Radiating Black Body as Sounding Piano

Keith Jarrett giving input energy to a sounding Steinway resonating like a radiating blackbody.

Here is more to think of in the hammock:

A piano is a black body and conversely the action of radiating blackbody is similar to the action of a piano. More precisely, a piano consists of

1. resonating soundboard (the back of an upright piano)
2. bridge on the soundboard in contact with
3. vibrating strings
4. excited by hammers connected to keys
5. subject to input energy from the pianist.

In short, the soundboard vibrates in resonance with the strings through the bridge,

the strings are excited into vibration by energy input from the hammers, and the soundboard

transfers its vibration into vibration of the air in the room around the piano which is perceived as sound by a listener.

A modern piano (referred to as a piano-forte) is a miracle in the sense that it combines a loud attack with a long sustain as an effect of multiple stringing, with two or three strings for each

note tuned to almost the same pitch (with a spread of about 1 Hz), except deep bass notes which have a single string:

• in attack the strings are in phase with maximal input of energy to the soundboard,
• in sustain the strings are out of phase with less transfer of energy to the soundboard.

A mathematical analysis of a model of a sound generator as a coupled multiple string - soundboard like a piano, is given in Near-Resonance with Small Damping, showing the close connection to the analysis of blackbody radiation in Computational Blackbody Radiation with

particular notice of the effect of multiple stringing at nearly the same pitch.

The analysis shows that we may think of the radiation from a blackbody as the sound from a piano with all the keys being struck at the same time with the same power by an 88-fingered pianist.

A blackbody thus acts like a soundboard (system of resonators) with the input from the strings

represented by incoming light absorbed by the blackbody.

There is an important difference between sound (piano) and light (blackbody), namely the form of the output energy: If the sound generator/blackbody is modeled as

• a second order wave equation combined with
• a dissipative term representing output and a
• forcing term representing input,

then the dissipative term has the following form

• radiation of light: - gamma x third order time derivative (Lamor's Law),
• sound waves: gamma x first order time derivative (viscous damping),

where gamma is a small positive coefficient.

The effect is that the output energy increases quadratically with the frequency (Planck's Law) in the case of radiation, while the output energy is independent of frequency in the case of sound, at equal temperature = energy of the resonator.

For a piano it means that all keys when struck with the same power will give the same output energy (in decibel).

For a blackbody it means that the energy increases with increasing frequency and a cut-off becomes necessary to avoid an ultraviolet catastrophe.

The cut-off in the piano case is simply that the piano does not extend infinitely to the right. In Computational Blackbody Radiation it is argued that similarly a blackbody has a built-in cut-off of high-frequencies.

So if you want while reflecting over blackbodies (and global climate) in your hammock, you may think of a blackbody as a piano with all the keys being struck: a complex chord.

PS Dec 2012 A piano consists of two coupled systems each consisting of a wave equation with damping subject to forcing:

1. vibrating string with damping from bridge and forcing from hammer,
2. vibrating soundboard with damping from surrounding air and forcing from the bridge. 

The input to the string comes from the hammer, and the output from string damping is the bridge force acting as input to the soundboard and the output from soundboard damping is the force on the  surrounding air generating the sound.

The Hen and the Egg of Gravitation

As summer holidays are coming up and string theorists gather in Uppsala, it may be time to reflect in the hammock about the origin of gravitation and then recall the knol The Hen and Egg of Gravitation reporting on the following idea exposed in Many-Minds Relativity:

The gravitational potential P is related to distribution of mass M by the differential equation

• - Laplace P = M

where Laplace is the Laplace operator. This equation can be read in two ways:

1. Given M, the potential P is obtained by solving the differential equation - Laplace P = M.
2. Given P, the mass distribution M is obtained by applying - Laplace to P.

Here 1 is a non-local operation (solving a differential equation) reflecting action at distance, while 2 is a local operation (differentiation) without action at distance.

While in the hammock the reader is invited to reflect about 1 vs 2: Which comes first, the potential (the hen) or the mass (the egg)?

The Mathematical Secret of Flight 4

The secret of flight is hidden in the above picture showing the flow separation at the trailing edge of a wing, as explained in detail in the article The Mathematical Secret of Flight and the upcoming book The Secret of Flight.

Mathematical analysis shows that the swirling flow separation shown in the picture results

from an instability of opposing flows meeting behind the trailing edge, and the swirling motion

allows the flow to separate with little retardation requiring high pressure. Instead low pressure

develops inside the swirling flow which does not like high pressure destroy the high lift/suction established on the crest of the wing, while causing only small drag because of the small diameter of the trailing edge.

This is the miracle of flight revealed by mathematical analysis, whether you like it or not in the words of Richard Feynman.

The swirling motion is similar to that used by noble men when backwards leaving the king after an audience; an elegant form of separation without high pressure destruction of what was gained during the meeting.

I have asked Antony Jameson about a comment but not received any response so far.

Compare with 50 ways to leave your lover: The most elegant and thus best way is with a swirling motion avoiding build up of high pressure.

