# Origins of sine waves

There is a lot of resistance amongst engineers and scientists to believing that the physics of electromagnetic signals and dynamic processes involving such signals is NOT based on sine waves. There is so much in engineering that has been and is still being invested into the mathematics and tools supporting harmonics based analytics and design activities of thousands of engineers. Equally, thousands of academics are teaching hundreds of thousands of electrical engineering students all this on and on.

But let’s think what we can learn from nature that communicates signals from one point in space to another point and does it in a most frugal and at the same time prodigal way.

Imagine that we need to send a signalling event from point A to point B in space. This clearly involves some sort of motion. What kinds of motions we know. Well, at least two. One directional – that would be, for our example, moving along a straight line between A and B. Another type of motion we know rotational. This latter one should go if there is a point in the circle or sphere around which the notion can revolve. So, if the signal event involves the change in say electric field E and magnetic field H, we know that their vector product ExH forms Poynting vector, which shows the direction of the signal propagation. This propagation can only happen at speed of light in the medium (as there has been no evidence of otherwise!). If the direction of the vectors E and H is not changing in time at point A, our signal, defined by the Poynting vector ExH, will travel from point A to point B directly, albeit, this direction may not be straight line as it may be determined by the surrounding environment’s properties of epsilon and mu. So, for example if that environment is a transmission line (TL) formed by two metallic plates, then the Poynting vector will travel in the direction defined by the TL.

Now, if E and H elements of the vector are steps, then clearly these steps will form pulses, both in time and space. Now if for example these components of the vector are themselves in the rotational motion, and this rotation of E and H happens with a particular frequency omega, then the phase of the rotation omega*time will form an instantaneous value of the angle whose sine and cosine will determine the value of the potential produced, say, by the E component, between the plates of the TL. Unfold this rotation of vector E in space in the direction of the propagation of the Poyning vector, and you get a spiral – which is a superposition of the rotational and directional motions. The projection of this spiral on the normal plane along the Poynting vector’s direction will give us a sine-wave for E, likewise a cosine-wave for H. Hence rotational shift of 90 degrees will manifest itself as a longitudal phase shift of 90 degrees.

So, to summarise, the sine-wave is a product of a combined effect of a rotational motion and longitudal motion. But what’s important is that this sine-wave has a clear point of the start and end, and it’s not being there as a fundamental element. It is a mathematical trace of the superposition of fundamental elements of motion.

Another possible source of sine-waves is a successive oscillation of step-wise process in the longitudal direction, between two points. This is something that we have already discussed in earlier blogs, where we talked about the physical processes of propagation of energy current in transmission lines. Indeed if we connect a TL which has an open circuit end with the TL which is short circuited end, we have the effect of C and L combined, and the process of reflection of the energy current in this system, will form a series of steps which can be approximated by a sine-wave. Again, the sin-wave is a mathematical product of more primitive physical process.

In my recent email exchange with a group of people arguing with Ivor Catt about the primal nature of sine-waves, I wrote:

“Mechanics tells us there are two basic motion types. Direct and rotation around a point. If we assume that ExH can follow both, the latter for example, under the influence of another force (gravity?), can help us the express the effect of “corkscrew propagation”, and hence spatio temporal sinewaves. Actually a corkscrew or spiral is a good explanation of an AC shaped energy current around a wire. Light can propagate from its source in myriads of corkscrews by the way! ”

“By the way a corkscrew-like propagation of signal (hence, information) via a medium with natural resistance or friction (usual epsilon and mu properties) would be most effective, again from mechanical analogy point of view. Nature, again, following Occam’s razor principle, would do it the same way we penetrate into a cork or a wall, and that’s what nature would always do if it needs to pass information from one point to another! Energy current screws up everything! Ha-ha!”

I also recommend to someone who wishes to visualise these corkscrew processes to have a look at these wonderful videos on youtube:

# Particles are secondary …

Ed Delian wrote today to Ivor Catt:

You ask me what happens inside an indivisible particle? “How does the particle’s right hand edge know of a collision of the left hand edge”? The answer is: Since it is an indivisible particle, there is no space for a “signal” to “travel” inside from “here” to “there” in order to transfer information. The whole indivisible particle as a continuum (!)  “knows” what its edges experience, all at the same time.

