interference.

### Example 1: Identifying the Resultant Wave of Two Waves Interfering

The two waves shown in the diagram have the same frequency, wavelength, and initial displacement. If the two waves interfere, which of the other diagrams—A, B, C, and D—best shows how the resultant wave compares to the two identical waves?

### Answer

The two waves that interfere can be modeled as two overlapping sets of equally spaced force vectors, as shown in the following figure.

Each pair of vectors at a point is summed, as shown in the following figure.

Each pair has the same direction and magnitude. This means that each vector retains the same direction and doubles its magnitude.

Options B and D show the magnitudes of the vectors decreasing, and option C shows the magnitudes of the vectors not changing.

Only option A shows a uniform doubling in the magnitudes of the vectors, and so it is the correct option.

Let us now look at another example.

### Example 2: Identifying the Resultant Wave of Two Waves Interfering

The two waves shown in the diagram have the same frequency and wavelength as each other, but with different initial displacements. If the two waves interfere, which of the other diagrams—A, B, C, and D—best shows how the resultant wave compares to the two waves?

### Answer

The two waves that interfere can be modeled as two overlapping sets of equally spaced force vectors, as shown in the following figure.

Each pair of vectors at a point is summed. Each pair has the same magnitude and opposite direction. This means that each vector becomes zero.

Options A, C, and D show nonzero magnitudes of the vectors.

Only option B shows zero magnitudes of the vectors, and so it is the correct option.

Up until this point, we have looked at wave pulses about one wavelength long. We can apply what we have seen concerning such pulses to waves that have any number of wavelengths.

Let us now consider an example involving waves consisting of arbitrary numbers of wavelengths.

### Example 3: Identifying the Nature of the Interference between Two Waves Interfering

Two waves with the same wavelengths and frequencies as each other move in the same direction, with one wave leading the other by one whole wavelength. Is the interference between the waves constructive, destructive, or neither constructive nor destructive?

### Answer

The two waves can be shown in a diagram, adjacent to each other, to make comparing them easier. This is shown in the following figure.

We can see that in the direction that the waves move, the point from which the wave shown in blue starts moving is 1 wavelength ahead of the position from which the wave shown in red starts moving.

We can see from the vertical lines connecting the two waves that the peaks and troughs of the two waves coincide.

The peaks and troughs of the waves represent the maximum magnitude of forces that the waves exert, where the direction of the forces corresponding to a peak are opposite to those corresponding to a trough.

From this we see that the forces exerted by the waves always act in the same direction, and hence, the resultant forces are always greater than those due to either of the waves alone.

An increase in the forces exerted along the entire region in which the waves exist indicates that the waves interfere constructively.

When waves of equal magnitude interfere destructively, the force exerted by the resultant wave at all points along its length is zero. The amplitude of the resultant wave is therefore zero.

The amplitude of a wave is the magnitude of the maximum displacement of the wave. A wave with a nonzero amplitude nevertheless has a displacement of zero at various points along its length.

Let us now look at an example that examines the relationship between the amplitude of a wave and the displacement at points on the wave.

### Example 4: Identifying the Nature of the Interference at Different Points between Two Waves Interfering

Two waves of the same frequency and wavelength pass each other, traveling in opposite directions. Both waves have an amplitude of 1 cm. The waves interfere to produce a resultant wave. The diagram shows the displacement of points on the resultant wave at an instant in time.

- Is the interference of the two waves at point A constructive or destructive?
- Is the interference of the two waves at point B constructive or destructive?

### Answer

**Part 1**

Shortly before any part of either waves occupies the same region of space, they can be represented as shown in the following figure.

Once the waves have started to overlap, they can be represented as shown in the following figure.

The parts of the waves that overlap interfere. The more time that passes, the greater the region within which interference occurs.

Only the region within which interference occurs is shown in the question. Approximately, 4 wavelengths of this region are shown by the diagram in the question.

We see that at point A, the two waves interfere to produce a resultant displacement equal to the amplitude of the resultant wave. The waves have equal amplitude, and the amplitude of the resultant wave is clearly not zero, and so the interference at A cannot be destructive. The only other option given is that the interference is constructive, so this is the correct option.

**Part 2**

At point B, the displacement is zero. This may give the idea that the interference at B must be destructive. This idea is however incorrect.

In a region where two waves of the same constant frequency overlap, the interference must be either constructive, destructive, or neither of these throughout the entire region. It is not possible for interference somewhere in the region to be constructive, while elsewhere in the region the interference is destructive. The phase difference between the waves is constant throughout the region.

The waves interfere constructively at A, and so they must also interfere constructively at B. The displacement of the resultant wave at B is zero. This means that the displacement of either of the contributing waves at B must be zero. A point at which these displacements are all zero is shown in the following figure.

Let us now summarize what we learned in this explainer.

### Key Points

- Light waves interfere when they occupy the same region of space at the same time.
- The resultant displacement of two interfering waves at a point is the sum of the displacements of these waves at the point.
- When interfering waves have a phase difference of zero, they interfere constructively.
- When interfering waves have a phase difference of , they interfere destructively.
- Two waves of equal frequency that interfere have a constant phase difference throughout the region in which they interfere.