Hallo Chris, I agree with your point about spurious correlation.
I was once asked to analyse the socio-economic factors stunting the growth of children from deprived backgrounds around Manchester in England. Some doctors had sponsored a study and the data, collected at some expense, showed two distinct groups. They had not noticed Group A were all boys and Group B were all girls.
I hope I haven't fallen into the same trap with my interpretation of Harold Conklin's paper on the generation of partials due to non-linear mixing.
Chris, I do not believe I have misapplied the results of the ASA paper or that my spectrograms are conflict with them. As I see it all the overtones we hear result from the non-linear response of the bridge to the vibrations in the string. You have to consider the longitudinal wave component at all frequencies, not only for longitudinal modes.
To my mind you have a reasonably complete model of a piano string when you include both longitudinal and transverse waves. I am not so sure you need look much farther unless you want to understand the mechanisms which dissipate energy or, more importantly, generate sound.
Please say if you think this wrong or if there are still some unwarranted assertions. In the hope of avoiding any further misunderstandings I think I'd better go back to square 1 and run through the whole argument.
My "experiment" was to see if anything would happen if I reseated the strings of D7 at the pressure bar. To my (untrained) ear, the note had livened up considerably in comparison with its neighbours. This was after I had revived that note by reseating them at the v-bar. I then repeated the exercise on the other 58 notes under the pressure bar, with similar results.
The question was why this occurred. As Del said, and as Emmery observed, the v-bar and the pressure bar are very effective at terminating the transverse vibrations of the string. It follows that what I did could not have directly affected the transverse vibrations in the speaking length.
Conventional wisdom is that what happens in the speaking length determines the sound. This is true of course for most practical purposes associated with tuning and voicing. Wave diagrams and formulae are about the speaking length. The non-speaking lengths do not feature in the Five Lectures on the Acoustics of the Piano you mention.
Earlier in this thread rxd mentioned transference of power from the hammer to the string; it is that energy which is at issue. Once the hammer has left the string its energy can either be dissipated or turn into sound. When you enliven a note by seating its strings you increase its energy by reducing the amount of energy being dissipated at one or more of the bearing and end points.
This is where conventional wisdom comes into the argument. This appears to be that all waves are in the speaking length and virtually none of their energy transfers across the termination points. Therefore making adjustments at the pressure bar, tuning coil and hitch pin can have no effect.
The elephant in the room had better come out into the open now; all the vibrational energy in the string is in the longitudinal vibrations. These waves are periodic stretchings and compressions along the string itself. They result from the hammer hitting the string and the transverse movements of the string that follow. Rather than take my word for it, read this short article on Longitudinal Waves
quoted from "Physical Audio Signal Processing" by Julius O. Smith III.
By the way, longitudinal modes are only one aspect of longitudinal waves.
The mathematics associated with waves in taut strings are moderately complex and have been the subject of some debate over the years. However a paper published in 2011 may have resolved the matter, "The potential energy density in transverse string waves depends critically on longitudinal motion", David R Rowland 2011 Eur. J. Phys. 32 1475, see the abstract here
. I do not have a copy of this paper but the general conclusion makes sense. This is that [to derive] the correct formula for the potential energy density in transverse waves on a taut string ... the longitudinal motion of elements of the string needs to be taken into account, even though such motion can be neglected when deriving the linear transverse wave equation
In other words, when you are talking about the behaviour of the speaking length, i.e. the transverse wave equation, you do not need to worry about the longitudinal waves. On the other hand, you should when talking about the behaviour of the piano, especially the bridge and the soundboard, as Giordano and Korty's work showed.
The next question is whether the longitudinal waves are mainly reflected at (a) the v-bar and bridge pin or (b) the tuning pin and the hitch pin. It seem obvious to me that much of their energy must reach the tuning pin ...
It honestly depends. For a normal person, probably three months. For a professional pianist, maybe a month. For a concert at Carnegie Hall, probably until intermission.
... and the hitch pin but in this forum such it seems such a conjecture needs solid proof.
Giordano and Korty measured the longitudinal motion at the bridge and then Conklin went one step further and measured the force of the longitudinal waves at the hitch pin. His experiment shows that a significant amount of longitudinal wave energy, if not all, is travelling along the whole length of the string. This is key point from Conklin's paper for this thread.
Once one knows that there is wave energy in the non-speaking lengths one can see why adjusting strings at the tuning coil, hitch pin, pressure bar, and bridge pins might affect the tone. As might any other bearing points where energy can be lost.
The point that extra longitudinal wave energy enlivens a note by increasing the power of the higher partials originally came from Giordano and Korty. This was a general observation they made without mentioning phantom partials. Conklin says he spotted the phantom partials in Giordano and Korty's spectrograms and, in that sense, his work extends theirs.
You will see in the following diagrams that there is longitudinal wave energy in the partials as well as the phantom partials. In fact he goes on to discuss how the phantoms smear the partials, see Fig 18 in the paper. I am including the diagrams here because it will be easier to compare them one above the other than spread about in the pdf. Sound Transverse waves at the bridge Longitudinal waves at the bridge