I responded:

Dear Ed,

Starting the discussion of electromagnetism or even gravity with particles, which are indivisible in one sense or another, is fraught with many problems and issues sticking out that seems impossible to put in order. This is especially problematic if one needs to explain the phenomenon of passing information in digital systems.

That’s why Ivor and us following his views propose to begin with the simple and clear foundational concept of energy current that travels and can only do so with a speed of light in the medium.

The medium is naturally resistive to this motion, in a manner analogous to friction. These two properties of the medium is the only assumptions we need to make (this “minimalism” is justified by the natural tendency of nature to follow the Occam’s razor principle, as well as experimental facts). The other basic assumption is that nature has different mediums occupying fragments of space and thus has boundary conditions, between which we have no instantaneous action because those boundaries involve distances. So, any particles are the result of the division of space between boundaries, in which energy current is trapped. These entrapments are often ‘leaky’ which allows energy current not just be confined within these particles (due to reflections) but also travel outside the particles and this create levels of interaction. So this way we have elements of mass formed with nonzero volume.

Now, states or levels of the energy current density that are inside and outside those fragments of the mediums form something that traditionalists prefer to called fields, and they can be associated with forces, electromagnetic and gravitational.

So, clearly, talking about particles is possible but not at the fundamental level of passing information in nature.

Kind regards

Alex

# Ordering notions in Electromagnetics: what comes first?

Today, I asked some people interested in Electromagnetics the following question:

Harry, All:

Do you have a clear idea how the values of epsilon and mu are obtained/measured for mediums? Are they obtained from measuring the velocity of light in the medium and from measuring the characteristic impedance of the medium, and then solving the two equations for c and Z0?

Alex

Alex,
No! Do you?
Harry

Followed by me:

No, I don’t, Harry. What I see on wiki is a sort of dead circle. Plus it involves the use of electron charge and other constants.Alex

Then Harry continued as follows:

Alex,
Regarding your second question. If we are talking about SI units, they are defined units. The mu is defined and  then the epsilon is defined by using the velocity of light. The ratio is the velocity of light. I think the definitions were established so as to keep the previous definitions of the ohm voltage and current. Frankly I am not up on this. My view is that free space impedance ought to be 1 ohm and the rest of the units based on that. But that is only my opinion.
The subject of units and measurement is very highly specialized and is fixed based upon international treaties. That has a long history behind it. The main idea being to try to keep the common units in use the same while bringing in new definitions of units based upon more stable reference standards.
Harry

And this has led me to proposing the idea of establishing some precedence order between various key notions in Electromagnetics:

Harry,

I agree units and measurements are connected. But, putting various notions of constants and parameters, such as epsilon and mu ‘on the table’ along with c and Z0, as well as electron charge, etc, could be done in some order of precedence, and that precedence could be aligned with the order in which the relevant physical notions are put into theory.

The latter (order of precedence of physical notions) should follow (at least) two key principles:

– experimental evidence, and

– Occam’s razor.

Perhaps also some basic geometric relationship of space and time, such as velocity will need to be used as a guide.

I don’t want to get deeper into natural philosophy here. But, to me, Ivor’s notion of energy current being most fundamental in EM, which has the two key attributes, the velocity of light (or generally, of EM energy current) and impedance (analogous to some sort of viscosity or friction, or some counter-action to progressive action) is most natural, and meets the above principles. So, the order of precedence seems to originate in first having c and Z0, which can be measured by existing equipment, is also natural. Those are then split into two principal components, epsilon and mu, which are in some sense are more primary if we talk about Electrical and Magnetic as the two key aspects of Electromag. So from the experimental point of view our “Adam and Eve” are c and Z0, which we can measure. From the theory point of view, our “Adam and Eve” are epsilon and mu.

That’s how I see it to myself in some logical order.

Alex

# Electron is a two-faced Janus of Electrical and Magnetic aspects of energy current trapped in it …

Can you give me a proof that what you have inside an electron is any different from the so-called ‘empty space’. Any finite section of space has the right to say – look, I have my one epsilon and mu hence I have my own speed with which ExH travels in me. What’s wrong with this approach? The fact that electron is tiny doesn’t deprive it from the privilege of having its own ExH trapped in it. Then this electron can have both electric and magnetic Januses to turn to us in the form of its charge and spin!

