Physics Tutorial: Interference of Waves

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In this Physics tutorial, you will learn:

  • What is interference of waves?
  • Which are the conditions to produce interference?
  • How many types of interference are there in waves?
  • What does the principle of waves' superposition say?
  • What happens to the amplitude and frequency when interference occurs?
  • How does the path difference affect the interference of waves?

Introduction to Interference of Waves

Why you can't understand well when two people are talking at the same time?

What do you call the phenomenon when you hear any undesired sound while you are listening to music?

What does a microphone do? Does it increase the frequency or amplitude of sound? Why?

In this tutorial, you will learn more about a phenomenon called interference, which is very common in waves. Also, you will learn about the two types of interference and how they affect the waves behaviour.

What is Interference?

By definition, interference is the combination of two or more waveforms to form a resultant wave in which the displacement is either reinforced or cancelled.

Look at the figure.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

The resultant wave Z is obtained by the overlapping of waves X and Y. Therefore, we can say that interference is the process of the overlapping of two or more waves.

Interference is produced as a result of the principle of superposition, a key principle in waves theory, which says:

The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements.

The principle of superposition may be applied to waves whenever two (or more) waves are travelling through the same medium at the same time.

There are two conditions the waves must meet to produce interference patterns.

  1. Waves must be of the same kind. Obviously, a light wave cannot interfere with a sound wave as they are of different nature. Light is an EM wave and it can spread only in transparent media, regardless whether they are mechanical or not, while sound waves are mechanical waves that can spread only in material media. Furthermore, the method of propagation is also different; sound waves propagate in a longitudinal way while light is a transverse wave. Hence, it is impossible for these two kind of waves to overlap.
  2. Waves must have the same frequency in order to produce a standing interference. Standing waves have features that do not change with time. If it was not so, waves could interfere at one point but not in the others. Therefore, the effect of interference would be not visible as the shape of the resulting wave would be irregular.
Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

The above case occurs when we hear two people talking simultaneously, or when waves of a radio channel interfere to another, so we cannot hear well.

In this regard, we can say that it is wavelength (and therefore, frequency) the quantity that determines the interference pattern. This means only like waves with the same frequency (wavelength) can produce interference in certain conditions. This last phrase, i.e. "in certain conditions" represents another restriction for the waves to produce interference. This condition is explained in the next paragraph.

Types of Interference

Interference may occur in all types of like waves. However, the interference pattern is more visible in transverse waves as the distinction between amplitude and wavelength is clearly observable in those waves because amplitude is perpendicular to wavelength. Therefore, we will focus only on transverse waves to explain the interference pattern and its properties.

Basically, there are two types of interference. They are:

1. Constructive Interference

In this kind of interference, waves enforce each other as they overlap at the same phase. We say the waves are coherent, i.e. they have the same behavior. As a result, we obtain a resultant wave whose amplitude is the arithmetic sum of the amplitudes of each single wave as shown in the figure below.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

In this figure, the resultant wave has the same horizontal features (phase, wavelength, frequency, speed and period) as the two constituent waves but it has a greater amplitude as the waves overlap constructively. As a result, a stronger wave will be produced.

Mathematically, we can write for the resultant amplitude of the wave produced during a constructive interference:

Ares = A1 + A2

As an example of constructive interference, we can mention the sound produced when you turn on two loudspeakers emitting the same song simultaneously. As a result, you will hear a louder volume if you are in between.

2. Destructive Interference

When two like waves have a phase shift of half a cycle, a destructive interference is produced, as shown in the figure below.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

If we assume the first wave has a slightly greater amplitude than the second wave, we obtain for the resultant amplitude

Ares = A1 - A2

In the special case when the amplitudes of the constituent values are equal, the resultant amplitude is zero and therefore, the waves cancel each other. As a result, no resultant wave will exist anymore.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

As an example of destructive interference, we can mention the modern electronic automobile muffler. It senses the sound propagating down the exhaust pipe and creates a matching sound with opposite phase. These two sounds interfere destructively, muffling the noise of the engine.

Example 1

What is the amplitude of the resultant wave produced by interference in the following cases if the amplitude of the first wave is 12 cm and that of the second wave is 8 cm?

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Solution 1

a) From the figure, we can see that waves are out of phase. Therefore, the resultant amplitude is

Ares = A1 - A2
= 12 cm - 8 cm
= 4 cm

The shape of the resultant wave is:

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

The resultant wave will be in phase with the wave which has the greater amplitude. In this case, it will be in phase with the first wave.

b) In this case, the interference is constructive because both waves have the same phase. Therefore, the resultant amplitude will be

Ares = A1 + A2
= 12 cm + 8 cm
= 20 cm

As a result, a stronger wave with similar features will be produced, as shown in the figure.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

Conditions for the Interference

When two coherent waves are parallel, they will behave as described in the previous paragraph, i.e. they can produce constructive interference when they are at phase, destructive interference when they are out of phase or they may not produce any interference at all when they are neither in phase, nor out of phase.

However, when two coherent waves pass through one or more narrows gaps comparable to the amplitude, they (albeit initially in phase) may behave differently (may deviate) after leaving the gap because they often deviate their original path as shown in the figure.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

Due to this deviation, waves will travel different paths after leaving the gap (d2 > d1). If we place a screen at the position these waves meet, we will observe on the screen one of the three options mentioned above (constructive, destructive or no interference). The type of behavior depends on the path difference of the two waves.

a) If the path difference d2 - d1 is a whole multiple of wavelength, i.e.

d2 - d1 = N × λ

where N = 0, 1, 2, ., the waves will be in phase when they fall on the screen. As a result, a constructive interference is produced on that point as the waves enforce each other.

b) On the other hand, when the path difference of the two waves is half a multiple of wavelength, i.e. when

d2 - d1 = N × λ + 1/2 × λ
= 2N/2 × λ + 1/2 × λ
= 2N + 1/2 × λ

where N = 0, 1, 2, ., there will be a destructive interference produced at that point of the screen, as waves try to cancel out each other.

