# Why water is a dispersive medium?

larsa
Water waves with larger wavelengths travel faster. Why water is a dispersive medium?

Mentor
Every medium is dispersive. Why are you surprised that water is?

Also, which waves do you mean? Gravity waves?

larsa
larsa
Every medium is dispersive. Why are you surprised that water is?

Also, which waves do you mean? Gravity waves?

Yes, gravity waves. What i meaned was, what is the physical reason for this property of water. What is the cause of dispersion?
But my question was badly formulated.

nasu
It's not really a propriety of water itself but rather of this kind of waves.
What I mean is that no matter what the medium is, this type of wave will show dispersion.
If you look at the basic dispersion relation (just gravitational, no surface tension effect), it does not contain any medium specific parameter. Just the depth of the water in equilibrium and g (gravitational constant).
If you take into account surface tension, then it will depend on the values of this too. But again, there will be dispersion in any medium not just water.

For sound waves the dispersion in water is very little and usually neglected.
And so is in most common homogeneous media. So again, nothing special about water.

larsa
larsa
It's not really a propriety of water itself but rather of this kind of waves.
What I mean is that no matter what the medium is, this type of wave will show dispersion.
If you look at the basic dispersion relation (just gravitational, no surface tension effect), it does not contain any medium specific parameter. Just the depth of the water in equilibrium and g (gravitational constant).
If you take into account surface tension, then it will depend on the values of this too. But again, there will be dispersion in any medium not just water.

For sound waves the dispersion in water is very little and usually neglected.
And so is in most common homogeneous media. So again, nothing special about water.
Thank you very much

muscaria
Just thought I'd add a few points to the discussion given the nature of the original post, by trying to give a "general" idea of what is going on. If you understand what dispersion is and where it comes from, you wouldn't be asking such a question, so hopefully this may help somewhat.. As mfb and larsa have pointed out already, dispersion is an inherent feature of wave-like phenomena. I'm guessing your real question is what does dispersion even mean and where does it come from? So I'll answer in that spirit. You got me thinking about this by the way :) and made me realize something I hadn't appreciated fully before, and others also I'd imagine too! Unless there's something I'm neglecting.. But it might make more sense to start another thread about it and include a link to this one.

There are two main points I'd like to make which I feel are important in this regard and should make all this clearer, the second following naturally from the first. I'll state them first just to give you a sense of the narrative and then illustrate by considering a very simple and intuitive example system which captures the essence of the whole thing. The crucial point is that wavelike phenomena occurs in a medium/field/continuum (fluids, lattices, Electromagnetic field, most generally all gauge fields) , due to the ("internal or body") interaction forces between the elements which make up the medium. The second point is that the dispersion of the medium depends then on the form of these interactions. In other words it depends on the manner in which the particles feel each other (I am considering a particle picture here to illustrate the main point, in a continuum we would generally require a description in terms of tensor fields etc).

To illustrate what I mean by this, let's imagine we had a 2d flat sheet of point masses which were interconnected by springs, along the lines of the following image:

but repeated out to ##\pm\infty## forming an infinite 2d lattice. For simplicity and to illustrate the basic point we'll assume they all have the same mass, the stiffness of all the springs are also the same and the masses can only move in and out of the screen, normal to the 2d surface. The springs are there simply to define some kind of force between these masses and we could have chosen some other form of interaction, as well as some other form of lattice geometry etc..

OK, now let's say none of them are moving initially, but then I start pulling one of the masses in the direction normal to the surface (out of the screen), then the 4 springs attached to that mass start to become stretched in this normal direction so that the 4 masses on the other ends of the 4 springs start getting pulled up also etc.. If the initial pulling was a sudden but intense tug, i.e. an impulse, the disturbance propagates giving rise to wave like motion, where masses get pulled up and then back down, oscillating on the spot so to speak. For example, the mass i suddenly jerked upwards, pulls its 4 neighbouring masses upwards but is nonetheless experiencing a restoring force due to the stretched springs which pulls it downwards and at some point its neighbours become higher than him and pull him upwards again, never as high as initially though of course because of the knock on effect of this process as each mass is pulling its 4 neighbours up and down, so that energy is being displaced throughout the system and the initial mass never returns to the same energy - you get the picture..

