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.