I'm still trying to get better and better data - it's tricky given the size and speed of the movement. But my working model is something like this. Cutting is made of of two phases: cut initiation, and cleaving.
Cut initiation is all about the cutting edge. In an ultrasonic blade, that edge oscillates and the tiny imperfections on the blade edge act like a saw to break the linking fibers in food. It's just like using a human-scale slicing motion, but at 40kHz, and with a microscopic stroke length.
Cleaving is mostly about friction. Cutting a block of cheddar is pretty much all cleaving, and a very sharp cutting edge doesn't provide much advantage. My blade vibrates along the blade face, so foods experience the coefficient of kinetic friction, not static friction. This reduces cutting forces, and does so in a way that's totally independent of the sharpness of the edge.
We experience different foods as more cut-initiation-centric or more friction-centric. Tomatoes are all about piercing the skin. Hard squash is a cleaving game. Bread is layers upon layers of initiating cuts in the bubbles of the crumb.
If you're interested, I published my testing on regular knives in the Quantified Knife Project by strapping 21 chef's knives to a robot arm and collecting data on cutting forces. The data are open-source on github, too. https://youtu.be/GUQy0Sdp8Hc
If any home users cut on glass, they'll be rewarded with an unpleasant screeching sound. I can't say that was by design, but it's not undeserved ;-)
The ultrasonic motion acts as an amplifier for physical sharpness. So, it's sharpest when it's got a geometrically great edge, but even as that edge dulls, it behaves sharper when on vs. off. This is reflected in BESS testing, and also in robotic cutting.
Moreover, a huge amount of the force required for cutting normal foods is actually a function of friction, not just bevel sharpness. So the reduced friction on the blade faces from ultrasonic motion remains just as effective even if the cutting edge is dull. In fact, commercial ultrasonic cutting machines don't use sharp blades at all!
Hey there, Scott here. I'm driving the knife at actually only 10W. When in resonance, this produces a stroke amplitude of 10-20 microns (depending on the spot on the blade) which is large enough to have a measurable impact on the ease of cutting. 50% reduction in peak force for tomatoes (as measured quantitatively with a robot arm), and I've seen even higher in other foods.
At this power level, there's no heating of the blade like the small blade tools you're describing. And firmware in the handle adjusts the operating frequency continuously to stay in resonance.
This all works because the ultrasonics aren't moving the blade like a reciprocating saw -- that would indeed require huge power. They're sending longitudinal shockwaves through the blade itself that cause the metal to expand and contract. Check out minute 2:30 in the video here to see that motion in action: https://youtu.be/cXjbSVt9XNM