bottle...contd:

I just wanted to add one point: energy is a scalar.
Even a half-full bottle lying on its side with the water in the bottom half, like so:

the above analysis still applies (with the caveat about the actual hydrodynamics), the only difference being that the length of the bottle is replaced with its width.

Moving on, somewhat:

That's the top view of a donut - assumed to be torus, with a circular cross-section, and a circular hole.
Why am I looking at donuts? Because:“Between the optimist and the pessimist, the difference is droll.

The optimist sees the doughnut, the pessimist the hole!” – Oscar Wilde               

Firstly, many doughnuts do not have holes at all. That would imply that pessimists would, perforce, have to eat only doughnuts with holes. Secondly, the sizes of doughnuts – even in the age of cookie-cutter standardization – are not fixed. Nor are their shapes clearly toroidal: or, not clearly having circular cross-sections.

Contrast Wilde’s statement with the more common characterization of the optimist as seeing a glass as half-full, and the pessimist as seeing it as half-empty.

http://www.wou.edu/~burtonl/courses/math494.594/HowDoesYourDoughnutMeasureNoack.pdf

Typical dimensions of a doughnut:

outer diameter: d_o = 9.4 cms

inner diameter: d_i = 2.7 cms

fill factor ff = (d_o)²/[(d_o)² + (d_i)²]

= 0.924

Note: one could calculate a volume ratio, and this would give a different number. The volume of the doughnut is: V = Ac, where A is its cross-section and c is its circumference. The cross-section is:

A = pr², where r = (r_o – r_i)/2 = 4.7 – 1.35 = 3.35 cms.

A = 35.26 cm²

The circumference is: c = 2pR, where R = (r_o + r_i)/2 = (4.7 + 1.35)/2 = 3.025 cms

C = 19.01 cm

V_d = (35.26)(19.01) = 670.29 cm³

Total volume is: V_t = (pR’²)(2r), where R’ = R + r = 3.025 + 3.35 = 6.375 cms

 = 855.43 cm³

Volume fill factor = (V_d)/( V_t) = 0.783

Clearly, Wilde has slanted his definition against a putative pessimist, making his/her job much more difficult. His definition is, of course, the area fill factor and not the volume fill factor.

Not that Wilde was an unabashed optimist: the author of “The Portrait of Dorian Gray” or “Salome” had a clear view of the dark side. It is more likely that he just couldn’t resist the aphorism. 

Of course, the volume fill factor is ambiguous: it could probably be defined differently.

But then, ambiguity was Wilde's forte! 

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!