The Mathematical Secret of Flight 3

After my talk The Mathematical Secret of Flight at Svenska Mekanikdagar 2011 an remark was made by Laszlo Fuchs recalling early attempts to compute the lift and drag of a wing by solving the Euler equations by Antony Jamseon (left) and Art Rizzi at KTH in the 1980s.

In essence what Jameson and Rizzi did was to solve the Navier-Stokes equations with a slip boundary condition at very high Reynolds number, essentially the same as we are doing, and observing the appearance of unsteady fluctuating solutions.

However, these solutions were regarded with suspicion by the fluid dynamics community as some kind of ghost solutions with unclear physical significance. In particular the slip boundary condition without any boundary layer was against the dictate of Prandtl that no-slip is the only physically correct boundary condition and that boundary layers have to be resolved because the truth is to be found there.

The result was that the (incompressible) Euler solvers of Jameson and Rizzi became marginalized, just as Birkhoff had been in the 1950s when asking if there were any steady solutions at all.

What we have shown is that Birkhoff, Jameson and Rizzi were on the right track and that the effective suppression of their ideas has delayed the advancement of computational fluid mechanics by several decades. The solutions computed by Jameson and Rizzi were not ghost solutions but true turbulent solutions carrying important information of real physics.

The suppression of correct science is often more harmful than the promotion of wrong science.

Mathematical Secret of Flight 2

Computed turbulent flow velocity around a NACA0012 wing at 15 degrees angle of attack in beginning stall with separation on top of the wing.

An updated version of my talk on June 15 at Svenska Mekanikdagar 2011, is now available for preview as

• The Mathematical Secret of Flight

describing joint work with Johan Hoffman and Johan Jansson.

This work shows that computation of mean values such as lift and drag of an airplane, car or boat can be accurately computed without resolving thin boundary layers. Based on the computations a new mathematical theory for flight is presented which is fundamentally different from that by Kutta-Zhukovsky-Prandtl filling text books. The new flight theory was first published in Normat 57:4 (2009) as The Mathematical Secret of Flight.

The spell of Prandtl as the father of modern fluid mechanics of attributing both lift and drag as effects of thin boundary layers requiring unreachable quadrillions of mesh points for computational resolution, can thus be broken. This opens a wealth of applications of computational fluid dynamics suddenly reachable with millions of mesh points.

The new theory of flight is presented in Mathematical Simulation Technology, a book which was officially banned by KTH in November 2010, as described in posts on KTH-gate. The reader can act as referee and decide if the ban is motivated from a scientific point of view. The last time a math book was banned was in 1632.

Mathematical Secret of Flight 1

Computed Lift and Drag of a 3d NACA0012 wing for different angles of attack by Unicorn (blue) compared with different experiments.

My talk on June 15 at Svenska Mekanikdagar 2011, is now available for preview as

• The Mathematical Secret of Flight

describing joint work with Johan Hoffman and Johan Jansson.

Based on accurate solution of the incompressible Navier-Stokes equations we identify the true mechanism for the generation of large lift L at small drag D of a wing with lift to drag quotient L/D of size 10 - 50, which is not described in the literature.

We combine the Navier-Stokes equations with a slip boundary condition on the wing motivated by the experimental fact that the skin friction is small for a slightly viscous fluid such as air or water, and we exhibit the role the slip condition in two crucial aspects:

• prevention of separation at the crest of the wing generating large lift
• 3d slip-separation at the trailing edge not destroying large lift and causing small drag.

Text books claim following Prandtl, named the father of modern fluid mechanics, that both lift and drag result from a boundary layer arising from a no-slip condition.

We obtain lift and drag in full accordance with experiments by solving the Navier-Stokes equations with a slip condition, which does not generate any boundary layer, and we thus present strong evidence that lift and drag do not originate from any boundary layer.

In short, we show that solutions to the Navier-Stokes equations with slip are computable and

correctly capture the physics of (subsonic) flight. See also

• The Mathematical Secret of Flight (Normat 57, 2009)
• Why It Is Possible to Fly (Google Knol) .

Compare with state-of-the-art as expressed by Moin and Kim in Tackling Turbulence with Supercomputers:

• To solve the Navier-Stokes equations for, say, the flow over an airplane requires a finely spaced computational grid to resolve the smallest eddies.
• Consider a transport airplane with a 50-meter-long fuselage and wings with a chord length (the distance from the leading to the trailing edge) of about five meters. If the craft is cruising at 250 meters per second at an altitude of 10,000 meters, about 10 quadrillion (10^16) grid points are required to simulate the turbulence near the surface with reasonable detail.

Kim and Moin express the necessity dictated by Prandtl to resolve thin boundary layers to correctly compute lift and drag of a wing or an entire airplane, which would require 50 years of Moore's law to increase the computing power with a factor 10^10 to reach the dictated 10^16 points.

We show that this is possible already today using 10^6 points by using slip without boundary layers to resolve.

Monstrosity of Quantum Mechanics 6: Collapse of Wave Function

Is quantum mechanics a physics beauty contest with all possibilities collapsing into one actuality upon observation? Who would you choose?

Since the multi-dimensional wave function of quantum mechanics is supposed to represent a probability distribution over all possibilities, the high dimensionality has to be drastically reduced to become an actuality of some interest.

