Question about wave interference and coherence

[Fionn Munnelly]

I have encountered the following definition of interference:

Interference is a wave phenomenon in which two or more waves from coherent sources meet and superpose to form a resultant wave such that the amplitude of the resultant wave at any point is the vector sum of the amplitudes of the constituent waves at that point.

If constructive interference occurs, then the amplitude of the resultant wave is greater than the amplitude of anyone of the constituent waves.

If destructive interference occurs, then the amplitude of the resultant wave is less than the amplitude of anyone of the constituent waves.

If the amplitude of the resultant wave is greater than the amplitude of one constituent wave but less than the amplitude of another constituent wave, is the interference constructive or destructive?

In addition, why must the sources of the waves be coherent? Is this only so that any interference pattern produced is stable and unchanging with time? Is the superposition of waves from non-coherent sources to form a resultant wave not still, technically, interference?

 

[sophiecentaur]

Is the superposition of waves from non-coherent sources to form a resultant wave not still, technically, interference?

There will be a resultant of phasor addition of the two sources at all points in space. If the sources are of slightly different but constant frequencies then, at all points, this resultant will constantly vary in amplitude and phase. If you take a snapshot of the pattern at anyone time, you will see an interference pattern. This pattern will be for ever changing - marching over the field at the rate determined by the frequency difference and the speed of the wave.

If you insisted, I could possibly agree that there will be a pattern for any pair of waves but it wouldn't be easy to detect and it becomes a bit of a nonsense. Consequently, we reserve the term Interference Pattern for situations when the two sources are either precisely the same frequency or when their frequencies are near enough to identify an interference pattern which 'crawls' very slowly.

The other point about coherence is that every source has phase noise and that gives a finite length over which the phases of two sources maintain the same relationship any point. This gives rise to the idea of coherence length which, as a practicality, limits the width of the optical two slits pattern because of the significantly different path lengths from the two slits relative to this coherence length. The fringes at the edges get fuzzier and fuzzier.

 

[phinds]

If the amplitude of the resultant wave is greater than the amplitude of one constituent wave but less than the amplitude of another constituent wave, is the interference constructive or destructive?

This is where the English language descriptions get a bit fuzzy so you should just stick to the math. Nature doesn't really care what we call it. Perhaps an English language term that would satisfy you as being more consistently in line with the math over the full range would be simply "interaction" between the waves.

 

[DaveE]

simply "interaction" between the waves

Yes. You pretty much fully described the situation when you said
"the resultant wave at any point is the vector sum of the amplitudes of the constituent waves at that point."
Sometimes when you add vectors they get bigger, sometimes smaller, that is all.

 

[sophiecentaur]

@phinds and @DaveE This is yet another result of 'teaching by numbers'. Students are taught in terms of categories rather than how to understand what's actually going on. They are (or at least they feel they are) assessed in terms of sound bytes. The result is confusion when in the real world.

 

[jtbell]

Perhaps an English language term that would satisfy you as being more consistently in line with the math over the full range would be simply "interaction" between the waves.

Yes. You pretty much fully described the situation when you said
"the resultant wave at any point is the vector sum of the amplitudes of the constituent waves at that point."
Sometimes when you add vectors they get bigger, sometimes smaller, that is all.

Among physicists, the commonly used term for this is "superposition" of waves.

 

[Mister T]

If the amplitude of the resultant wave is greater than the amplitude of one constituent wave but less than the amplitude of another constituent wave, is the interference constructive or destructive?

It's neither.

 

[Mister T]

Students are taught in terms of categories rather than how to understand what's actually going on.

I'm sure that that happens quite often. But even if the teacher emphasizes sense-making, many students will still revert to that answer-making strategy as their go-to learning scheme.

 

[sophiecentaur]

students will still revert to that answer-making strategy

Especially when so many questions are multiple choice.