onsdagen den 29:e juni 2011

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.

tisdagen den 28:e juni 2011

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)?

torsdagen den 23:e juni 2011

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.

onsdagen den 15:e juni 2011

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.

måndagen den 13:e juni 2011

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
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.