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A Galilean Cannon is a toy similar to the famous basketball-and-tennis-ball demonstration. You take a tennis ball, balance it on top a basketball, and drop them both. The tennis ball will bounce up to a height several times as high as it was dropped from. The Galilean Cannon simply takes a series of several balls rather than two balls, allowing you to shoot the smallest ball many times the height it was dropped from.

My question is about making a modified version from two pieces of steel. I'm assuming there's a base of mass $M$ and a cap of mass $m$, and $M>>m$, although the ratio is fixed. The base has some sort of cut-off conical/horn shape going from a radius $R$ at the bottom to $r$ at the top. The cap has radius $r$ at the bottom and is rounded off at the top, and sits on top the base.

The physics of this device is now wave propagation. It's dropped onto a hard surface from a small height, causing a compression wave to travel up through the base. The wave will bounce off the sides of the base and hopefully concentrate to high energy density as it reaches the cap, ultimately shooting to cap off to a very high height.

My question is, suppose the base is cylindrically-symmetric with a radius $radius(h)$ where $h$ is the height above the bottom of the base and $radius(0) = R$ and $radius(h) = r$. For a given $M,m,R, r, h$ bulk modulus ($K$) and density ($\rho$), what is the optimal $radius(h)$ to shoot the cap as high as possible?

I would guess it is either exponential (constant ratio of adjacent "slices") or a straight cones (constant angle between boundary and net propagation direction), but I'm not sure. Please make any reasonable assumptions you need about boundary conditions for waves at the edge of the base, etc.

Additionally, how sensitive is this shape to having the initial conditions be perfect? If it's dropped at a slight angle or surface it lands on isn't perfectly flat, how messed-up does wave propagation become?

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There is a significant variable that you omit from your list, except as etc.: the variation of bulk modulus and density through the bulk of the material. The basketball has nontrivial variation between the surface material and the air contained within it. As fiber optics show, engineering dispersion by varying internal structure is both possible and crucial for shaping and preserving the structure of impulse wave-forms. With this addition, however, the optimal shape will presumably depend on what material variations can be engineered. –  Peter Morgan Jun 24 '12 at 13:02
    
Meta-comment. I'm sorry my comment above is somewhat unhelpful. I think your question is interesting, and I can't immediately see how to address it, but it struck me quite forcefully that the degree of engineering of internal structure that you require/assume seems quite artificial, and, qua engineering, counterproductive. The basketball does have nontrivial internal structure, which I think contributes to the Galilean cannon effect by shaping the internal compression wave, so to rule out internal structure variations seems not to be the way to go. Again, sorry to be so Meh. –  Peter Morgan Jun 24 '12 at 13:10
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You are unfortunately asking about bomb design. The question of focusing a shock wave to impart the most momentum to the other side is the main question of explosive implosion, and the optimal design for a material which does this is an explosive lens. This is probably why you don't see this question talked about much. The waves used in bombs might be outside the linear regime, I don't know, it's not relevant.

Regarding the first part, the assumptions you make are not ideal--- none of the shapes you propose, the cone and the exponential thing, are any good. To understand why, you need to know why the basketball/tennis-ball device works. That's because the basketball is hollow. This means that the wave propagation is along the surface of the ball, and the impact point is at the bottom. The surface of the sphere has equal time along all geodesics to the antipode, so the geometric-optics sound flow is refocused on the other side of the sphere to make the antipode contain nearly all the shock wave at the impact point, before it rebounds to distribute the momentum over the sphere. But if you have a tennis ball at the antipode, it absorbs a sizable fraction of the basketball momentum and flies off at great speed.

The same design doesn't work with solid balls, because there are waves propagating on the interior. No matter what the shape, you don't get focusing, because the different paths to the other side, with reflection at the boundary, don't take an equal time. So this won't work as a Galilean cannon.

But there is something you can do with the parameters you gave--- you can make a very fast oscillating radius, out than in, so that it is a bunch of disks connected with very narrow necks, with a little ball on the bottom center. Then the impact shock wave will travel through each of the disks, amplifying at each successive neck. If each disk is narrower by a large factor than the previous one, you can make each disk have a greater concentration of momentum by a factor, until you reach the tippy top. I didn't work out the exact shape, because it isn't a calculus of variations thing, the limitations are the speed of sound in the material, and the result in the limit of very large stiff material is just to make the disks impossible large, so that there is no real convergent nice shape limit. It's like a tennis ball on a basketball on a larger basketball and so on.

If you allow hollow material, or materials with verying density and speed of sound, the problem becomes more interesting, but completely classified.

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