I apologize for this delayed response to the many interesting comments in this thread I started. I’ve been going out of town and tending to important issues. For those who don’t know, I’m in Complete Response for Multiple Myeloma, a cancer of the plasma cells in the bone marrow. I have medical treatments (infusions) two days of every other week, and they give me headaches and hangover, but I’m not complaining, it’s not too bad. If by chance, anyone else in the group is going through the same thing, I invite you to contact me privately and we can compare notes, even though this disease is extremely variable.
Now being able to provide a more energetic description, I must first state clearly that the suggestion I proposed in the OP is really only the simplest theoretical step in trying to understand why different tongue materials might produce different acoustic effects, and I want to correct a sentence in the “Quote” part of that post, which was a misstatement:
According to these theoretical considerations, both tongues should produce the same
acoustic sound vibration: fundamental plus overtones.
The Euler-Bernoulli wave equation for a vibrating bar is the simplest formulation for such behavior, and it is considered very accurate when rotational inertia and translational shear can be neglected, and that is the case if the thickness of the bar is not too great and the vibrations contain only small angles, which often occurs with free reed tongue vibration.
Attached is a .docx file that gives the E-B bar equation and the Timoshenko bar equation. I was not permitted to upload this file. Anyone?
Shown is the simplest form for the E-B bar equation, which does not include a forcing function (external force, such as a bellow’s pressure), nor aerodynamic drag (friction) terms. Mathematically, it is called a non-dissipative (frictionless), homogeneous formulation, and the utility of it is that it (along with its boundary conditions) provides the eigenfunctions for any type of bar vibration consistent with its underlying simplifications. In other words, the complete solution when you do include a forcing function with friction (the complete formulation) is made up of these same eigenfunctions, which give the axial dependence of the shape of the bar (its curve). The time dependence of the vibration in this complete case is then determined by the time dependency of the forcing function with the complete formulation.
Thus, the simple suggestion in my OP will reliably predict accurate frequencies and general axial beam shapes for the fundamental and overtones that are experienced in actual operation (with dissipation and bellows pressure). And it does not depend much on how accurately the tongue and slot (entire reed) is made. However, it cannot give a complete description of the oscillations in air pressure (the acoustic sound) that the vibrating tongue produces. Let’s focus now on the acoustic sound, which is our prime interest.
In order to predict a complete description of the acoustic sound of the reed, we need to know how the vibrating tongue motion translates to oscillatory air motion, and this air motion needs to be understood in the near field (close to the vibrating tongue) and the far field (after the sound waves move to a region away from the reed – say to a region that is more than about ten tongue lengths away, which is the sound we hear). With a complete formulation, we will get some information on the acoustic near field, and here, we may be in luck, at least in so far as making conclusions about how different tongue materials might compare in their acoustic sound (volume and frequency spectrum, or timbre). These conclusions would be enabled because of the addition of a forcing function and dissipation terms into the E-B bar equation, and scrutinizing those terms. It’s the same way I suggested in my OP, only now we have more terms in the equation. The boundary conditions (B.C.) remain the same in this complete formulation (fixed at one end, free at the other).
In order to accurately determine what these terms are, we need to develop an accurate physical model for the tongue motion, and how this motion interacts with air movement. I developed such a model after I was invited to deliver a paper at the Acoustical Society of America 2017 meeting this last Dec 4 – 8 in New Orleans. I delivered the paper, with the published Abstract:
New Orleans was fun, and this paper is a work in progress. I have completed the physical model and have conjured a mathematical method of solution for the resulting governing equation and B.C. I now have to finish the formal solution - which at this point, is mostly a lot of Algebra - and to perform calculations and graphical results, check agreement with experiment, etc. But because of further travel plans out of the country, work on this project will be put off for more than a month.
I’m explaining all this in the hopes that I can convince a reed maker to first take up the simplest suggestion in my OP. By Spring, I should have completed the analysis and could hopefully make some statements about what geometry would be required to cajole two tongues of different material to not only vibrate with the same frequency and have the same overtones, but also to produce the same acoustic sound, if possible. It may not be possible. And of course, it may not be possible to find such simplistic generalizations from only this study, in which case, we would have to rely on an acoustical analysis of the air sound field. But let’s not yet give up hope on the simplest approaches first.
I wasn’t sure how to present all this, and I hope I haven’t confused things with my attempted explanation here. I’m glad to answer any questions, if I can. In the coming days, I plan to respond to the comments by others in this thread.