Interference is a technical word for the overlapping of waves. In this lesson, we'll take a close mathematical look at how two sine waves interfere with each other to form a larger sine wave (constructive interference) or cancel each other out completely (destructive interference).

Where does interference occur in nature?

Interference is a fundamental phenomenon which occurs everywhere in nature and technology where waves are involved. Many things around us consist of waves.

Water waves on the sea.

Sound waves that allow us to communicate verbally with other people.

Light waves emitted by our sun.

Matter waves of smallest particles, like electrons, which also show interference phenomena.

Other applications include anti-reflective coatings, such as anti-reflection coating for eyeglasses. But also interferometers, with which the smallest changes in length can be measured. These include, for example, Michelson interferometers, Mach-Zehnder interferometers and Sagnac interferometers which you may learn about later in life as you follow the path of physics.

In physics, you can find interference in the double-slit experiment, in acoustic beat or Bragg reflection.

How is interference generated?

To understand what happens when waves are overlap, you first have to create waves.

For example, you can throw two stones next to each other into the water. The stones create water waves, the waves collide and interfere.

You can turn on two speakers. These generate sound waves that can interfere with each other.

To keep things from getting too complicated, let's look at interference using a very simple but very basic example:

One-dimensional sine wave propagating along the \(x\) axis.

The sine wave is a sequence of wave crests and wave troughs and has two properties important for interference:

Wavelength \( \lambda \) ("Lamb-da!") - this is the distance between two wave crests (or troughs).

Amplitude \( A \) - in the case of a sound wave it indicates how loud the sound is.

If you turn on a loudspeaker sine tone, the sound is emitted in different directions so that everyone in the room can hear it. Let's set the amplitude to the value \( A ~=~ 1\) so that we can easily overlap two waves with it.

One wave alone is not enough to study interference. We need another wave for this purpose. This can be generated, for example, by another loudspeaker standing next to it. Let's switch on exactly the same sine wave at the second loudspeaker. This has the advantage that the wavelengths of the two sine waves are the same, making the calculation a bit easier. Of course, you can also consider any other waves.

Constructive interference of two sine waves

As soon as the two sine waves meet, they overlap. We sent them off in such a way that a wave crest of one sine wave meets a wave crest of the other sine wave. Also each wave trough meets corresponding wave trough at the same point.

To perform the interference mathematically, you proceed as follows:

Select any position \(x\) where you want to superimpose the two waves.

Add the amplitude \( A_1 \) of one wave to the amplitude \(A_2\) of the other wave at the same position \(x\). The sum of the two amplitudes gives the amplitude \(A(x)\) of the resulting wave at the considered location: \( A(x) = A_1(x) + A_2(x) \).

Plot the value of the new amplitude in an amplitude-position graph at the considered position \(x\).

For example, take two maxima (crests) at the same location \(x\). Since both crests have amplitude 1, the resulting amplitude is:

Proceed analogously with all other locations \(x\). If you plot all the resulting points in a diagram and connect them, you will get a resulting wave.

In this example, it was a so-called constructive interference because the amplitude of the new sine wave has become twice as large as the amplitude of the original waves. The resulting sound described by this sound wave is twice as loud!

Path difference of two sine waves

Whether two waves interfere constructively depends on how the two waves are positioned relative to each other. In the above example, one wave was exactly "above the other" wave. The one wave does not precede and does not lag behind the other.

But you also have constructive interference if you shift a wave by \(\lambda\) to the right or to the left. In this case there is also a wave crest above a wave crest. A further shift by \(\lambda\), i.e. by a total of \( 2\lambda\) results again in a case where wave crests lie on top of each other and constructive interference occurs.

Constructive interference also occurs at all other integer shifts \( 3 \, \lambda \), \( 4 \, \lambda \) and so on. But also at negative shifts by \( -1 \, \lambda \), \( -2 \, \lambda \), \( -3 \, \lambda \) and so on.

This value of displacement is called path difference \( \Delta s \).

