Secret of Laser vs Secret of Piano

There is a connection between the action of a piano as presented in the sequence of posts The Secret of the Piano  and a laser (Light Amplification by Stimulated Emission of Radiation), which is remarkable as an expression of a fundamental resonance phenomenon.

To see the connection we start with the following quote from Principles of Lasers by Orazio Svelto:

• There is a fundamental difference between spontaneous and stimulated emission processes. 
• In the case of spontaneous emission, the atoms emit e.m waves that has no definite phase relation with that emitted by another atom... 
• In the case of stimulated emission, since the process is forced by the incident e.m. wave, the emission of any atom adds in phase to that of the incoming wave...

A laser hus emits coherent light as electromagnetic waves all in-phase, and thereby can transmit intense energy over distance. 

The question is how the emission/radiation can be coordinated so that the e.m. waves from many/all atoms are kept in-phase. Without coordination the emission will become more or less out-of-phase resulting in weak radiation. 

The Secret of the Piano reveals that the emission from the three strings for each note in the middle register, which may have a frequency spread of about half a Herz, are kept in phase by interacting with a common soundboard through a common bridge in a "breathing mode" with the soundboard/bridge vibrating with half a period phase lag with respect to the strings. The breathing mode is initiated when the hammer feeds energy into the strings by a hard hit.

In the breathing mode strings and soundboard act together to generate an outgoing sound from the soundboard fed by energy from the strings, which has a long sustain/duration in time, as the miracle of the piano. 

If we translate the experience from the piano to the laser, we understand that laser emission/radiation is (probably) kept in phase by interaction with a stabilising half a period out-of-phase forcing corresponding to the soundboard, while leaving part of the emission to strong in-phase action on a target.

An alternative to quick hammer initiation is in-phase forcing over time, which requires a switch from input to output by half a period shift of the forcing. 

We are also led to the idea that black body radiation, which is partially coherent, is kept in phase by interaction with a receiver/soundboard. Without receiver/soundboard there will be no radiation. It is thus meaningless to speak about black body radiation into some vacuous nothingness, which is often done based on a fiction of "photon" particles being spitted out from a body even without receiver, as physically meaningless as speaking into the desert.    

What Makes People March in Step?

The Secret of the Piano exhibits a basic resonance phenomenon where the vibrations of a collection of strings are kept in phase by all having a phase shift of half a period vs a soundboard in a "breathing" interaction between strings and soundboard with the strings moving "in" when the soundboard goes "out" and vice versa. The collection of strings and the soundboard thus take "opposite positions".

This phenomenon may also be seen in the interaction between human beings in a society, where a collection of people "march in step" e.g by sharing certain political views, and one may ask how this coordination comes about:

1.  Is it the idea of a leader that is adopted and followed?
2.  Is it a common "enemy" as the opposite of a leader, which unites people? 

The lesson from the piano suggests 2, the reason being that in this case there is massive energy exchange between strings and soundboard back-and-forth allowing coordination to occur. On the other hand, when strings and soundboard vibrate in phase and then without energy exchange, there is no correction mechanism and strings and soundboard will quickly go out-of-phase. 

Similarly, one may expect that "negative leadership" may be more effective than "positive leadership", because taking opposite views involves massive exchange of energy thus allowing correction to occur, while agreeing can be effortless without corrective measure. 

This observation may connect to my experience of many times ending up with the opposite view of a group of people all marching in step and with a lot of energy mobilized on both parts.      

The Secret of String Instruments (vs Planck's Radiation Law) 1

((This post is updated to a more correct analysis in The Secret of the Piano 2)

The new proof of Planck's radiation law offered by Computational Blackbody Radiation also reveals the secret of string instruments composed of:

• one or several strings for a given tone
• soundboard
• bridge connecting strings with soundboard.