This is supposed to happen in an interaction with an observer, referred to as collapse of the wave function, where the observer somehow picks one of all the potentialities and makes it into an actuality, as when Miss America somehow is chosen among many candidates by some educated physics observers.

Is then quantum mechanics a beauty contest? Well, ask your favorite physicist about the nature of the collapse of the wave function. Is it real? What is collapsing? Physical reality or our knowledge about reality.

Or is it quantum mechanics itself which collapses upon critical observation?

Monstrosity of Quantum Mechanics 5: Passive Observation Impossible

Is passive observation really impossible in the world of quantum mechanics?

David Albert, together with Barry Loewer inventor of a version of the Many-Worlds Interpretation referred to as Many-Minds (different from the one I suggest), tells us that the physical process of making an observation in the quantum world necessarily interferes with what is being observed. In other words, the ideal of fully passive observation of classical mechanics, cannot be upheld in quantum mechanics. The observer will always interfere more or less with what is being observed. Albert tells us that this is the big difference between classical and quantum mechanics.

But is this true? Is fully passive observation impossible in quantum mechanics? Maybe, or maybe not, depending on what is meant by an observation. A human being can make observations in different forms:

1. Inspection of an analog physical apparatus capabable of measuring some phenomenon.
2. Inspection of digital simulation of the phenomen.

Here 2. represents a digital simulation based on solving the Schrödinger equation describing the phenomenon, e g the ground state of an atom, and observing its energy, while 1. would be to directly observe the emission spectrum.

Th nice thing about 2. is that it is a completely passive observation, in the sense that the computational process is independent of the observer making the final observation of the energy as a number coming out of the computation.

So maybe passive observation is possible in quantum mechanics. Maybe quantum mechanics is not so different from classical mechanics. Not so mysterious?

Monstrosity of Quantum Mechanics 4: Quantum Computers

The belief of the modern physicist that the linear multi-dimensional Schrödinger equation

describes the quantum world of atoms and molecules, has led to the idea of the quantum computer:

• device for computation that makes direct use of quantum mechanical phenomena, such as superpositionand entanglement, to perform operations on data.
• Experiments have been carried out in which quantum computational operations were executed on a very small number of qubits (quantum bits).
• Government and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis.

I have noticed in previous posts that the linear multi-dimensional Schrödinger equation is a monster, which cannot be solved, not even on any thinkable supercomputer with any thinkable known microprocessor technique. The dimensionality is simply overwhelming.

We have noticed that the impossibility of solving the multi-dimensional Schrödinger equation

results from the fact the equation describes all possibilities rather than specific actualities, which is overwhelming for microprocessors limited to performing computations on specific data.

The Schrödinger equation is thus a monster computationally, and to handle such a beast a monster computer is needed, a computer which computes all possibilities rather than specific actualities, which computes on all data rather than on specific data: In other words a quantum computer is needed.

Are there any quantum computers? No, only with a few quantum bits.

Is it possible to construct a quantum computer? Nobody knows. Few seem to believe one can.

Does the multi-dimensional Schrödinger equation give a realistic description of the atomic world? Nobody knows because solutions cannot be computed and compared to experimental observation.

Can you solve a monster equation on a monster computer, that is a device which simulates a real analog monster by being a real digital monster?

What if the multi-dimensional Schrödinger equation is just an invented fictional monster, which will disappear as soon you stop talking about it?

Compare with the post today on The Reference Frame singing praise to the Copenhagen Interpretation of the multidimensional Schrödinger equation, as if it has a meaning. Read yourself and ask if you understand anything.

Monstrosity of Quantum Mechanics 3: Many-Worlds

The monstrosity of quantum mechanics is expressed in full bloom in Everett's Many-Worlds interpretation reflecting that solutions of the linear multi-dimensional Schrödinger equation can freely be superimposed.

The Schrödinger cat in its closed box thus can be in a state of superposition of both alive and dead and only upon opening the box for observation does the cat have to collapse into either alive or dead, as if there were two possible parallel universa prior to collapse into one actual universe.

The solution of the linear multi-dimensional Schrödinger equation thus is interpreted as a universal wave-function supposedly representing all possible universa, out of which a specific actual universe is singled out in one way or the other.

How to react to this breath-taking ocean of possibilities? In this case there seems to be two possibilities:

1. Accept the linear multi-dimensional Schrödinger equation as given by God.
2. Replace the linear multi-dimensional Schrödinger as a basic model of quantum mechanics with something more reasonable.

I would vote for 2. and I explore one possibility in Many-Minds Quantum Mechanics.

After all, it was Schrödinger and not God who wrote down the equation. It was Schrödinger who understood that his equation had serious flaws and should be replaced by a version describing actualities instead of possibilities.

What do you say? 1 or 2? One actuality or all possibilities? Would you prefer all possible lives

before one actual life.

Compare with the title of the biography: A Life of Erwin Schrödinger.

Nobody would be able to write a biography with the title All Possible Lives of Erwin Schrödinger, and even if somebody could, nobody would be interested in reading it.