My response to Akinbo’s email:

Regards

Alex

From: Akinbo Ojo <taojo@hotmail.com>
Sent: 14 January 2020 15:36
Subject: Re: Displacement Current in Deep Space for Starlight

Hi Alex,

When taking a medicine is worse than the disease one wants to cure I think it is wise to stick with the disease. I also ask you to take note of what Harry just posted concerning how you want to combine the equations.

The only place I see usefulness for the ExH concept is in transmission lines (co-axial cables) where E can travel in the core wire and H can travel alongside in the space between. But there are no such transmission lines or co-axial cables in space so this type of energy current cannot work in empty space.

Regards,

Akinbo

# The cacophony of particularities in Maxwell’s equations … at the end of the day, it’s only Catt’s Heaviside signal that puts things right!

Over the last couple of week I have been witnessing an interesting email discussion about Maxwell’s equations between 2-3 people trying to come to terms with the difficulty of accommodating the notion of displacement current in free space and ‘sorting out’ the Ampere’s law. The latter combines, for whatever reason, both the elements of propagating field (not requiring charged particles as this field can propagate without involving massed matter) and current density (implying the existence of massed particles).

I have drawn my own conclusions out of this discussion, which ended up with conclusions that the above mentioned difficulty cannot be easily resolved with the bounds the temple of the classical electromagnetics with its holy book of Maxwell’s laws.

Here are my comments on this:

To Ivor Catt:

Following your theory where the Heaviside signal travels (and can only do so) with the speed of light in the medium, such a speed is entirely determined by epsilon and mu. Thus, where we have an interface between very low epsilon dielectric and very high epsilon metal, from the point of view of energy current, we have the effect similar to friction (against the metal surface – like a rotating wheel goes forward on the ground thanks to experiencing friction against the ground). And thanks to this “friction” it prefers to trolley along the metal wire, or between the metal plates of the capacitor.

To David Tombe:

Catt’s theory works at a different level of abstraction. This is the level of fundamental energy current. This level underpins “charged particles”. The latter are the result of the ExH energy current trapped in corresponding sections of space. What’s important is that that trapped energy never stops inside those particles as it can only exist in the form of ExH slabs moving with speed of light in the epsilon-mu medium. So then, when you apply energy current travelling outside those particles, there is an interesting interaction with the energy current inside those particles.

The entire world is filled with energy current fractally sectioned into fragments determined by space sections.

All I can say is that in my opinion you misunderstand the domain of action of Catt’s theory. It does not consider static electric field. That’s it. There’s no such a thing as static EM energy. It can only move at speed c=dx/dt, in all directions.

And this energy fills up space according to its epsilon/mu properties.

There’s no need for Maxwells equations to be involved in Catt’s theory. All these equations are partial, like Greek gods.

To Ivor Catt:

I, perhaps surreptitiously,  was awaiting for your email, either in public or in private.

Coincidentally, about an hour ago, I typed a message intended to be sent to the whole list from that discussion (adding Malcolm, who I think is on the same wavelength with us), saying:

“Ivor, please, say something, because these people are facing an impossible task of ‘squaring the circle’ of a set of Maxwell’s laws into something coherent – but the reason why it is impossible is that no one actually knows exactly what Maxwell meant by that list of laws dressed into fairly sophisticated mathematics. So, Ivor, the same fate may be with you, unless you say something, in some 20 years from now {well, it looks like I miscalculated by 5 years from your estimate of 2045!} no one will know exactly what Catt meant by his energy current”.

Then some invisible force pushed me to discard that email! And now, I had an evening walk to my office to freshen up my mind, and here I see your email.

Sadly, people, don’t listen and can’t liberate themselves from the heavy chains of those (partial) laws – which are like, indeed, those separate gods of Greeks or Romans being responsible for one aspect of life or another or one phenomenon in nature or another. They can’t understand that the Occam’s Razor of nature wouldn’t tolerate having so many (purportedly, fundamental) relationships, with lots of tautology in them. All those relationships, taken individually, are contrapuntal and superposing.

People can’t understand that there is no need for stationary fields, no need for separate treatment of charged particles etc. Everything comes naturally as a result of energy current trapped in sections of space, where it continues to move.

