What is the definition of interference?

I think the difficulty appears because the terms "interference" and "superposition" are sometimes used as if they are interchangeable when that's not correct.

Superposition is the the more fundamental concept, and indeed any two waves will superpose - but this does not necessarily result in observable interference. As a very simple example, consider two electromagnetic waves with frequencies $\omega_1$ and $\omega_2$, the amplitude of their superposition is (for a point with the same spatial phase, to ignore said term)

$$|\psi_T|^2 = (e^{i\omega_1t}+e^{i\omega_2t})(e^{-i\omega_1t}+e^{-i\omega_2t})=1+1+2\cos(\Delta \omega t).$$

For high frequency waves such as visible light, $\Delta\omega$ may very well be in the THz range, such that it's essentially imperceptible for most/all instruments, including your eyes. But for other waves, like sound waves, this isn't necessarily the case, and the interference of two waves of different frequency is detected as a beat.

Interference of light and interference of matter waves are different in some important aspects. The term is poorly defined/taught in physics all the way from high school up to university, you are not alone.

Bear in mind this first very important concept: when waves superimpose in matter (water, air, violin string) we can observe nulls and crests but these are only temporary! In an ideal medium energy is never lost ... the energy is only lost when the waves crash on the beach, sound waves absorbed by furniture etc. If 2 tsunamis were to meet in the middle of the ocean and superimpose we would observe either a very high crest or a null .... but a few seconds later both tsunamis would remerge and travel along their merry way .... energy only lost when they impact land.

So for matter waves every teacher says the waves cancel at the nulls.... but they don't realize that the energy is stored in the medium , yes water is elastic, and air is elastic. Energy is never lost in the ideal medium (maybe water/air is 99% ideal).

The concept of interference is transferred to light in the famous double slit experiment ... but there is even more confusion here than in matter waves. In high level light physics the term interference is used whenever light seems to interact with light and there are many well known types.

Coherency is not a requirement for interference, any 2 waves can interfere. We can discover some interesting effects/patterns when waves are coherent .... if we have 2 sources/waves coherent it just means the sources are some how related/effecting each other (they are correlated in some manner) in a wave tank 2 sources can be driven by the same motor for example.

When you have your PhD in physics you can use the terms interference and superposition just as poorly as the rest of us do ... unfortunately it is not well defined!

I actually think you understand the idea of interference, and the issue with the definition you quoted is that "coherent waves" doesn't mean anything. (or, at least not without more explanation, that I think will likely invoke concept of interference or superposition at some point, for instance to define something like a coherence length or coherence time, creating a circular definition.)

Sometimes people talk about a "coherent" vs an "incoherent" superposition. In a coherent superposition, the phases or signs are important, while in an incoherent superposition they are random. So I would reword the definition you found into

interference is a phenomenon in which two waves are coherently combined by adding their intensities or displacements with due consideration for their phase difference.

An example of a coherent superposition would the double slit experiment, which you can think of two sources (at the two slits) emitting waves at a screen. The relative phase of those sources is very important for determining the pattern on the screen. Or if we think of sound waves, the phase of the waves is very important for noise cancelling headphones which intentionally produces sounds that cancel the sounds entering your ear.

An example of an incoherent superposition would be a lot of sources of different frequencies that have random phases, that just produce a random mess of a time series. For instance, when water falls on rocks and produces white noise, the specific phases of the sound waves aren't that important. It's often useful to analyze such signals in the frequency domain and only look at the power spectrum, or amplitude of the Fourier components, and not the phases. For white noise like water babbling on some rocks, you'll find a relatively flat frequency spectrum. This incoherent, scale invariant spectrum can be quite soothing for humans.

Interference is how when you add two numbers, sometimes the result is larger than the original numbers and sometimes it's smaller (specifically, when they have different signs). It's usually in reference to something wave like.

The coherence argument is that the interference patterns are typically quite uninteresting to look at in non-coherent light, but can be quite stark if the light is coherent. I'd choose to say coherence is needed for interference patterns. If you have an experience involving interference/diffraction using a laser, and you replace the light source with a flash light, you would not see the interesting patterns. It's not that the light isn't interfering, it's just that it's interfering in complicated ways that are hard to see.

In the case of light, we can do a good numeric example. Red light has a wavelength of 700nm and orange light has a wavelength of 650nm. At the speed of light, those correspond to frequencies of roughly 428THz and 461THz, which means the interference between them has "beats" at 33THz. Needless to say, you can't see something that fast, so there's nothing interesting to see in the interference.

You also need the light to be in phase to see interference patterns. If you have light coming in at multiple phases, you find that some of those patterns are increasing where others are decreasing, and the net result is very murky looking.