Again, we're neglecting transverse motion so that masses only move into and out of the screen, in the same way that your water waves result from water levels rising and falling, no water is being translated along the water surface even if energy is being translated. This is arises from the existence of ##\textit{internal}## forces between the water molecules which results in a ##\textit{surface}## tension over infinitesimal volumes of water, because all the molecules are fighting to stay close together so to speak, because it's energetically favourable, and also why a droplet of water resting on a table doesn't spontaneously smear out across the table but wants to stay as compact as it can (other internal repulsive forces/interactions come into play if they get too close - so there's a natural balance), which would be a sphere but takes an elliptic form due to the ##\textit{external}## gravitational force which reaches into and acts throughout the ##\textit{volume}##.

So the initial energy in stretching the 4 springs from earlier, or in deforming an initially level surface of water, gets dissipated or ##\textit{dispersed}## (hence the term dispersion) through the field as a result of the interactions between the masses. In fact, the effect of gravity in your water waves scenario is simply to flatten out, what would otherwise be a massive sphere of water and in my understanding, can play no part in giving water medium like qualities. I'll start a new thread for this discussion.
It is the existence of these internal type forces which gives rise to energy transmission/flow/dispersion through the system, i.e. momentum, and wavelike properties to the medium. If you then take the limit where the distance between the masses in the spring system goes to 0, you have a force field which acts across the boundary of elementary surfaces, so across the edges of tiny squares in 2d. For a 3d system it would be a field acting over the bounding surfaces of infinitesimal volumes. Again this is how energy can be dispersed through the system.

Now the second point is that the form of the interactions between the masses will determine the manner in which energy flows across the system. In the spring system, 2 neighbouring masses interact harmonically with interaction potential ##\frac{1}{2}m\omega^2(\Delta x)^2##, where ##\Delta x## is the extension of the spring in the normal direction. But we could have written down another form of attractive interaction of a higher even power, or included interactions between masses diagonal from each other (next nearest neighbour) as well as the nearest neighbour, or between masses separated by 2 lattice sites, 3, or whatever we want and giving whatever form we want to the interaction potential or changing the geomtery of the lattice. But once we have defined how they all interact, and set up an initial configuration of the masses i.e. set their positions normal to the 2d surface, energy will disperse/flow in a determined way.

Now for the final point.. Taking our our 2d harmonic spring system above, If you just randomly set the vertical positions of the masses, energy would flow differently from point to point over the system, meaning the 2 momentum components over the surface become a function of where you are in the surface. There are however, certain special configurations of the masses for which energy flows uniformly across the system. For the harmonic system, this corresponds to certain oscillating initial positions given to the masses - imagine the flat sheet now oscillates across one of the directions (fixed moment-snapshot in time), this is your wavelength. If you set the positions up that way to begin with, all the masses will then over time, just oscillate up and down at the same frequency and energy is being translated uniformly along one direction. Do you see where I'm getting at? These different initial perfect wavelengths are your modes corresponding to a definite and uniform flow of energy (i.e momentum) across your system, just like a steady fluid flow has the same momentum at every point. Every pure wavelength corresponds to a mode of flow of energy which is constant over the system. If energy is flow coming into a point at some rate and flowing away also at the same rate, then the energy remains constant at that point, and this is true of every point. Alternatively, if you look at any given mass over time, it simply oscillates up and down at a given frequency where kinetic and potential energy are getting converted into one another but the total energy of each mass is constant. So for every mode of uniform flow of energy (momentum), the total energy of the system is distributed uniformly over the system.. You can then build up a "curve" where you associate a definite energy (frequency) to configurations which will give you a definite flow of energy/momentum (wavelength), which you call a dispersion relation.. again, constant flow of energy/momentum everywhere -> energy stays constant everywhere.

This all came from the form of the harmonic/spring interactions between the masses.. and gave rise to these nice harmonic/pure oscillations over space and time. If you were to change the form of interactions between your degrees of freedom/masses, the form of the configurations giving rise to steady flow of energy will change and become some other shape instead of a sine wave with perfect frequency, and their energies also, but it is always these that you are seeking (modes/eigenstates etc) because steady energy flow/momentum gives a uniform energy density distribution which stays constant over time and also means that one flow mode doesn't exchange energy with another, although you can superimpose many flow modes to describe complicated general flow patterns, each one has a definite effect which doesn't affect the others (uncoupled, diagonalised etc..). Anyway, this is why dispersion relations are interesting and why band theory became significant, because they are reflective of the nature of the interactions between the elements of the system and can tell you a lot about the system by just looking at it, giving you the energies associated with these orthogonal modes of flow.
So I can only really answer the question, "what is the cause of dispersion", by saying that it emerges from the "introduction" of interactions in many particle systems or in the equivalent continuous case, from internal forces or stress of a fluid. The form of these internal forces/interactions give rise to certain definite energy flow modes of uniformly distributed energy density which determine how energy is dispersed within the system to which we associate a curve..
EDIT: forgot square for ##\omega## frequency

