To understand this scaling law, we need a formula reminiscent of the energy principle in quantum mechanics. This is beyond even undergraduate college physics, but middle and high school students can learn about normal modes in slinkys, coupled pendulums and Chladni plates. They can also understand the concept of scaling, as in the use of a wind tunnels and water tanks to test scaled down models of airplanes, sailboats and ships. A simple way to explain the Box Cello acoustical laboratory is that we are trying to build a scaled-down acoustical cavity which matches the loudness curve of real cello. This scaling law suffers from an important flaw which also limits the use of small models for the testing of airplanes, boat hulls, and even controlled fusion experiments.
None are exact.
My assumption is that a mediocre cello is one in which the normal modes match in frequency to those of a good cello. Most low-cost starter cellos are not even mediocre. I recently observed a cello in an elementary school to be missing its sound post, and apparently nobody had noticed. The final accomplishment of the Spring 1998 Box Cello Club was verification that a Plexiglas plate exhibits a reasonably narrow resonance for its consideration as the material to be used for our initial efforts. Most readers are probably hopelessly lost at this point, but one more advanced topic needs to be mentioned.