Hmmmm… PhD in physics here. This isn't true and even if it were, in a practical system (i.e. a real bass guitar) it would need to account for every part of the system, so changing any variable of the neck, body, bridge, etc would affect this. It's simply untrue that 37" is the "mathematically correct" scale length for a B string.
Sheldon didn't work out 37" with complex math, it was done with a jig and what sounded good to him. This doesn't discount that it sounds good, it simply is not mathematically optimal and without evidence that it is I don't think anyone's going to buy that.
I'll start by saying I'm not a physicist and yield to your field knowledge here. I have a lot of interest in engineering and physics concepts, and little stomach for the arithmetic classes. I sure am glad someone else has the stomach, so I don't have to.
To be clear I don't think Sheldon Dingwall sat down with a calculator and worked out the correct speaking length of a .130 Gauge string. He built the jig, 'stumbled' upon the speaking length, and expanded from there.
We both know that a real world application of physics, with gravity, atmosphere etc, is rife with variables that make it difficult, if not impossible to come up with the 'perfect' model to any complex motion.
Many years ago I attended a lecture at the Lockheed ATC discussing the 'Physics of the Violin'. Can't remember the name of the lecturer (this was in the late 90's) but he was an aerospace professional (engineer maybe) who also liked building violins. Harnessing frequency and nodes of vibration was a big part of the topic. Found a paper with similar information - this does not discuss scale length but the talk I attended did. The paper is worth looking at, interesting stuff here even if it doesn't directly relate.
My point is: yes with the variables in materials, contact points etc, one could not build a model to make the perfect scale length down to 0.xxxxxx". But getting a model that works to say 0.25" variance (huge and sloppy by physics standards)seems pretty possible.
Not really trying to argue, would like your thoughts

