I was walking in the park with a water bottle in my bag, and some of the water in it had already been consumed. I noticed that the half-empty bottle made a bit of a sloshing gurgling sound as I walked along. So naturally I wondered what fraction of the bottle should be full for it to make the most sound?
Note the proverb that "empty vessels make the most sound."
However, like a midgap trap has the highest recombination rate, so intuitively one might expect that a half-empty bottle would make the loudest sloshing sound.
The following argument is original - although I discussed it with my friend Tirthankar Haldar - and he gave me valuable feedback and suggestions. The usual rules apply: any remaining error is only due to me.

Consider the bottle is a cylinder of area A and length d.

The water in it is filled to a length x., and its density is r, and it is subject to acceleration g.

The sound produced by the water is proportional to the energy, and the energy is given by:

                                                   E = rgAx(d - x)

The motivation for this energy is the mass of the water (rAx ) multiplied by the acceleration g and by the distance the water can move: d-x.

The force is given by the negative of the derivative w.r.t.x:

                                                    F= - rgA(d - 2x)

Differentiating again and putting the derivative it to zero gives: x = d/2.

That is, the half-empty bottle would make the most sloshing sound.

The restoring force is of the form: F = kx, not the usual F= - kx, that is characteristic of simple harmonic motion. That implies that the sloshing frequency w = Ö(k/m) is imaginary and that the sloshing is damped i.e. it dies out exponentially.

a) Clearly the above argument rests - or falls - depending upon the validity of the energy expression given above for the sloshing motion of the water in the bottle. 

b) It is also assumed that the sloshing sound is a maximum exactly when the sloshing energy (as defined above) is maximum - which need not necessarily be the case.

c) The initial condition assumed above (see figure) is a bit odd. If a bottle is horizontal, the top half should be empty, not the right side! However, this kind of an initial condition could be achieved if one starts with a half-empty vertical bottle and turns it upside down. Or turns it to any angle other than the horizontal! So the diagram above should be rotated clockwise by 90 degrees - but I'm not gonna do it!

Ok, so this argument may be fatally flawed. 

Never mind, Milord, I rest my case - and if the defendant is guilty of stupidity as charged, so be it!