# Clarifications on interference of waves

Here is my understanding:

Superposition describes the effect of two waves, of the same type, coinciding at a point, stating that the resultant displacement is equivalent to the vector sum of the individual waves. Any two waves, given that they are the same type, will superpose. However, for two waves that happen to have a constant phase difference, they produce an interference pattern. So the way interference is defined, can only occur between two coherent waves, meaning they have a constant phase difference. Does this mean non-coherent waves cannot interfere, because of how interference is defined?

A subset of interference, is constructive and destructive interference, which occur when the constant phase difference is 0 and 180 respectively. Constuctive interference producing a resultant maximum displacement and destructive interference producing a resultant minimum displacement. Often the above is written using maximum amplitude and minimum amplitude, what is the difference? Are both terms the same?

Various definitions and explanations have become confusing, some contradicting each other, and a clarified answer would be great. Is my understanding stated above correct?

• Interference and superposition are the consequence of the linearity of these waves. Linearity simply means that there is no (relevant) self-interaction term in linear media and the physical vacuum . Instead of looking at these terms as descriptors of physical effects, it's better to understand them as the consequences of the absence of any effect at all. Jan 17 at 4:49
• Waves that intersect add and so always "interfere" with each other. However, to be observed, interference needs to persist over time. So, some of the most demonstrable interference effects can be seen using coherent sources. However, beat frequencies between slightly detuned sound waves is an example of an interference phenomenon between non-coherent sources. Of course, as the frequency difference increases the effect becomes harder to detect. Jan 20 at 21:38
• On displacement vs. amplitude, displacement applies to waves that involve, well, physical displacement, for example waves on a string. When one is talking about say electromagnetic waves, "displacement" isn't a particularly applicable term. Even with sound waves it is the pressure that is typically being considered so "displacement" isn't the right term. Amplitude is the general mathematical term for describing the "height" of a wave and so is applicable in any of these situations. Jan 20 at 21:48
• Example: if a tsunami started in Japan headed to the US coast and at the same time a tsunami of "opposite phase" started in the US headed to Japan, when the waves meet in the middle of the ocean they cancel .... BUT a few seconds later the waves reemerge and continue on their way (i.e they pass thru each other). Lesson: in an ideal medium (water and air are ok) energy is never lost by the wave interferences, the energy is only released by crashing on the beach! Jan 21 at 20:26

Does this mean non-coherent waves cannot interfere, because of how interference is defined?

Non-coherent waves can superpose and hence produce an "interference" pattern which is not stationary and which could be changing so fast so that it is not observable.
A good example of non-coherent sources interfering is two sources with frequencies $$f_1$$ and $$f_2$$, and a frequency difference $$\Delta f = |f_1-f_2|$$ which is very small compared to the two frequencies of the two sources.
Such a situation produces beats which can be thought of a moving interference pattern observed at one position, ie maxima and minima of intensity move over the observed position.

Constructive interference producing a resultant maximum displacement and destructive interference producing a resultant minimum displacement. Often the above is written using maximum amplitude and minimum amplitude, $$\dots$$
The better term to use is amplitude which is the maximum displacement from the equilibrium position.
The problem with using displacement is that displacement can be either positive or negative and so a wave with amplitude $$3$$ units ranges from a displacement of $$+3$$ to $$-3$$ and hence has a smaller minimum displacement that a wave of amplitude $$2$$ units which ranges from a displacement of $$+2$$ to $$-2$$.

Your understanding is mostly correct, but there are a few nuances to clarify:

• Superposition Principle: Yes, it states that when two waves of the same type meet, their total displacement at any point is the vector sum of their individual displacements at that point.

• Interference and Coherence: Interference indeed occurs when two waves superpose, but the persistence of a stable interference pattern (like in Young's double-slit experiment) requires coherence. However, non-coherent waves can still interfere; their interference just won't produce a stable, predictable pattern because their phase relationship varies with time. See my answer here in the context of wave optics.

• Constructive and Destructive Interference: Correct, constructive interference occurs when waves are in phase (phase difference of 0 or multiples of 360 degrees), resulting in maximum displacement or amplitude. Destructive interference happens when waves are out of phase (phase difference of 180 degrees or odd multiples of 180 degrees), resulting in minimum displacement or amplitude.

• Displacement vs. Amplitude: These terms are often used interchangeably in the context of waves. Displacement refers to the distance a point on the wave is from its rest position. Amplitude is specifically the maximum displacement of points on a wave from the rest position. In the context of interference, they essentially refer to the same concept: the height of the wave crest (or depth of the trough) from the rest position.

The terms "maximum amplitude" and "minimum amplitude" in the context of wave interference refer to the following:

• Maximum Amplitude: This occurs in constructive interference, where the waves are in phase (their peaks and troughs align). The amplitudes of the individual waves add together, resulting in a wave with greater amplitude than either of the original waves. This is the "maximum amplitude" – the highest point reached by the wave, which is greater than the amplitude of either individual wave.

• Minimum Amplitude: In contrast, minimum amplitude occurs in destructive interference, where the waves are out of phase (a peak aligns with a trough). The amplitudes of the waves effectively subtract from each other, and in the case of perfectly destructive interference (where the waves have equal amplitude but opposite phase), this can result in complete cancellation. The "minimum amplitude" is the lowest point reached by the wave, which, in the case of complete destructive interference, can be zero.

So, "maximum amplitude" and "minimum amplitude" in this context refer to the extremes of wave amplitude resulting from the interference of two waves, with "maximum" denoting the highest possible amplitude (from constructive interference) and "minimum" denoting the lowest possible amplitude (from destructive interference).

Superposition describes the effect of two waves,… stating that the resultant displacement is equivalent to the vector sum of the individual waves.

For the sake of completeness, it should be noted that in addition to the longitudinal wave (that up and down of a water wave) there is always a transverse component, the molecules of the wave also move sideways (depending on the pressure conditions in the medium). If two waves meet in destructive interference, the kinetic energy contained in the two waves does not disappear. The molecules are moved sideways and lose their kinetic energy in favour of friction.

Any two waves … superpose.

The better word ist „interfer“. Interference is the resulting wave pattern due to the mutual effect between two waves. In the special case of destructive interference, the kinetic energy of both waves is completely converted into friction and heat.

Does this mean non-coherent waves cannot interfere, because of how interference is defined?

See above. Any two waves interfere. The definition of interference is not limited to coherent waves.

A subset of interference, is constructive and destructive interference, which occur when the constant phase difference is 0 and 180 respectively. Constuctive interference producing a resultant maximum displacement and destructive interference producing a resultant minimum displacement. Often the above is written using maximum amplitude and minimum amplitude, what is the difference? Are both terms the same?

Yes, both terms are the same.