I think the next big leap where Catt’s vision will show its power will happen in high-speed computing and a massively parallel scale – truly high-speed! Until then we will probably fight against the Windmills of stale minds and deaf ears …

This confused discussion between David Tombe and Akinbo (not sure if Malcolm has seen that) is an illustration of the fact that behind the mathematically elegant façade of Maxwell’s laws there is a massive mess of physical concepts, a cacophony of man-made and contrived ‘pagan-like’ beliefs and disbeliefs (e.g., did Maxwell mean that or not?), which may work in special cases. The fact that these beliefs can work in special cases of success in growing crops or hunting/herding animals in some regions of the world, or in a more modern terms, wiring up a Victorian mansion and sending data to Mars rovers. But how they are going to succeed in the future when needs Terabit/s  data rates or picosecond latency in accessing storage, nobody knows. I am less inclined in dividing people into true scientists, careerists or other categories. Historical materialism (which we had to study back in the USSR) gave pretty good explanation of all kinds of folk under the sun. Nobody is saint here. It’s just a personal comfort matter …

To Akinbo Ojo (in reply to his email attached below):

Just combine ∇2E = µε(∂2E/∂t2) or ∇2H = µε(∂2H/∂t2)  into one eqn, replacing E and H with ExH, and you’ll have the Heaviside signal (aka energy current) propagating in space with speed of light in the epsilon/mu medium, and that’s all what is needed by Catt’s theory, and that what fills up fragments of space, in order to form transmission lines of particular Z0, capacitors, inductors, elementary particles, etc.

Everything in the world is filled up with this energy current, and any such an entrapment of energy current turns sections of space into elements of matter (or mass)!

From: Akinbo Ojo <taojo@hotmail.com>
Sent: 14 January 2020 14:20
Subject: Re: Displacement Current in Deep Space for Starlight

Hi David,

I didn’t say there was an error. I said given the Ampere and Faraday equations when you follow the curls and substitutions you will confront something that would be unpalatable to you and which you must swallow before you can get ∇2E = µε(∂2E/∂t2) or ∇2H = µε(∂2H/∂t2).

Regards,

Akinbo

# Static vs Dynamic when referring to the electric field in capacitor

I wrote in my paper “Energy current and computing” (https://royalsocietypublishing.org/doi/10.1098/rsta.2017.0449 ):

“there is no such a thing as a static electric field in a capacitor. In other words, a capacitor is a form of TL in which a TEM wave moves with a single fixed velocity, which is the speed of light in the medium”.

This statement causes some controversy – Ivor Catt refers to it as “heresy”.

Here I would like to explain what is meant here by static/dynamic:

One of the important aspects of considering the distinction between ‘static’ and ‘dynamic’ is that of what we mean by dynamic/static in the first place.

I think that the notion of dynamic/static, first of all, concerns as to whether a particular value (say, electric field intensity E) changes in time or not, i.e. whether dE/dt is non-zero or not. Another notion of dynamic/static is about the movement of the value in space (and, necessarily in time because movement in space cannot be instantaneous!), so if we talk about the electric field E, we can be talking about dE/dx being non-zero, and here is the critical notion of the link between dE/dt and dE/dx, which MUST be mediated by dx/dt (speed of light in the medium!). The latter MUST BE ALREADY SET UP, ab initio, and that’s what Ivor Catt’s Heaviside signal is about. So, even if we have an impression that something is static – like electric field in a fully charged or fully discharged capacitor, this impression will be viewed in the form of contrapuntal dE/dt=0, we somehow need to retain the notion of c=dx/dt being constant and non-zero. But then the immediate question arises of: what is there that is moving in a longitudal direction at speed c? And the answer is the Heaviside signal! What else? So, my understanding is that THIS MOVING THING is what makes me state that that there is no such a thing as a static electric field in a capacitor!

# “Contrapuntal superposition” of Heaviside signals unravelled as a lookalike state coding problem in asynchronous circuit design

This article http://www.ivorcatt.co.uk/x267.pdf by Ivor Catt – published (now more than) 40 years ago – proposed looking at transverse electromagnetic (TEM) wave by means of the so-called Heaviside signal. Heaviside signal is basically EM “energy current”, described by Poynting vector ExH (E and H are electric and magnetic field intensities, respectively), that travels and can only travel in space with a speed of light in the medium, fully determined by its fundamental parameters permittivity (epsilon) and permeability (mu) – i.e., c=1/sqrt(mu*epsilon). The key point here, I should again stress, is that ExH cannot stand still – it can only travel with speed of light. One might ask, where does it travel? It travels where the environment – i.e the combination of materials – leads it to, and in practice it predominantly goes where the effective impedance of the medium is smaller. The effective or characteristic impedance of the medium, Z0, is also fully determined by the permittivity (epsilon) and permeability (mu), i.e. Z0=sqrt(mu/epsilon). Moreover, Z0=E/H – this is sometimes called the constant of proportionality of the medium.