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larsa
larsa
Every medium is dispersive. Why are you surprised that water is?

Also, which waves do you mean? Gravity waves?
Sound waves in air are not dispersive

larsa
Just thought I'd add a few points to the discussion given the nature of the original post, by trying to give a "general" idea of what is going on. If you understand what dispersion is and where it comes from, you wouldn't be asking such a question, so hopefully this may help somewhat.. As mfb and larsa have pointed out already, dispersion is an inherent feature of wave-like phenomena. I'm guessing your real question is what does dispersion even mean and where does it come from? So I'll answer in that spirit. You got me thinking about this by the way :) and made me realize something I hadn't appreciated fully before, and others also I'd imagine too! Unless there's something I'm neglecting.. But it might make more sense to start another thread about it and include a link to this one.

There are two main points I'd like to make which I feel are important in this regard and should make all this clearer, the second following naturally from the first. I'll state them first just to give you a sense of the narrative and then illustrate by considering a very simple and intuitive example system which captures the essence of the whole thing. The crucial point is that wavelike phenomena occurs in a medium/field/continuum (fluids, lattices, Electromagnetic field, most generally all gauge fields) , due to the ("internal or body") interaction forces between the elements which make up the medium. The second point is that the dispersion of the medium depends then on the form of these interactions. In other words it depends on the manner in which the particles feel each other (I am considering a particle picture here to illustrate the main point, in a continuum we would generally require a description in terms of tensor fields etc).

To illustrate what I mean by this, let's imagine we had a 2d flat sheet of point masses which were interconnected by springs, along the lines of the following image:
View attachment 99045
but repeated out to ##\pm\infty## forming an infinite 2d lattice. For simplicity and to illustrate the basic point we'll assume they all have the same mass, the stiffness of all the springs are also the same and the masses can only move in and out of the screen, normal to the 2d surface. The springs are there simply to define some kind of force between these masses and we could have chosen some other form of interaction, as well as some other form of lattice geometry etc..

OK, now let's say none of them are moving initially, but then I start pulling one of the masses in the direction normal to the surface (out of the screen), then the 4 springs attached to that mass start to become stretched in this normal direction so that the 4 masses on the other ends of the 4 springs start getting pulled up also etc.. If the initial pulling was a sudden but intense tug, i.e. an impulse, the disturbance propagates giving rise to wave like motion, where masses get pulled up and then back down, oscillating on the spot so to speak. For example, the mass i suddenly jerked upwards, pulls its 4 neighbouring masses upwards but is nonetheless experiencing a restoring force due to the stretched springs which pulls it downwards and at some point its neighbours become higher than him and pull him upwards again, never as high as initially though of course because of the knock on effect of this process as each mass is pulling its 4 neighbours up and down, so that energy is being displaced throughout the system and the initial mass never returns to the same energy - you get the picture..

Again, we're neglecting transverse motion so that masses only move into and out of the screen, in the same way that your water waves result from water levels rising and falling, no water is being translated along the water surface even if energy is being translated. This is arises from the existence of ##\textit{internal}## forces between the water molecules which results in a ##\textit{surface}## tension over infinitesimal volumes of water, because all the molecules are fighting to stay close together so to speak, because it's energetically favourable, and also why a droplet of water resting on a table doesn't spontaneously smear out across the table but wants to stay as compact as it can (other internal repulsive forces/interactions come into play if they get too close - so there's a natural balance), which would be a sphere but takes an elliptic form due to the ##\textit{external}## gravitational force which reaches into and acts throughout the ##\textit{volume}##.

