I don't find the explanations above very satisfying. They don't answer some very essential quesitons, like,

"How does one design the shape of the hull to ensure it will not porpoise in its design speed range?"

"What is the range of center of gravity locations for stable operation of a given design?"

"What is the critical trim angle for a given design?"

I think

this paper gives a much more satisfying explanation of porpoising. What the author is saying is, porpoising starts out as an instability for small motions that grows in amplitude until limited by nonlinear effects. So if you want to avoid porpoising, it's sufficient to look at the linear conditions for stability - you don't need to be able to predict the amplitude of the oscillations that ensue if you get it wrong.

Starting from a trimmed, steady running condition, if the craft is disturbed in heave or pitch, it will be stable if the following four conditions are met:

- increasing depth (heave) creates a force that wants to accelerate the craft back up (C33 is positive in his equation),

- increasing pitch (bow up) creates a moment about the c.g. that wants to pitch the bow down (C55 is positive),

- increasing the pitch tends to accelerate the craft upwards (C35 positive),

- increasing the depth tends to pitch the craft bow down (C53 positive) [this one actually sounds backwards to me, so maybe I'm misinterpreting the paper]

The craft will be unstable when C53 is negative, meaning the that for an increase in depth, it tends to pitch the craft up. So the question becomes, "What conditions will drive C53 negative?"

Here's my application of Ikeda and Katamaya's princples to the situations cited in previous messages:

The pressure distribution on a planing surface is heaviest just behind the leading edge. As the trim angle is reduced without changing the center of gravity, the leading edge moves forward.

Eventually the heavily loaded portion gets to be far enough ahead of the center of gravity that it causes a bow-up pitching moment when the depth increases. This causes the craft to be unstable, and it reaches the onset of porpoising.

Moving the center of gravity forward without changing the trim angle will reduce the bow-up pitching moment from the planing surface and make the craft more stable.

But in general, moving the c.g. forward will also reduce the trim. So if the leading edge of the planing region moves forward slower than the c.g. moves forward, the c.g. shift will be stabilizing. If it moves forward faster than the c.g. shift, it will be destabilizing.

The same calculations that one does to estimate the wetted area and running trim for performance could be used to get the variation in the forces and moments about the c.g. for small changes in the heave and pitch angle. Then you can apply Ikeda and Katamaya's static stability criteria (C35 and C53 have the same sign) to find out if the craft is stable or not.

Better yet, one could estimate all the matrix elements in the equation, including inertias and damping coefficients, and get the four eigenvalues to see if their real parts are all negative.