torsdag 17 december 2015

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? 

12 kommentarer:

  1. "What coordinates the atomic oscillations underlying the radiation of a radiating body?"

    Why coordination? Thermal radiation isn't especially coordinated.

  2. I think some coordination is needed to give an output: with all phases present to the same degree, the net is zero, right?

  3. No, the net is not zero. I can not do the full derivation here, of course, but from the standard treatise of superposing N waves the resulting intensity is

    E_0^2 = \sum_i^N E_{0i}^2 + 2 \sum_{j>i}^N\sum_{i=1}^N E_{0i}E_{0j} \cos(\alpha_j - \alpha_i),

    where \alpha_i is the phase. When the phases are uncorrelated, that is present to the same degree, the double sum vanish since there are as many positive as negative contributions and we are left with

    E_0^2 = \sum_i^N E_{0i}^2.

    The sum of squares of the individual intensities adds upp to the square of the resulting amplitude.

    So I don't see how there can be a puzzle.

  4. Well, I guess it comes down to what "uncorrelated" means. It is clear that full cancellation is possible, and one thing is then to explain why this case does not appear, or does indeed appear.I will return to this...

  5. With uniformly distributed phases and equal amplitudes the energy of the superposition/sum of waves is much much smaller than the sum of the energy of the individual waves, as a result of massive cancellation. The question I seek to answer is if there is a mechanism coordinating phases so that this case does not arise in reality.

  6. "With uniformly distributed phases and equal amplitudes the energy of the superposition/sum of waves is much much smaller than the sum of the energy of the individual waves, as a result of massive cancellation."

    But if there are uniformly distributed phases and equal amplitude the total energy is just proportional to N*E_{01}^2 if there are N added waves with equal amplitudes E_{01}. The energy just adds the contributing parts.

    The cosine will take all values between -1 and 1. Since the amplitudes are the same the double sum vanishes and the result is just the addition of all amplitudes squared.

    Do you disagree?

    See first random set of slides from a google search showing the expressions.
    Especially slides 4 to 8.

  7. The difference between in-phase and random-phase is a factor N which can be very large. Maybe this says that coherent laser light is much more energetic than incoherent light, rather than saying that incoherent light does not transfer any energy. But then there is a whole range of light with different coherence and energy transfer capability. Right?

  8. What are you talking about?

    You wrote

    "I think some coordination is needed to give an output: with all phases present to the same degree, the net is zero, right?"

    So you seem to have thought that random phases evens out and don't transfer any energy. That was the thing I objected to.

    But you seems to agree now. There is nothing strange with the fact that a solid with thermal energy radiates.

  9. I understand what you are saying. But if you think that you understand the interaction matter-light as something trivial without anything strange whatsoever, then you may be missing something essential, and that is what I am searching to understand.

  10. No I agree.

    Light-matter interactions in general are very complicated under some settings. Thats the reason why there are so many active fields doing research on the matter. But in the case of "ordinary" transfer where the statistical limits holds I don't think it's so interesting.

    There are many, many interesting and cool things to investigate. Take near-field radiation transfer for instance where one can get a blackbody to transfer many order of magnitudes above the Planck-limit that you get from Plancks law. How cool isn't that?!

  11. Hi Claes should have wished you a happy Christmas but late so wish you a happy and healthy New Year.
    Something interesting for you. I was pointed to an article on the Catt question in Progress in Physics by Stephen Crothers See here (the article by Weng -a classical model of the photon is also interesting)I wondered about the Catt question and found Ivor Catt's articles on his site here . His questioning of basic electrical theory made me think of you with your questioning of fluid dynamic theory (and of course heat AGW hypothesis) Something about Ivor Catt can be found here on Wiki

  12. Interesting! Maybe there is a coupling with my idea that "transmission" of light or EM is not performed by busy little photons traveling from source to goal, but rather is a resonance phenomenon carried by standing EM waves capable of two-way propagation of disturbances but with the effect of one-way transfer of heat from hot to cold, for example.