The secret is hidden in the following dynamic wave model representing an instrument composed of $N$ strings connected to a common soundboard by a common bridge: For $n=1,..,N,$ and $t>0$

1. $\ddot u_n + f_n^2u_n=B(U-u_n)$ 
2. $\ddot U + F^2U+D\dot U=B(u_n-U)$

where $u_n=u_n(t)$ is the displacement of string $n$ of eigen-frequency $f_n$ at time $t$ and the dot represents time differentiation, $U$ is the displacement of the soundboard with eigen-frequency $F$ and damping coefficient $D$ representing outgoing sound, and the right hand side represents the connection between strings and soundboard through the bridge as a spring with spring constant $B$. We consider a case of near-resonance with $f_n\approx F$ for $n=1,...,N$, with a difference of about 1 Hz in a basic case with $F=440$ Hz say.

We can think of this model as composed of $N+1$ masses each connected to a fixed support by elastic springs ($N$ strings and 1 common soundboard ) joined by elastic springs connecting each string to the common soundboard/bridge through an elastic spring.

Recall that for a piano up to three strings are used for each single tone. 

The performance of the instruments is expressed by the following energy balance obtained by multiplying 1. by $\dot u_n$ and 2. by $\dot U$:

• $\dot E=-D\dot U^2$
•  $E=\frac{1}{2}\sum_n(\dot u_n^2+f_n^2u_n^2+\dot U^2+F^2U^2+B(u_n-U)^2)$,  

where $E=E(t)$ is the total energy of the instrument at time $t$ as the sum of the string energy, soundboard energy and "bridge energy" $\frac{1}{2}\sum_n(u_n-U)^2$.

A tone is initialised by setting the strings in motion by plucking (guitar), by bow (violin) or hammer (piano) and we now focus on the interaction of the strings and soundboard after initialisation as a sound is generated from the vibration of the soundboard into the surrounding air. In a subsequent post we  will consider the initialisation with near-resonance as one key to the secret.

The key to the secret of the sound production is revealed by the following observation:

• The displacements of strings and displacement of soundboard is maintained with a phase shift of one half period through interaction via the common bridge, although the eigen-frequenices of the strings are not exactly equal to the eigen-frequency of the soundboard.  
• In other words, the strings and soundboard vibrate in coordinated motion with maximal mutual displacement $(U-u_n) with strings moving up/down when soundboard is moving down/up in a "pumping motion" and thus with substantial bridge energy. 
• In the real case of a guitar, violin or piano, the pumping motion with substantial force exchange between string and soundboard, is reflected by zero motion of the bridge with string and sound board pulling in opposite directions.  

The secret of the sound production of the instrument is hidden in the following question:

• What sustains sound production by coordinated string-soundboard motion with all strings with a half-period phase shift with a string-soundboard eigenfrequency difference of 1 Hz?   

The answer comes out by subtracting 1. and 2. to get for $w_n=U-u_n$ for $n=1,...,N$

• $\ddot w_n+\tilde F^2w_n\approx 0$, 

where $\tilde F^2\approx F^2+2B\approx F^2$ if $B\le F$. The difference $U-u_n$ thus comes out as the same eigen-function for all $n$ with the phase shift of all strings coordinated to a common half-period phase shift vs the soundboard.

On the other hand, adding 1. and 2. gives for v_n=U+v_n

• $\ddot v_n+F^2v_n\approx 0$,

as an eigen-function of frequency $F$ with $F^2<\tilde F^2$, representing motion with $u_n$ in-phase with $U$.

It then remains to explain why the mode $w_n$ with half-period phase shift and substantial bridge force is preferred by the instrument before the mode $v_n$ with a full period (or zero) phase shift and zero bridge force. I will return to this question in the next post starting with a study of the initialisation dynamics.

The model tells in the half period phase shift case that the sound dies quickly as soon as the strings are damped, because that means that both the string energy and the bridge energy is put to zero leaving only a the minor portion of soundboard energy for continued sound production.

