I am working on a little analysis with FEM to see just how using this ratio actually works with respect to the distribution of natural frequencies. But I wanted to be sure that I was using the "rule" properly before I tried assessing it.
So here is what the finite element modeling says about how the "k" factor from the Herger patent performs with respect to the distribution of natural frequencies.
First, I modeled an isotropic plate (stiffness is the same in both directions) with a aspect ratio of 9/5, as prescribed in the Herger patent. The vertical series of dots on the far left (k=1) represent the frequency distribution for that case. Note that the absolute value of the modal frequencies is arbitrary, but their relationship to each other along the frequency axis will be the same regardless of the plate thickness, total area, etc., so long as the plate is isotropic and the aspect ratio is 9/5.
On inspection, it's far from clear to me why this distribution would be particularly good. Herger says it is, so maybe it works just fine for him. To me, it looks like a poor choice, particularly looking at the odd, odd modes, that radiate the most efficiently. Note how far apart the first two of these are (1,1 and 1,3). And then, the 5,1 3,1 and 3,3 are all bunched up at near 500 Hz.
But the question I was really wanting to look at was this: Does applying the "k" correction factor to the aspect ratio result in the same distribution of the natural frequencies along the frequency axis? The answer seems to be "yes", for a few of the modes, but "no" for the majority of modes.
I modeled two additional cases, one where the stiffness ratio (k) was 1.25 and another with k=2.0. The aspect ratios in those case was (9/5)*1.25=2.25, and (9/5)*2=3.6, in the two cases, respectively.
The results show that the relationships if the 1,1 2,2 3,3 and 4,4 modes all stay exactly the same. But all other modes either increase or decrease significantly, depending on the k factor, changing their relationships with each of the other modes.
I should add that the patent never claims that using the k factor to adjust the aspect ratio would keep the relationships of the natural frequencies from changing, but it is natural to assume it's what they were trying to achieve. In any event, it seems to work that way for a small subset of modes, but not the majority of them.
Eric