So the initial energy in stretching the 4 springs from earlier, or in deforming an initially level surface of water, gets dissipated or ##\textit{dispersed}## (hence the term dispersion) through the field as a result of the interactions between the masses. In fact, the effect of gravity in your water waves scenario is simply to flatten out, what would otherwise be a massive sphere of water and in my understanding, can play no part in giving water medium like qualities. I'll start a new thread for this discussion.
It is the existence of these internal type forces which gives rise to energy transmission/flow/dispersion through the system, i.e. momentum, and wavelike properties to the medium. If you then take the limit where the distance between the masses in the spring system goes to 0, you have a force field which acts across the boundary of elementary surfaces, so across the edges of tiny squares in 2d. For a 3d system it would be a field acting over the bounding surfaces of infinitesimal volumes. Again this is how energy can be dispersed through the system.

Now the second point is that the form of the interactions between the masses will determine the manner in which energy flows across the system. In the spring system, 2 neighbouring masses interact harmonically with interaction potential ##\frac{1}{2}m\omega^2(\Delta x)^2##, where ##\Delta x## is the extension of the spring in the normal direction. But we could have written down another form of attractive interaction of a higher even power, or included interactions between masses diagonal from each other (next nearest neighbour) as well as the nearest neighbour, or between masses separated by 2 lattice sites, 3, or whatever we want and giving whatever form we want to the interaction potential or changing the geomtery of the lattice. But once we have defined how they all interact, and set up an initial configuration of the masses i.e. set their positions normal to the 2d surface, energy will disperse/flow in a determined way.

Now for the final point.. Taking our our 2d harmonic spring system above, If you just randomly set the vertical positions of the masses, energy would flow differently from point to point over the system, meaning the 2 momentum components over the surface become a function of where you are in the surface. There are however, certain special configurations of the masses for which energy flows uniformly across the system. For the harmonic system, this corresponds to certain oscillating initial positions given to the masses - imagine the flat sheet now oscillates across one of the directions (fixed moment-snapshot in time), this is your wavelength. If you set the positions up that way to begin with, all the masses will then over time, just oscillate up and down at the same frequency and energy is being translated uniformly along one direction. Do you see where I'm getting at? These different initial perfect wavelengths are your modes corresponding to a definite and uniform flow of energy (i.e momentum) across your system, just like a steady fluid flow has the same momentum at every point. Every pure wavelength corresponds to a mode of flow of energy which is constant over the system. If energy is flow coming into a point at some rate and flowing away also at the same rate, then the energy remains constant at that point, and this is true of every point. Alternatively, if you look at any given mass over time, it simply oscillates up and down at a given frequency where kinetic and potential energy are getting converted into one another but the total energy of each mass is constant. So for every mode of uniform flow of energy (momentum), the total energy of the system is distributed uniformly over the system.. You can then build up a "curve" where you associate a definite energy (frequency) to configurations which will give you a definite flow of energy/momentum (wavelength), which you call a dispersion relation.. again, constant flow of energy/momentum everywhere -> energy stays constant everywhere.

This all came from the form of the harmonic/spring interactions between the masses.. and gave rise to these nice harmonic/pure oscillations over space and time. If you were to change the form of interactions between your degrees of freedom/masses, the form of the configurations giving rise to steady flow of energy will change and become some other shape instead of a sine wave with perfect frequency, and their energies also, but it is always these that you are seeking (modes/eigenstates etc) because steady energy flow/momentum gives a uniform energy density distribution which stays constant over time and also means that one flow mode doesn't exchange energy with another, although you can superimpose many flow modes to describe complicated general flow patterns, each one has a definite effect which doesn't affect the others (uncoupled, diagonalised etc..). Anyway, this is why dispersion relations are interesting and why band theory became significant, because they are reflective of the nature of the interactions between the elements of the system and can tell you a lot about the system by just looking at it, giving you the energies associated with these orthogonal modes of flow.
So I can only really answer the question, "what is the cause of dispersion", by saying that it emerges from the "introduction" of interactions in many particle systems or in the equivalent continuous case, from internal forces or stress of a fluid. The form of these internal forces/interactions give rise to certain definite energy flow modes of uniformly distributed energy density which determine how energy is dispersed within the system to which we associate a curve..
EDIT: forgot square for ##\omega## frequency

