The Secret of the Piano 2

                                   Damping of two out of three strings when tuning a piano tone.

We return to the model of The Secret of String Instruments 1 with $N$ strings connected to a common soundboard by a common bridge: For $n=1,..,N,$ and $t\gt 0$

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

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 small damping coefficient $D$ representing outgoing sound, $u=\frac{1}{N}\sum_nu_n$  and the right hand side represents the connection between strings and soundboard through the bridge as a spring with spring constant $B$ (with $B\le F$ say). 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 are interested in the phase shift between $u_n$ and $U$ in the two basic cases: (i) zero phase shift with strings and soundboard moving together in "unison" mode and (ii) half period phase shift with strings and soundboard moving in opposition in "breathing" mode.

We have by summing over $n$ in (1), concentrating on the interaction between strings and soundboard thus omitting here the damping from outgoing sound setting $D=0$:

• $\ddot u + F^2u+\frac{1}{N}\sum_n(f_n^2-F^2)u_n=B(U-u)$      (3) 

Introducing $\phi = U+u$ and $\psi =U-u$ representing the two basic modes, we have by summing and subtracting (2) and (3):

• $\ddot \phi + F^2\phi = -\frac{1}{N}\sum_n(f_n^2-F^2)u_n\approx 0$
• $\ddot \psi + (F^2+2B)\psi =\frac{1}{N}\sum_n(f_n^2-F^2)u_n\approx 0$          

with $\phi\approx 2U$ and $\psi\approx 0$ in case (i), and $\phi\approx 0$ and $\psi\approx 2U$ in case (ii), and $F^2+B\approx F^2$.

The difference between the two cases comes out in (1): In  case (i) the average of $B(U-u_n)$ is small while in case (ii) the average of $B(U-u_n)\approx 2BU$.  The right hand side $B(U-u_n)$ in (1) therefore acts to keep the different $u_n$ in-phase in case (ii), but does not exercise this stabilising effect in case (i), nor in the case of only one string.

The result is that the "breathing" mode of case (ii) can sustain a long aftersound with a sustained energy transfer from strings to soundboard until the strings and soundboard come to rest together.

On the other hand, in case (i) the strings will without the stabilising effect quickly go out of phase with the result that the energy transfer to the soundboard ceases and the outgoing sound dies while the strings are still oscillating, thus giving short aftersound.

You can follow these scenarios in case (i) here and in case (ii) here. We see 10 oscillating strings in blue and a common soundboard in red with strings in yellow,  and staples showing string energy in blue and soundboard energy in red. We see strings and soundboard fading together in case (ii) with long aftersound, and soundboard fading before the strings in case (i) with short aftersound.

We saw in The Secret of the Piano 1 that the hammer initialises case (ii) and we have thus now uncovered the reason that there are 2-3 strings nearly equally tuned for most tones/keys of the piano: long aftersound with strings and soundboard fading slowly together.

More precisely, initialising the soundboard from rest by force interaction through the bridge with already initialised string oscillation, will in start-up have the soundboard lagging one quarter of period after the strings with corresponding quick energy transfer, and the phase shift will then tend to increase because the soundboard is dragged by the damping until the "breathing" mode with half period phase shift of case (ii) is reached with slower energy transfer and long aftersound. The "unison" mode with a full period (zero) phase shift will thus not be reached.

The analysis of the interaction string-soundboard may have relevance also for radiative interaction between different bodies as exposed on Computational Blackbody Radiation by suggesting an answer to the following question which has long puzzled me:

• What coordinates the atomic oscillations underlying the radiation of a radiating body? 

Humanity Saved by Empty COP21 Treaty

The empty treaty (agreement) of COP21 without measurable reductions of CO2 emissions, saves humanity from the threatening disaster of a treaty asking for drastic reductions of CO2 emissions all the way to zero by 2050.

No wonder that the world leaders behind this monumental rescue operation raise there hands under massive ovations from all the people around the world, who can now go back to business.

