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A while back there was a post on this subject but after reviewing it I realized it was way too mathy and technical. So, here is a slimmed down version that avoids any technical side trips:

https://youtu.be/beSQSpLwakk

Thanks Dave.

I am interested what you would predict if the bridge termination point were able to flex, move laterally or pivot in response to the sideways force of the string? Would this reduce the separation and hence inharmonicity, and would there be other consequences?

Hi Chris.

That is an interesting question that I had not thought of and, at least for the time being, beyond the capabilities of my model.

However, thanks for asking.

Dave

It is just that I have wondered if inharmonicity could be reduced by a termination point that gives in some way, and your models suggests to me that it could. If so then I suspect there may be loss of power but a model may shed some light on the possibilities.

Thanks Dave!
The only suggestion I have is changing the word use. "Bend" should be "flex". Bends are permanent deformation. Hammer strikes on piano wire rarely permanently deform the wire.

Interesting implications regarding generating false beats in the rapid spreading of the inharmonic forces at the bridge.

How true to life do you think your model is on this?

Hi Ed.

Thanks for the tip on terminology.

Regarding the relationship of the simulation results with reality, I think the math models are correct from the physics point of view but are also idealizations. My goal is insight, not replication of reality.

The simulation results tell me that pulses do travel back and forth along the string. The results also suggest that there is some sort of interaction between the string and the bridge which is going to depend on the degree of elasticity. Now, do the real pulses look like those in the simulation? I suspect that the real pulses are not as "clean" as those in the simulation simply because the model is ideal.

The acceleration that the simulation shows was unexpected and enigmatic when I first saw it a couple of years ago. The simulation suggests that it comes from the inversion and reflection. I have further suggested that there is an exchange of force between the string and the bridge. This last step may be a bit of a stretch.

Speculating beyond this is a bit risky.

However, it certainly is a pleasure to hear constructive comments like those from you and Chris.

Dave

The bridges do give. That is what is hypothesized to cause the Weinreich effect - the effective length of the string changes, the center of pivot moves forward or backward depending on the stiffness of the system.

I tried to listen all the way through but I couldn't grasp what the point was. Could you condense the information and post just the conclusion? I saw a double force but can't understand why there are two force lines.

Thanks, Dave. Interesting to see an alternative take on the standard model.

Does your simulation mean that there is an increase in the frequency of the wavelets of the stiff string, as energy is reflected at the bridge? If that's true, then iH could be measured to quickly build/increase if we had the ability to measure under 10ms increments?

Interested to hear how you think this dovetails with the existing model..

Thanks Dave. Very nicely done. Clear and concise explanations. It is interesting to add bridge motion to the existing restoring forces on a stiff string: tension and stiffness.

A Wordy Summary (probably not as condensed as you might want):

The simulation shows that the pulse resulting from a hammer strike moves back and forth along the string. When the pulse reaches a speaking length end point, either at the agraffe or at the bridge, it is inverted and reflected. Here is a snapshot of the pulse 3.29 ms after the hammer struck the flexible string. The bottom trace shows a short history of the force that the flexible string transmitted to the bridge.


The force of the string on the bridge (and hence the sound board) is proportional to the negative slope of the string (at the bridge) which changes greatly at the moments when inversion and reflection take place. This force on the bridge is transmitted directly to the soundboard thereby generating longitudinal vibrations in the air. The model assumes that the spectrum of the force on the soundboard is proportional to the spectrum of the sound waves generated by the soundboard.

(I have been working on simulating the combination of the hammer, string, bridge and soundboard for several months. It is a much tougher nut to crack but I have made some progress and the results so far support the simpler approach taken in this video…but I digress…)

If the string is flexible, forces are exchanged with the bridge BUT there is no effect of the bridge on the string when the reflection and inversion occurs (because the string is flexible and does not resist deformation). If the string is elastic and therefore resists deformation, there is a force transmitted from the string to the bridge and an equal and opposite force transmitted from the bridge to the string. This equal and opposite force of the bridge on the elastic string causes the pulse to accelerate such that the elastic string “springs ahead” of the flexible string. Because of the elastic string’s resistance to deformation, small ripples in the shape of the elastic string’s pulse start to emerge. These ripples change the slope of the string during reflection and inversion, causing the force transmitted to the bridge to take on a quirky shape rather than a relatively well behaved standing wave:


At the start of the video I showed a spectrum of a real C4 note from my Knabe grand where I pointed out the inharmonicity and the attenuation at the 9th partial.


At the end of the video I showed a spectrum of the force of the simulated C4 string on the bridge and suggested that the two spectra were similar in many ways.


This similarity supported (I think) my contention that inharmonicity arises from the reflection in and inversion of an elastic string.

Could we describe the wire stiffness as coupling the initial displacement further down the string because the ends of the initial pulse are producing a rocking effect ahead of the impulse?

Dave,

I haven't looked at the video yet, but I've read your word summary.

I'm a bit confused by your juxtaposition of "flexible" and "elastic". The one does not preclude the other - or does it? I mean, real piano wire is both flexible (capable of being flexed) and elastic (tending to return to its original shape when being flexed). Should one perhaps substitute "stiff" for "elastic"?

Not trying to split hairs, but trying to clarify for myself (and possibly others).

Hi Mark.
A flexible string does not resist deformation. If you "bend" (to use Ed's nomenclature) a cloth string it will not return to its pre-bent shape. If you bend no. 15 piano wire it will spring back to its pre-bent shape upon release. In the simulation both the flexible and elastic (or stiff) strings are under tension.

A string that is elastic is considered "stiff". While a string that has no elasticity is considered "flexible". The degree of stiffness or elasticity is characterized by the Young's modulus.

Dave

Ed.
We could indeed.
Dave

Just so you know, a "flexible" string is usually known as a "limp" string by physicists and engineers.

In engineering school when we were deriving the equations for harmonic motion, we started with a theoretically limp string, where the only restoring forces were the tension forces, and then added complexity by considering stiffness and damping.

These animations are very nice. Another way of looking at it is that for a limp string all frequencies move at the same speed but in a stiff string higher frequencies move faster so from a time point of view the higher frequencies "see" a shorter string and they resonate at a higher frequency.

Kees

Dave,
Can you add the simulation for longitudinal mode into your model? And have you read Conklins papers on "Phantom Partials"?

Thanks for that alternative way of looking at it, Kees.

Ed,

No and yes.

No. Adding the longitudinal component is not a trivial extension of the model that I use but it is an interesting challenge. Thanks for mentioning it. It is now on my bucket list.

Yes. I have read those articles. Interesting. After reading them I took a look at the spectra of several keys and I could find the phantom partials. Neat stuff. There are some comments about them in chapter 5 of a book that was advertised in PW a month or so ago.

Dave