Your post is awesome. One question; why you wrote forget square for frequency? And please post the thread that you were thinking about

muscaria
Your post is awesome. One question; why you wrote forget square for frequency? And please post the thread that you were thinking about
Ah cool, glad you liked it then! I know it was a bit long to read, but don't know I guess I just went with the flow :p
I hope it was helpful in any case, let me know if there's anything obscure you feel needs more/deeper explanation/justification. Or any part that seems unclear, happy to bounce ideas around :). The edit comment, was just to say that in the original post I had forgotten the square over the ##\omega## which appears in the harmonic potential equation.. So edited the file to correct that and include the square, as it appears now. Should have phrased it differently though probably!
Oh and yes, will start the new thread soon, just got distracted there by an interesting thread and have a fair amount of work.. In the next 10h say ;)
All the best

larsa
Mentor
Sound waves in air are not dispersive
They are, just less than many other things. Dispersion is more relevant in the ultrasound range, but no medium is completely free of dispersion. See this old publication PDF, for example (page 17 there, and reference 10).

larsa
nasu
So the initial energy in stretching the 4 springs from earlier, or in deforming an initially level surface of water, gets dissipated or ##\textit{dispersed}## (hence the term dispersion) through the field as a result of the interactions between the masses. In fact, the effect of gravity in your water waves scenario is simply to flatten out, what would otherwise be a massive sphere of water and in my understanding, can play no part in giving water medium like qualities. I'll start a new thread for this discussion.

You can then build up a "curve" where you associate a definite energy (frequency) to configurations which will give you a definite flow of energy/momentum (wavelength), which you call a dispersion relation.. again, constant flow of energy/momentum everywhere -> energy stays constant everywhere.

It is a long and interesting post. If I understand you correctly, you are describing normal modes in a system, basically.
However, I am afraid you are a little confused about the term dispersion in the context of the OP.
Even though in the common language dispersion may signify all these synonyms (flow, dissipate), here it means a quite different thing.
Maybe am not understanding you properly. I am not quite sure what you really mean in your second quote.
But definitely dispersion is not about energy flow or dissipation.
Energy flows or spreads in non-dispersive situation as well. And dissipation occurs in non-dispersive media as well.

This happens when the basic dispersion relation ω(k) is not linear in k. If this happens, a wave-packet composed of components with different frequency will spread out (disperse) as it propagates through the medium because the components have different speeds.

A well know example of dispersion is the decomposition of white light into components when is refracted. This is because the speed in glass (and so the index of refraction) varies slightly form red to blue light.

PS. Interesting that one of the simplest systems to show (real) dispersive behavior is the 1D chain of balls and springs, a simpler version of your example.

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larsa
They are, just less than many other things. Dispersion is more relevant in the ultrasound range, but no medium is completely free of dispersion. See this old publication PDF, for example (page 17 there, and reference 10).

twiz_
Is light in wave form dispersive?

nasu
Light is always in wave form.
Dispersion is rather a property of the medium. In some media there is more dispersion, in others very little. In vacuum there is no dispersion.