One thing though: The treaty limits global warming to plus 1.5 C, but says nothing about global cooling. Hopefully that can be fixed in a revision: A limit to minus 0.5 C of global cooling would prevent another Ice Age, and stable global temperatures (and inflation of 2%) has shown to be the right climate for global business (and global happiness).

Another thing: The $100 billion dollars per year of transfer from rich to poor hinted at in the treaty, represents about 0.2% of the world GNP or 1% of China's GNP and thus like the warming of CO2 in the atmosphere, has a non-measurable empty effect.

Stunning Complete U-Turn at COP21

BBC reports under the title: COP21: Final push for climate deal amid 'optimism':

Negotiators at the Paris summit aim to wrap up a global agreement to curb climate change on Saturday - a day later than expected.

"We are nearly there. I'm optimistic," said French Foreign Minister Laurent Fabius, who is chairing the summit.

Efforts to forge a deal faltered on Friday, forcing the talks to over-run.

UN Secretary General Ban Ki-moon said the negotiations were "most complicated, most difficult, but, most important for humanity". 

Mr Fabius told reporters in Paris that he would present a new version of the draft text on Saturday morning at 0800 GMT, which he was "sure" would be approved and "a big step forward for humanity as a whole". 

"We are almost at the end of the road and I am optimistic," he added.

The new version to be presented on Saturday morning by Mr Fabius, firmly determined to lead the 195 countries on Mother Earth "to do something" to save Mother Earth from humanity, has however leaked and shows a stunning complete U-turn, apparently resulting from a sudden insight in the brain of Mr Fabius that CO2 is beneficial to both Mother Earth and humanity and that it would be completely insane to lead the world into a fossil free Hell by 2050:

Summary Statement

Over thirty years of intense (and extremely expensive) research has totally failed to produce any evidence that human emissions of CO2 are driving climate. CO2 is not a danger to but a benefit for all life on our planet.

We call on governments, NGOs and universities to stop pursuing policy and dogma based ‘evidence’ gathering.

• That they stop scaremongering.

•  That they dissolve the IPCC and the UNFCCC.

•  That governments focus instead on encouraging means of ensuring that under-developed and developing nations have full access to the cheapest reliable energy (particularly electricity), regardless of whether fossil fuels are used, so as to improve their access to clean water, low pollution cooking facilities and good medical services.

• That once respected academic institutions and scientific publications put their own houses in order and once again allow the free exchange of scientific ideas and results without prejudice.

• That those involved in alleged cases of scientific fraud, which have resulted in huge financial costs, causing greater poverty and many deaths among the poorest, be brought before the relevant Court of Law.
10th Dec 2015
Further, more detailed statements, references and videos of presentations at the Conference are available on the website: www.pcc15.org

PS Der Spiegel reports that the U-turn so to speak has turned everything upside-down into a complete mess:

• China, India, USA, EU and the oil producing nations downright crashed against each other with full force in the negotiations. 

The Secret of the Piano 1

We continue our study of the interaction of string and soundboard through a bridge of a piano with focus on the initialisation where the string is hit by the hammer and through the bridge transfers energy to the soundboard.

We model the string-soundboard-bridge system by the following coupled wave equations: Find functions $u(x,t)$ and $U(x,t)$ representing displacements of string and soundboard from initial flat configuration, such that for $0\lt x\lt 1$ and $t\gt 0$

• $\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0$, 
• $\frac{\partial^2U}{\partial t^2}-\frac{\partial^2U}{\partial x^2}=0$,

combined with the following boundary conditions for $t>0$

• $u(0,t)=U(0,t)=0$,
• $\frac{\partial u}{\partial x}(1,t)=-\frac{\partial U}{\partial x}(1,t)=S(u(1,t)-U(1,t))$,

where $S$ is a spring constant representing a springy connection of string and soundboard through a bridge located at $x=1$.

We initialise by setting $u(x,0)=U(x,0)=0$, $\frac{\partial u}{\partial t}(x,0)=1$ for $0.4\lt x\lt 0.6$ and $\frac{\partial u}{\partial t}(x,0)=0$ else, and $\frac{\partial U}{\partial t}(x,0)=0$ for $0\lt x\lt 1$, corresponding to hitting the string with the hammer, and watch the result here with the string red and soundboard blue.