New Proof of Planck's Radiation Law vs (Italian) Piano Tuning

Computation Blackbody Radiation presentats a new approach to Planck's radiation law based on finite precision computation applied to a wave model consisting a set of harmonic oscillators with small damping subject to near-resonant forcing, each one of the oscillators of the form

• $\ddot u(t)+\nu^2u(t)+\gamma\dot u(t)=f(t)$ for time $t>0$,  

where $\dot u=\frac{du}{dt}$ and $\ddot u=\frac{d^2u}{dt^2}$, $\nu >> 1$ is the eigenfrequency of the oscillator, $\gamma $ is a small positive damping coefficient with $\gamma\nu\le 1$, and $f(t)$ is a near-resonant forcing, for example given by

• $f(t)=\sin((\nu -0.5)t)+\sin(\nu t)+\sin((\nu +0.5)t)$

with a total frequency shift of 1 Hz.

A basic aspect of this model connects to the so called Italian tuning of a piano, where the three strings in the middle high register for each key/tone are tuned with a total frequency shift of about 1 Hz as above. 

Let us now seek to understand in what sense Italian tuning is different from standard tuning with the three strings for each key/tone tuned to exactly the same frequency or pitch. We then identify in the above model 

• the three strings are represented by the forcing $f(t)$,
• the sound board of the piano is represented by the oscillator,
• the outgoing sound from the sound board is represented by the damping.  

Let us first consider standard tuning with strings and soundboard all with the same eigen-frequency, that is a case with perfect resonance. A pressed key activates a hammer with hits the strings which start to vibrate and thereby as forcing transfers energy to the sound board, which in turn starts to vibrate and produces a sound. In this case the forcing $f(t)$ will stay in phase with the velocity $\dot u$ over time, which means quick transfer of energy from strings to sound board as the integral of the positive product $f(t)\dot u(t)$ as the work performed per unit time. The result may be an outgoing sound of relatively high volume but short duration.

On the other hand, with the Italian tuning, the forcing from each of the three strings with slightly different frequencies cannot all be in phase with the common sound board velocity $\dot u$ over time, which means less quick energy transfer to the sound board with $f(t)\dot u(t)$ of changing sign and thus slower string energy loss as compared to the standard case. The result may be a sound of less volume but longer duration (sustain) than in the standard case, and also with slight "beat".

Of course, in reality it may be difficult to clearly separate the two cases, because perfect resonance does not really occur for a real piano with standard tuning, because of the complexity of the sound board, and also because the damper stops the string vibration before the tone has faded.

In any case, the distinction between perfect and near resonance is fundamental in the new proof of Planck's radiation law offered as by Computation Blackbody Radiation, a proof without reference to mystical statistics.

PS It is possible to change the setting by letting $f(t)$ represent the outgoing sound and the damping the input from the strings to the sound board. This is of relevance in stationary periodic state with sustained sound over long time without damping.

Blackbody Radiation as a Generic Emergent Phenomenon

A body of temperature T emits radiation with an intensity E similar to that of an ideal blackbody given by Planck's law (Rayleigh-Jeans law with cut-off):

• E = gamma T f^2,
• with a high frequency cut-off proportional to T,

where f is the frequency and gamma is a constant.

The radiation spectrum thus only depends on the temperature and not of the material of the body. This indicates that blackbody radiation is a generic emergent phenomenon resulting from collective atomic vibrations and not from individual atoms or molecules which have different line spectra.

This idea is explored in the upcoming book Mathematical Physics of Blackbody Radiation (and in one of my chapters in Slaying the Sky Dragon) by an analysis of resonance in a wave equation with radiation.

In this model blackbody radiation can be thought of as the sound of a piano with all keys being struck at the same time with the same force: a complex chord which sounds the same for all pianos. An generic emergent phenomenon which cannot be understood by looking at just one key.