larsa
muscaria
It is a long and interesting post. If I understand you correctly, you are describing normal modes in a system, basically.
Yes, long in any case! And yes I am describing normal modes, but the intent of the post was to build up to this, making sure that the distinction between the action of external and internal forces in media was clear.
Energy flows or spreads in non-dispersive situation as well
And it was this point I wanted to make clear but may not have at all! :p This is why I started out attempting to make a clear distinction between external and internal forces but without being too technical because I feel one can grasp the essence of it without explicitly writing equations for the change in mechanical density as a integral over the infinitesimal bounding volume of the external force field + the intergal over the surface of the stress tensor. If there is no stress tensor field, there is no effective medium which carries energy stored in the medium is basically what I meant and also the reason I started with a spring system at equilibrium and then stretch, as energy becomes stored in the field in the form of work and then flows through the system depending on the form of the internal interactions, no external field. So all talk of energy flow in the post was initially stored in the system in the form of work.
However, I am afraid you are a little confused about the term dispersion in the context of the OP.
Even though in the common language dispersion may signify all these synonyms (flow, dissipate), here it means a quite different thing.
Yes I agree reading back over I used the terms disperse and dissipate, wrongly, dissipation especially given it physically has nothing to do with what I meant, I actually thought I had replaced it but obviously not, wrote it all pretty quick and loosely I guess, but thanks for bringing that out! There was no intention of implying any for of propagation of disorder, all the energy I have alluded to was meant to be mechanical! I do feel I understand the idea behind dispersion though and I was also using it in a loose sense but felt that as things were progressing through the text, an understanding of what I was getting at would be clear. I did get the (hence dispersion part wrong though!)
Maybe am not understanding you properly. I am not quite sure what you really mean in your second quote.
But definitely dispersion is not about energy flow.
But in a sense I feel it is, and I probably should have brought out what you said more:
If this happens, a wave-packet composed of components with different frequency will spread out (disperse) as it propagates through the medium because the components have different speeds.
As opposed to what I said
If you just randomly set the vertical positions of the masses, energy would flow differently from point to point over the system, meaning the 2 momentum components over the surface become a function of where you are in the surface.
There are however, certain special configurations of the masses for which energy flows uniformly across the system.
I think the spirit in which I approached the question was from my interpretation of the OP's
Why water is a dispersive medium?
What is the cause of dispersion?
And so I felt that to he may have not much intuition as to why excitations of a medium may travel at different velocities or what is going on other than some things of some form are traveling faster than things of another form. This needs explanation I feel, which can be provided. Even if I say OK well I take some wavepacket which is just a linear combination of modes it still doesn't get to the heart of what is going on, namely what are these special waveforms which travel at uniform velocity, how do they arise etc.. that seemed to me to be what one would actually want to understand, what gives rise to these modes. The point being that states of definite mometum then correspond to uniform/steady energy flow across the system. The flow of energy is momentum! (Noether type I mean, not canonical momentum, i.e.not momentum in field space which is the momentum of one of the mass degrees of freedom). And the flow of momentum is stress. And because the momentum associated with these modes is constant throughout, the rate of energy flow into a point is the same as the rate of flow out of the point, i.e. no divergence of current etc, meaning the energy density is uniformly distributed with a definite energy associated with the mode. I guess this only applies to transnationally invariant systems though, right?
The type of curve I was talking about would be something closely related to the following diagram, which is a plot of the dispersion relation for a 2-d square lattice tight binding model ##\epsilon(k_x,k_y)##

A well know example of dispersion is the decomposition of white light into components when is refracted. This is because the speed in glass (and so the index of refraction) varies slightly form red to blue light.
Yeah I see your point of looking at dispersion that way by building up a wavepacket, and seeing how it decomposes/disperses :p (got it right there!).. I guess I just felt that going the other way and building up all your components and trying to give a rough idea of how these come about, seemed like a nice take on it :)
All the best

larsa
twiz_
Light is always in wave form.
Dispersion is rather a property of the medium. In some media there is more dispersion, in others very little. In vacuum there is no dispersion.

I don't want to derail this into quantum physics or something, but I was hoping to refer to the double slit experiment. Anyway, with the recent insight about white light refraction, does this mean that dispersion of light only happens outside of a vacuum? In the previous post, it isn't clear whether it's the light or the prism that disperses.

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muscaria
Light is always in wave form.
Dispersion is rather a property of the medium. In some media there is more dispersion, in others very little. In vacuum there is no dispersion.
Actually,come to think of it, I guess there are some many body interacting systems which one would naturally consider a medium which can become completely dispersion-less! Under critical filling of certain lattice structures you can get completely localised eigenstates with a flat band dispersion.

larsa
Mentor
In the previous post, it isn't clear whether it's the light or the prism that disperses.
There is no "either or". Different frequencies of light have a different speed in glass. This effect is called dispersion.

There is no dispersion in a perfect vacuum. At least according to current theories and within the experimental uncertainties, some models predict tiny deviations from that.

nasu
I don't want to derail this into quantum physics or something, but I was hoping to refer to the double slit experiment. Anyway, with the recent insight about white light refraction, does this mean that dispersion of light only happens outside of a vacuum? In the previous post, it isn't clear whether it's the light or the prism that disperses.
Dispersion does not play a role in the double slit experiment. At least I don't see it.
Why would you want to refer to it in this thread?

twiz_
Why would you want to refer to it in this thread?

Sorry, that's not what I meant. When I said I was hoping to refer to it, I meant that I was hoping others would understand that I was referencing the double slit experiment. Like I said, that's off topic and I don't want to derail.

muscaria
of the tight binding model it's simply $$\epsilon(k_x,k_y) = \epsilon_0 - 2t_x\cos{k_xa} -2t_y\cos{k_ya}$$