We compare with initial data for $u$ changed to $u(x,0)=\sin(\pi x)$ and $\frac{\partial u}{\partial t}(x,0)=0$ with somewhat different response here.

We see that the motion settles into periodic modes of string and soundboard with a phase difference of half a period with the bridge basically at rest: when the string deflects upward the soundboard deflects downward and vice versa with zero net force on the bridge. 

We have thus recovered the "breathing" motion with the bridge at rest of the previous post as the basic resonance mechanism of a piano allowing long aftersound with slow transfer of energy from string to soundboard. 

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.

Digitaliseringskommissionens Kraftlösa Slutrapport

Digitaliseringskommissionen lämnade igår sin slutrapport Digitaliseringens Transformerande Kraft med följande huvudbudskap:

1. Sverige ska vara bäst i världen på att använda digitaliseringens möjligheter. 
2. För att individer ska kunna arbeta på en digitaliserad arbets-marknad som kontinuerligt kommer kräva nya kunskaper och förmågor ställs nya krav på utbildningssystemen och formerna för fortbildning och kompetensutveckling.
3. Digitaliseringens möjligheter att skapa ett hållbart samhälle behöver tas tillvara.
4. Sveriges har en stark position i internationella jämförelser av länders it-användning och digitalisering.
5. Det offentliga måste vara proaktivt för att förverkliga digitaliseringens möjligheter. 
6. Som ledare är det angeläget att vara medveten och ha kompetens om på vilka sätt den egna verksamheten kan digitaliseras eller kommer att påverkas av andras digitalisering.
7. Digitaliseringskommissionen anser att den transformerande kraft som digitaliseringen innebär för samhället gör att det finns ett behov av att staten genom ett kontinuerligt och långsiktigt statligt engagemang utvecklar mjuka former av styrning för att främja digitaliseringen i samhället. 
8. Livslångt lärande, utbildning, kompetensutveckling och omskolning blir allt viktigare för individer, företag och samhälle. Som samhälle behöver vi också kompetens om och hur digitaliseringen kan användas för verksamhets utveckling, värdeskapande och innovationer för att ge fortsatt välstånd och tillväxt. 
9. För att kunna dra nytta av hela kompetensen i samhället behöver könsbalansen inom it-området bli jämnare.
10. Regeringen bör säkerställa en kontinuitet i nationellt främjande stöd till digitalisering. 
11. Regeringen bör tillsätta en utredning i syfte att göra en kartläggning av digitaliseringsförsvårande lagstiftning. 
12. Regeringen bör tillsätta en utredning som ser över om arbetsrätten och konsumentlagstiftningen behöver anpassas till nyttjande- och delningsekonomin. 
13. Ett incitamentsprogram bör inrättas för att öka könsbalansen på högre it-utbildningar. Studerande med underrepresenterat kön, mindre än 15 procent, som påbörjar och fullgör högre it-utbildningar får sina studiemedel för sex terminer, motsvarande kandidatnivå, avskrivna efter erlagd examen längst till dess att en köns-balans på 30/70 uppnås. 
14. Regeringen bör i överenskommelse med Sveriges kommuner och landsting (SKL) genomföra ett digitalt kompetenslyft för ledare i kommunal verksamhet. 
15. Regeringen bör inrätta ett samverkansråd som utvecklar it-utbildningar på högskolenivå. 

Sammantaget en soppa av Guds nåde: Hållbarhet, könsbalans, ledarskap, samverkansråd, statlig styrning, utredning...

Men ingenting om skolans roll för att förbereda dagens elever för dagens IT-samhälle. Ingenting om den enastående resurs som skolan utgör för detta ändamål. Ingenting om Regeringens uppdrag till Skolverket att införa programmering i grundskolan.  Ingenting som kopplar till Matematik-IT. 

Bara omkväden att Sverige är bäst i världen.

Det enda konkreta som framförs är att kvinnor skall få gratis studiemedel för it-studier. Ett barockt förslag utan reson och legalitet.