If you count the number of wavelengths in the figure above, you will notice that from the gap to the meeting point the first wave makes 5.5 cycles and the second wave makes 6.5 cycles. Therefore, the difference between the two waves' paths d1 and d2 is

d2 - d1 = 6.5 × λ - 5.5 × λ
= 1 × λ

we have a constructive interference in the given point of the screen as the path difference between the two waves is a whole multiple of wavelength.

If we had d2 - d1 = 0.5λ or 1.5λ or 2.5λ and so on, we would observe a destructive interference in the given point of the screen. For all the other values for the path difference d2 - d1 (for example for d2 - d1=0.7 × λ, 1.3 × λ, 5.27 × λ and so on), no interference would be observed.

If the waves 1 and 2 were light rays, a pattern with dark and bright regions is formed on the screen, where the maximum brightness is obtained on the central maximum and fringes around it, that become dimmer when moving away from the central maximum, as shown in the figure below.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

Example 2

Two parallel and coherent waves emitted by the same source fall on a screen after passing through a narrow gap as shown in the figure (for convenience, waves are represented by straight lines).

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

If the distances d1 and d2 travelled by the two waves after passing through the gap are 21.6 cm and 18.2 cm respectively and the wavelength is 4 mm, what kind of interference (if any) will occur at the point A on the screen?

Solution 2

We must calculate the path difference d1 - d2 (as d1 is longer) and see how many times the wavelength it is. Thus,

d1 - d2 = 21.6 cm - 18.2 cm
= 3.4 cm
= 34 mm

This path difference is 34 mm / 4 mm = 8.5 times greater than the wavelength. This is a half multiple of wavelength; therefore, we will have a destructive interference at the point A on the screen.

Physics Revision: Interference of Waves Summary

By definition, interference is the combination of two or more waveforms to form a resultant wave in which the displacement is either reinforced or cancelled.

In other words, interference is the process of the overlapping of two or more coherent waves.

Interference is produced as a result of the principle of superposition, a key principle in waves theory, which says:

The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements.

The principle of superposition may be applied to waves whenever two (or more) waves are travelling through the same medium at the same time.

There are two conditions the waves must meet to produce interference patterns.

  1. Waves must be of the same kind. Obviously, a light wave cannot interfere with a sound wave as they are of different nature.
  2. Waves must have the same frequency in order to produce a standing interference.

It is wavelength (and therefore, frequency) the quantity that determines the interference pattern. This means only like waves with the same frequency (wavelength) can produce interference in certain conditions.

Basically, there are two types of interference. They are:

a) Constructive Interference. In this kind of interference, waves enforce each other as they overlap at the same phase. As a result, we obtain a resultant wave whose amplitude is the arithmetic sum of the amplitudes of each single wave, i.e.

Ares = A1 + A2

b) Destructive Interference. When two like waves have a phase shift of half a cycle, a destructive interference is produced. As a result, we obtain a resultant wave whose amplitude is the arithmetic difference of the amplitudes of each single wave, i.e.

Ares = A1 - A2

The resultant wave will be in phase with the wave which has the greater amplitude.

In the special case when the amplitudes of the constituent values are equal, the resultant amplitude is zero and therefore, the waves cancel each other. As a result, no resultant wave will exist anymore.

When two coherent waves pass through one or more narrows gaps comparable to the amplitude, they (albeit initially in phase) may behave differently (may deviate) after leaving the gap because they often deviate their original path. Due to this deviation, waves will travel different paths after leaving the gap (d1 ≠ d2). If we place a screen at the position these waves meet, we will observe on the screen one of the three options mentioned above (constructive, destructive or no interference). The type of behavior depends on the path difference of the two waves.

a)If the path difference d2 - d1 is a whole multiple of wavelength, i.e.

d2 - d1 = N × λ

where N = 0, 1, 2, ., the waves will be in phase when they fall on the screen. As a result, a constructive interference is produced on that point as the waves enforce each other.

b)On the other hand, when the path difference of the two waves is half a multiple of wavelength, i.e. when

d2 - d1 = (2N+1)/2 × λ

where N = 0, 1, 2, ., there will be a destructive interference produced at that point of the screen, as waves try to cancel out each other.

If the waves 1 and 2 were light rays, a pattern with dark and bright regions is formed on the screen, where the maximum brightness is obtained on the central maximum and fringes around it, that become dimmer when moving away from the central maximum.

Physics Revision Questions for Interference of Waves

1. What kind of interference will be formed by the two waves in the figure, which move towards each other? The initial distance between the two waves is 23 cm and the wavelength is 3 cm.

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves
  1. Constructive interference
  2. Destructive interference
  3. Neither constructive nor destructive
  4. There is not enough information

Correct Answer: C

2. Which of the following pairs of waves can produce interference?

  1. A high water wave and a short water wave
  2. A loudspeaker and a light source
  3. A rope wave and a sound wave
  4. Two light waves with different frequencies

Correct Answer: A

3. Two parallel and coherent waves emitted by the same source fall on a screen after passing through a narrow gap as shown in the figure (for convenience, waves are represented by straight lines).

Physics Tutorials: This image provides visual information for the physics tutorial Interference of Waves

If the distances d1 and d2 travelled by the two waves after passing through the gap are 9.8 cm and 8.2 cm respectively and the wavelength is 4 mm, what kind of interference (if any) will occur at the point A on the screen?

  1. Constructive interference
  2. Destructive interference
  3. Neither constructive nor destructive
  4. There is not enough information

Correct Answer: A

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