onsdag 29 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.

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