It can also be thought of as the complex sound of a big gong with a big range of frequencies. There is an interesting experiment showing that a gong can be made to sound by a short laser pulse kicking the gong atoms into an emergent collective vibrational motion producing sound waves hitting your ear.

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.

Mathematics of Blackbody Radiation, Without Backradiation

Blackbody in the form of a grand piano as a system of vibrating strings of different eigen-frequencies, including a "cut-off" damping mechanism transforming high overtones into heat.

Computational Blackbody Radiation presents an analysis of the following model of blackbody radiation:

• U_tt - U_xx - R U_ttt - D^2 U_xxt = F
  
• E_t = int (F^2 - RU_tt^2) dx

where U(x,t) is the amplitude of a vibrating string at position x at time t, with the subindices indicating differentiation. The string is subject to radiative forcing of intensity F^2 (incoming waves) and responds by emitting radiation (outgoing waves) with intensity RU_tt^2. The difference between incoming and outgoing radiation acts as a heat source to the internal energy

• E = vibrational energy + heat
• vibrational energy = int (U_t^2 + U_x^2) dx.

Finally, - D^2U_xxt represents a dissipative mechanism grinding waves of high frequency into heat. The coefficient D = H/T with H a (small) spatial mesh scale, represents a cut-off length or smallest coordination length, which is the smallest wave length which can be carried by the vibrating string as coordinated wave motion. Further E ~T^2 with T temperature.

As discussed in Computational Blackbody Radiation the dissipation can take different forms with more or less sharp cut-off. We here consider the simplest case including essentials.

The cut-off length D decreases with increasing temperature T: The hotter the more fine details can be represented and emitted. The colder, the "dumber" is the string.

There is a corresponding cut-off frequency T/H conforming with Wien's Displacement Law.

We assume that R is small expressing that the emission is a small perturbation on top of carrier wave of temperature T ~ 1/R. Incoming waves trigger resonances in the system from which waves are re-emitted. The effect is that an incoming blackbody spectrum below cut-off

can be fully absorbed and re-emitted as the same blackbody spectrum.

A blackbody spectrum is characterized by equal temperature of all frequencies (below cut-off).

The model can be seen as a collection of (atomic) resonators with a range of frequencies from small to medium to large, the motion of which is sustained by incoming waves (with blackbody spectrum). The resonators absorb an incoming blackbody spectrum and react by

• re-emitting a blackbody spectrum below cut-off
• transforming incoming spectrum above cut-off into heat (as part of internal energy).
  

The transformation of high frequency input comes from an inability of the system to correctly absorb certain input, because the required coordination length is too small for the

available precision, and the system therefore distorts incoming high-frequency waves into incoherent high-frequency motion kept as internal (heat) energy, which is not radiated. The heating is similar to blushing from an inability to properly respond to a (sharp/nasty) remark.

The model shows two basic features of blackbody radiation:

• low incoming frequencies are re-emitted without causing heating.
  
• high incoming frequencies are transformed into heat (=incoherent high frequencies).

The net result is that a warm blackbody can heat a colder blackbody, through incoming frequencies above cut-off. But a cold blackbody cannot heat a warmer, because incoming frequencies below cut-off will be re-emitted without heating effect.

Since "backradiation" refers to the latter case, the model indicates that "backradiation" is not physical.

The question is now to what extent the model captures real physics? Is a blackbody an analog computer performing some form of analog computation of finite precision, when absorbing and emitting radiation acting as a system of resonators with finite coordination length?

It is like a system of crickets able to emit variable pitch sound with the top pitch

increasing with "temperature" or "excitation level". Or the frequency of "the wave" in the stadium increasing as the excitation of the public increases. Or like politicians delivering increasingly high pitch coordinated messages as the campaign temperature increases towards election.

Note that the above model is deterministic with the quanta statistics of Planck's classical model (created in an "act of despair" from an apparent collapse of classical mechanics), being replaced by finite precision computation (as an "act of resurrection of the hope" of classical mechanics).