So let's summarize: for constructive interference of two waves to occur, the path difference must be a multiple of the wavelength:

Formula anchor$$ \begin{align} \Delta s ~=~ m \, \lambda \end{align} $$

Here \(m\) is a number which can take negative and positive integers: \( m ~=~ ...-2,-1,0,1,2... \).

Destructive interference of two sine waves

But what if one sine wave precedes the other wave not by a multiple \( m \, \lambda \) , but by exactly half the wavelength, i.e. by \( \lambda/2 \) ? Then each wave crest meets a wave trough. The deflections at the wave trough and wave crest always add up to zero at each location. At the zero points the deflections also add up to zero: 0 + 0 = 0. And also at every other place the deflections add up to zero. The two waves cancel out completely. In this case, we have destructive interference. It always occurs when one wave precedes the other by \( \lambda/2 \). Or also by \( \lambda/2 +1\lambda \). Or \( \lambda/2 +2\lambda \), or \( \lambda/2 +3\lambda \) and so on.

Condition for destructive interference of two equal waves with multiplied out wavelength

Formula anchor$$ \begin{align} \Delta s ~=~ \lambda/2 ~+~ m \, \lambda \end{align} $$

Formula anchor$$ \begin{align} \Delta s ~=~ \left( 1/2 ~+~ m \right) \, \lambda \end{align} $$

Here \( m \) is again an integer with negative and positive values: \( m ~=~ ...-2,-1,0,1,2... \).

Partial interference

So far we have considered two special cases: where one wave crest has perfectly met the other wave crest, i.e. constructive interference, and where one wave crest has perfectly met a wave trough, i.e. destructive interference. But this does not always have to be the case! The waves could just as well interfere only partially with each other. This would mean that the wave crest does NOT exactly meet the other wave crest... But these cases are rather unimportant, because in applications mostly complete amplification or complete cancellation are useful.

Write path difference as phase difference

Sometimes the path difference \( \Delta s \) is given as phase difference \( \Delta \varphi \). What is a phase, you may ask?

We can express our sine waves mathematically as follows:

Equations for two sine waves

Formula anchor$$ \begin{align} y_1(t) ~=~ A_1 \, \sin\left( \omega \, t ~+~ \varphi_1 \right) \\\\
y_2(t) ~=~ A_2 \, \sin\left( \omega \, t ~+~ \varphi_2 \right) \end{align} $$

Here \(A_1\) and \(A_2\) are their amplitudes, \(\omega\) is the angular frequency. In our case it is the same for both sine waves and it contains the information about the wavelength. The angle \(\varphi_1\) of one wave and the angle \(\varphi_2 \) of the other wave gives the displacement of the wave to the left or to the right - with respect to a previously defined starting position. This angle is called the phase of a wave.

The phase difference, as the name implies, is the difference of the two angles (phases):

Each path difference \( \Delta s \) corresponds to a certain phase difference \( \Delta \phi \). You can specify the phase difference either in radians (from \( 0 \) to \( 2\pi \)) or in degrees (from 0 to 360°).

You can calculate the phase difference in degrees using the path difference as follows:

Phase difference in radians by using path difference

We have already derived an expression for the path difference for constructive (eq. 3) and destructive interference (eq. 5). You can now simply substitute the condiitions into the equations for phase difference in degrees or radians.

Phase difference for destructive interference in degrees, looks like this, for example:

Formula: Phase difference with destructive interference

Formula anchor$$ \begin{align} \Delta \phi &~=~ \frac{ \left( 1/2 ~+~ m \right) \, \lambda }{\lambda} ~\cdot~ 360^{\circ} \\\\
&~=~ \left( 1/2 ~+~ m \right) ~\cdot~ 360^{\circ} \end{align} $$

+ Perfect for high school and undergraduate physics students + Contains over 500 illustrated formulas on just 140 pages + Contains tables with examples and measured constants + Easy for everyone because without vectors and integrals