I suppose the answer depends on context, and here: I am keeping it classical.

For me, "interference" means coherence: equal frequency (and polarization, if necessary) and a constant phase difference, a la: Young's Double Slit Experiment. (Your first definition).

Your second definition is just superposition; that is: a linear system. I usually explain this as if you have radio and flashlight, those EM waves add linearly, and the radio doesn't cast a shadow while the flashlight doesn't affect your radio show. You can have a CAT scan while someone is using a UV light for sterilization and a heating element is flooding the room with IR as you listen to radio in the Earth's magnetic field: different frequencies just not care.

What made me think about this was someone calling "beats" interference (see @agaminmon answer). I think it is not interference; I call it superposition, the two signals pass through each other and do not care...but lets's table that for now.

If you think about water-waves (gravity waves in water)..it just breaks the language. They are non-linear, so two crossing wave trains--say standard weather driven waves breaking near the shore and the wake of a huge ship--they "interfere" with each other, superposition fails, but no one would call this wave "interference".

With beats, you have a linear system and two eigenstates:

$$ |\psi\rangle = |\nu_1\rangle + |\nu_2\rangle $$

which you now express as a product:

$$ |\psi\rangle = |\nu_+\rangle|\nu_-\rangle $$

with

$$ \nu_{\pm} = \frac 1 2 (\nu_1 \pm \nu_2) $$

For particles, you can replace frequency with momentum, Each particle is in a momentum eigenstate. Then compare with angular momentum (ok, I've gone quantum): two particles in angular momentum eigenstates are not in eigenstates of total angular momentum.

The angular momentum case is much more complicated since rotations don't commute, while translations do.

But that's a whole other thing.

It seems to me: a factor common to all instances of visible interference effect is consistent phase relation.

I will go over a series of cases of interference effect, setting up for a discussion of the notion of consistent phase relation.

Thin film interference:
A well known instance of thin film interference effect is that some drops of gasoline have spilled into a puddle of water. The gasoline spreads out over the surface, creating a thin film, the thickness is in the order of micrometers.

The interference effect from the thin film gives rise to bands of colors; areas of equal color are areas of equal film thickness.

The thin film interference color effects occurs with straightforward daylight. As we know, daylight is wide band, and on an overcast sky all daylight is scattered light. So: for the source of the light we have: no temporal coherence, no spatial coherence.

The factor that makes it possible for interference effect to occur is that the reflection against the top surface and the reflection against the gasoline-water interface occur in very close proximity. (Close to the wavelength of visible light.)

A double slit interference demonstration with sunlight as light source.
Setting up a double slit interference demonstration with sunlight is challenging, but doable.

The demonstration is set up in a darkened room. The only light entering the room is through a small hole. I will refer to that as 'the source hole'. From the source hole the light travels to the screen with the two slits. The reason that the source hole needs to be small is as follows:
-all of the light that reaches the left hand slit must have traveled the same distance, to within the wavelength of the light.
-all of the light that reaches the right hand slit must have traveled the same distance, to within the wavelength of the light.
The size of the source hole must be small enough such that as seen from the screen with the slits the source is in effect a point source of light. When the source is (relative to the target) a point source then as the light reaches the target it has a property called 'spatial coherence'.
For the light that passes through the slit there is diffraction. The property of spatial coherence carries over to the light that makes it through the slit.

By contrast: if the light that reaches the two slits comes from a wide source then no visible interference pattern can be obtained. It is when all of the light that reaches the slit has traveled the same distance (to within the wavelength) that a visible interference pattern is possible.

As we know, sunlight is a spectrum of frequencies. Because of that: the interference pattern that is obtained that way is sharply defined only for the central line. Away from the central line there are chromatic effects.

There is of course the famous Michelson-Morley experiment. According to the wikipedia article: for alignment light from a Sodium lamp was used, and for taking the measurements, which was done by looking into an eyepiece, white light was used.

So the light source that was used did not have the property of temporal coherence.

Not having a source of light with temporal coherence made the Michelson-Morley experiment very challenging, but still doable.

Just as in the case of a double slit setup: for the Michelson-Morley experiment the light source must be set up such that the light entering the interferometer has the property of spatial coherence.

The light travels two different paths through the interferometry setup. An interference pattern is obtained if and only if the two paths have the same length, to within the coherence length of white light, which is several micrometers. (The wavelength of visible light is from 0.4 to 0.7 micrometers.)

A Mercury vapor lamp produces light that is suitable for interferometry. The emission lines include 404.7 nm, 438.8 nm, and 546.1 nm lines. Filters are used to remove the wavelengths other than the 546.1 nm wavelength.