Why is this look at the TEM wave more advantageous than some other looks, such as for example, the so called “rolling wave” of the alternating concentrations of magnetic energy 1/2*mu*H^2 and electric energy 1/2*epsilon*E^2 in the direction of propagation? As Catt shows in the above article, this more conventional way is actually meta-physical, because it is based on the assumption of causality between the electric field and magnetic field and vice versa. The latter is a form of tautology because it creates a non-physical, but rather, mathematical or equation-based “feedback mechanism”, which does not make sense in physics.

Another important issue that calls for the use of Heaviside signal is that it retains the notion of the travelling EM “ExH slab” in each direction where it can travel, and hence its change-inducing geometric causality between points in space. As exemplified by the effects of travelling TEM waves in transmission lines (TLs), this look, for example, naturally separates the incident wave from the reflected (of the interface with another medium) wave, or from another wave that may travel in the opposite direction. As a result, the analysis of the behaviour of the TL becomes fuller and can explain the phenomena such as superposition of independent waves in cases such as cross-talk between TLs. Here is another paper by Ivor Catt – published more than 50 years ago – http://www.ivorcatt.co.uk/x147.pdf and subsequent clarifications – http://www.ivorcatt.co.uk/x0305.htm of the superposition of the even and odd modes (modes of TEM travelling with different speeds of light in the medium due to different epsilon and mu conditions arising between adjacent pairs of metal lines).

As shown in these papers, the view provided by the conventional theory is necessarily contrapuntal – it looks at the combined EM field in every point in space and in time. As a result it simply overlays the travelling ExH signals. And that’s what one can see by measuring voltage and current in points of interest on the TL. Or, equally, what one could see on the oscilloscope’s waveforms at points in space. Interestingly that looking at the same time at a number of points, in a spatially orderly way, leads to a conjecture that there is an interplay of several travelling TEM waves, but the conventional rolling wave approach would not explain the physics behind them properly!

What is remarkable in this for me is that this reminds me the difference between two types of models in asynchronous control circuits and how one of them obscures the information revealed by the other. One type of model that is based on recording purely binary encoded states of the circuit (akin to the contrapuntal notion). The other is based on a truly causal model (say Signal Transition Graph – or STG – called Signal Graph or Signal Petri Net in my early publications: https://www.staff.ncl.ac.uk/alex.yakovlev/home.formal/LR-AY-TPN85.pdf or https://www.staff.ncl.ac.uk/alex.yakovlev/home.formal/AY-AP-PN90.pdf), where we have the explicit control flow of signal transitions or events running in the circuit. The difference between these two looks is often manifested in the so-called Complete State Coding problem (cf https://www.researchgate.net/publication/2951782_Detecting_State_Coding_Conflicts_in_STGs ). If we only look at the contrapuntal notion of the state without knowing the pre-history of the event order we cannot distinguish the semantically different states that map onto the same binary code provided by the signals. To distinguish between such states one needs additional information or memory that should be either provided in the underlying event-based model (the marking of the STG) or by introducing additional (aka internal or invisible) signals (in the process of solving the CSC problem).

I am not claiming that the above-noted analogy leads to a fundamental phenomenon, but it reflects the important epistemic aspect of modelling physical world so that important relationships and knowledge are retained, yet in a minimalist (cf. Occam’s razor) way. Some more investigation into this analogy is needed.

# Clearing my way through quantum entanglement – things are actually rather trivial …

The end of year 2019 was marked for me by a sudden revelation about the entanglement (aka EPR) “paradox”. Here is my confession, first. For a long time I had been thinking that the “superposition or entanglement paradox” consisted in the following:

Two particles (possessing a certain probabilistic characteristic such as a spin) originating from the same source were entangled, i.e. connected by, say, one being in phase alpha, while the other one in opposite phase (180-alpha). Then the particles were be sent into different directions of travel and remained entangled and what’s important their alpha parameter would still be unknown. Then at some point in time and space one of the particles would have been measured, and say found to be equal to 1 (I suppose we can use binary encoding without loss of generality). Now, here comes my misinterpretation: Then at the very same moment in time the other particle would be disentangled and its value would be exactly opposite, i.e. equal to 0. Therefore the paradox (in my interpretation) was as follows: (1) the state resolution in time is simultaneous for both particles, i.e. while forcing the measurement of the first particle we immediately have the measurement of the second particle; and (2) the state resolution on terms of value of one particle would completely determine that of the value of the second particle.

