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.

String Instrument as Model of Blackbody Radiation

A string instrument like a guitar or piano offers a conceptual model of blackbody radiation which can help to remove the mystery surrounding this phenomenon. The sound of a string instrument is generated by plucking strings in contact through bridges with a soundboard which generates sound waves in the surrounding air.  The basic mathematical model takes the form:

• wave equation for soundboard + acoustic damping force = string force,

where the acoustic damping force models the sound force output from the instrument and the string force is the force on the soundboard transmitted from a plucked string through bridges. 

The analysis presented on Computational Blackbody Radiation shows the following fundamental relation as a consequence of resonance between sound board and string: 

• output sound energy = string energy 

which is to be compared with a case of non-resonance:

• output sound energy < < string energy.

We see that a soundboard in resonance with a string transmits the full string plucking energy into output sound energy, while in the case on non-resonance only a small fraction is transmitted, see PS below for some more details.

In blackbody radiation this phenomenon comes out as high emissivity in the case of resonance and low emissivity in the case of non-resonance. 

For example, CO2 has a main resonance at wave number 667, which gives high emissivity for wave numbers close to 667 independent of concentration, but low emissivity away from 667.

CO2 alarmism is based on high emissivity of atmospheric CO2 in the whole wave number band 600 - 800, which however most likely is an incorrect assumption.

PS The analysis on Computational Blackbody Radiation exhibits a phenomenon of near-resonance under small acoustic damping with the string force being in-phase with the soundboard displacement (and thus out-of-phase with the soundboard velocity), as the key to a good instrument with string and soundboard working together to produce a good sound.   

Near Resonance with Small Damping

In my article Computational Blackbody Radiation in the Sky Dragon book I derived the Rayleigh-Jeans and Planck Laws of Radiation from a wave equation model with small damping, as a consequence of a phenomenon of near-resonance in a resonator with small damping.

This phenomenon appears to also be fundamental in the acoustics of string instruments with strings representing the damping and the body or soundboard of the instrument the resonator.

The phenomenon can be studied in the most basic of all models of physics, the harmonic oscillator, subject to small damping, as I do in the new article Near-Resonance with Small Damping. The article is a good complement to the Sky Dragon article.

The key point is that in near-resonance the forcing is balanced only to a small part by the damping force, the main part being balanced by the resonator, which reflects that forcing and velocity are out-of-phase. In this case the resonator acts as an amplifier of the damping (the soundboard amplifies the sound of a string).

Near-resonace is fundamentally different from perfect resonance with forcing and velocity in-phase and the damping force balancing the forcing without amplification from the resonator.

(Near-resonance is of course also different from the case of no resonance).

The importance of near-reonance is well-known to a piano tuners who tunes the two or three strings of a tone (except the single deep bass string) at slightly different pitches to create a longer sustain and singing quality of the piano.

The analysis in the article exhibits the interaction of the vibrating string and the vibrating resonating body, with the string pumping energy into the body during a start-up phase and

then changing role to sustain output from the body, all the time with the string vibrating in-phase with the body with out-of-phase output from the body in equilibrium.

The model suggests that there is a principal similarity between the radiation spectrum of a radiating body and the acoustic spectrum of a multi-string instrument from repeated arpeggios over the strings (with a capo d'astro in high position so that fundamental low frequencies are not involved).

For radiation the spectrum scales with the frequency squared as the result of a damping related to accelleration, while the corresponding spectrum in acoustics is flat in frequency because the damping in this case relates to velocity.

Near-resonane amplification conforms with the experience that the resonating body of a string instrument functions over a wide spectrum of string frequencies.

Near-resonance connects to broad resonance with a frequency band of larger width than that of sharp resonance scaling with the damping.