In the Kennedy-Thorndike experiment a Mercury vapor lamp was used as source, because the experiment was set up to use arms of different length. There is a relation between spectrum width and coherence length. Wide spectrum light, such as daylight, has short coherence length, the closer to monochromatic a light source is the longer the coherence length.

The Mercury 546.1 nm emission line has a coherence length of about 32 centimeter.

About 'temporal coherence':
Temporal coherence is about consistent phase relation. Given the degree of narrowness of the 546.1 nm emission line: there is over a longer length a sufficient phase relation so that there is still a visible interference effect.

While temporal coherence is a temporal property it is usually encountered in the form of managing length. How much difference in pathlength there can be while still obtaining an interference pattern.

Finally, we get to interference setups with a laser.

The light exits the laser cavity through a very small hole. The hole is so small that you get a divergent beam (diffraction). A collimating lens is used to refract that divergent beam to a parallel beam.
A laser source, when properly constructed, has spatial coherence. So that takes care of spatial coherence.

The large temporal coherence that a laser offers is a great asset; it makes interferometry setups much easier to accomplish; the paths can have large length difference..

Consistent phase relation

Returning to the case of thin film interference effect:
the incoming light is scattered light: in effect we can think of the incoming light as originating from an exceedingly large amount of individual sources, spread across the sky. For the thin film interference efffect: single source or many sources makes no difference: what counts is phase relation.

With a thin film interference effect the two reflections occur very close together. The thickness of the thin film is within the coherence length of daylight.

About double slit setup: a more performant version is a diffraction grating (When instrument makers succeeded in making high quality diffraction gratings astronomers started using them for spectral analysis of starlight.)

Obtaining interference fringes with a diffraction grating is more demanding in the sense that in order to obtain consistent phase relation the light source that illuminates the two diffraction grating must be (relative to the diffraction grating) a point source.

If a star is far enough away then the entire star is effectively a point source. The condition of consistent phase relation is met: an interference pattern is obtained.

As you have probably gathered from the collection of answers so far, there is no precise and universally used definition of the word "interference".

In general, "interference" refers to situations where different waves are added together. Sometimes people just mean this scenario in full generality; in this case, it very similar to the concept of "superposition". All waves can "interefere" in this sense, regardless of how coherent they are. Your second quoted sentence is using the word in this sense.

But other times, people might use the word to mean one of several more specific concepts. For example, sometimes people use the word "interference" to emphasize that there is the potential for both constructive and destructive interference, because the individual constituent waves can take on negative or complex values. In this case, they use the word "interference" to distinguish from (e.g.) the Kolmogorov axioms, which constrain the ways that you're allowed to "combine together" classical probabilities (e.g. via the "and" and "or" constructions) and do not allow for destructive interference.

There is another sense of the word "interference" that is more restrictive than the fully general sense. This sense is not entirely precise, but roughly corresponds to "interference phenomena that are simple enough that we can understand them using simple and/or reasonably universal concepts". Basically, this refers to phenomena that we can develop some intuition and/or analytic handle over. Examples include beats, interference fringes, nodes of standing waves, Fresnel diffraction, etc. As several other answers have pointed out, this usually requires coherent constituent waves and/or a small number of sources. Your first quoted sentence is implicitly using the word in this more restrictive (and admittedly somewhat vague) sense.

Interference is the addition of the energy potentials of two disturbances encountering each other in a medium. Sounds complicated? But it covers all cases and excludes none.

• First of all, subtraction is also taken into account – one disturbance has a positive potential (wave crest) and the other disturbance has a negative potential (wave trough(. If both energies are equal in magnitude, except for the sign, total cancellation occurs (for the sake of completeness: the energies disperse into laterally evading longitudinal waves in the medium).
• Secondly, the presence of waves of the same frequency or a specific alignment of the two waves in relation to each other is not a prerequisite for interference. Only the drifting apart of two waves is not an interference: if I throw a stone into a pond and consider two semicircles as separate waves, then these do not interfere.

Electromagnetic radiation do not interfere in a vacuum. Two radio waves pass each other unhindered. (The situation becomes critical at high energies, where high-energy photons can lead to pair production.)

Why is EM radiation mentioned at all? This is due to the flippant interpretation of the experiments on edges. Edges lead to intensity distributions with a wave-like character. This has nothing to do with interference. It is about a deflection (diffraction) of the radiation at the edges. In the dark area, there is NO destructive interference (cancellation) and in the brightest area, there is no frequency doubling (wavelength change). Quite simply, as the radiation passes the edges, more photons are deflected away from the dark area towards the bright area.