My problem was not in understanding why (2) was true. That was quite clear to me for a long time, especially after my discussions with experts like Professor Werner Hofer, who explained to me, a non-expert on quantum theory, that both particles, once entangled, would retain their phases and in that respect would remain information-wise connected. My problem was in accepting and understanding (1), which I my view violated temporal causality and no action at distance. I could not accept the fact that there would be no delay between the initiated measurement of the first particle and simultaneous resolution of the second particle. The reason for my conundrum was that I am a firm believer in causality of related events in time, and I could not accept (1).

But, thanks again to good old Werner (!), with whom we talked a few days before Christmas 2019, I realised that there is actually no paradox at all here. What actually happens is that – there is no issue (1) involved! The resolution of the second particle in fact can happen concurrently or independently of that of particle one! And the whole pathos of the EPR was only in part (2). This gave me an enormous relief and peace of mind needed for the coming festive season. The expected asynchrony and delay-insensitivity of the physical world had been restored! And, as far as part (2) is concerned, that was a trivial thing to me – this is purely a combinatorial (non-sequential in terms of automata) issue of one value being statically opposite to the other value – what’s the big deal !?

Now, what annoyed me in all this conundrum, well, obviously my own naivety and my inaccurate reading about the EPR paradox. On the other hand, I think that the lack of clarity in separating the issues of timing from value, i.e. when and what, that is quite symptomatic of the 20th century mathematical physics, involving complex quantum mechanical constructions, is what makes engineering-minded people like me – who expect both these issues to be properly addressed – confused and misled!

Happy days!

# Correction on my previous blog and some interesting implications …

Andrey Mokhov spotted that to satisfy the actual inverse Pythagorean we need to have alpha=1/2 rather than 2. That’s right. Indeed, what happens is that if we have alpha = 1/2 we would have (1/a)^2=(1/a1)^2+(1/a2)^2. This is what the inverse Pythagorean requires. In that case, for instance if a1=a2=2, then a must be sqrt(2). So the ratio between the individual decay a1=a2 and the collective decay is sqrt(2). For our stack decay under alpha = 2, we would have for a1=a2=2, a=1/2, so the ratio between individual decay and collective decay is 4.

It’s actually quite interesting to look at these relations a bit deeper, and see how the “Pythagorean” (geometric) relationship evolves as we change alpha from something like alpha<=1/2 to alpha>=2.

If we take alpha going to 2 and above, we have the effect of much slower collective decay than 4x compared to the individual decay. Physically this corresponds to the situation when the delay of an inverter in the ring becomes strongly inversely proportional to voltage. Geometrically, this is like contracting the height of the triangle in which sides go further apart than 90 degrees – say the triangle is isosceles for simplicity, and say its angle is say 100 degrees.

The case of alpha = 1/2 corresponds to the case where delay is proportional to the square root of Voltage, and here the stack makes the decay rate to follow the inverse Pythagorean! So this is the case of a triangle with sides being at 90 degrees.

But if alpha goes below 1/2, we have the  effect of the collective decay being closer to individual decays, and geometrically the height of the triangle where sides close up to less than 90 degrees!

Incidentally, Andrey Mokhov suggested we may consider a different physical interpretation for inverse Pythagorean. Instead of looking at lengths a, b and h, one can consider volumes Va, Vb and Vh of 4-D cubes with such side lengths. Then these volumes would relate exactly as in our case of alpha=2, i.e. 1/sqrt (Vh)=1/sqrt(Va)+1/sqrt(Vb).

Cool!

# Charge decay in a stack of two digital circuits follows inverse Pythagorean Law!

My last blog about my talk at HDT 2019 on Stacking Asynchronous Circuits contained a link to my slides. I recommend you having a particular look at slide #21. It talks about an interesting fact that a series (stack) discharge rate follows the law of the inverse Pythagorean!

It looks like mother nature caters for a geometric law of the most economic common between two individual sides.