sethoflagos

Thank you for your intervention - I appreciate it +1 Can we put the trumpet stuff to one side? It's what initially got me thinking about this topic but the OP is about building the correct general mathematical framework. The rest of your analysis is correct though the introduction of temperature via the Equation of State now needs isentropic conditions to be established via Eqn 02.02, and the introduction of a variable sonic velocity calls in Newton Laplace (Eqn 00.04) so there's 7 independent equations for 7 variables . The choice of spherical coordinates certainly helps eliminate a lot of terms that might otherwise be awkward.

It was based on Eqn. 02.06 given in the document posted above complete with a full derivation from thermodynamic fundamentals. Suggest you try reading it before you make any further inflammatory statements.

In the absence of a well developed mathematical model, these questions cannot be quantified with authority. You seem to be putting the cart before the horse. The eleven year-old trumpeter in my avatar was already getting regular paying gigs, and he says 'Citation, Please'. Or words to that effect that I'm not going to repeat in a public forum. Newtonian gravity is a far better approximation for anything you or I are likely to experience first hand. I guess this has been answered by default.

It's in the 'Speculations' section. If there was a citation it would be a speculation would it. Do you have anything to say on the meat of the OP or are you happy just sniping at the peripheral stuff?

Being 1 metre downwind of a trumpet bell will put you in the middle of a soundfield oto 1 m2. And no, the proportion of energy input that becomes acoustic energy is a matter of a musician's skill among other factors. For trumpet players above a certain standard, the kinetic energy of the DC component of flow is much less than that of the AC component. These issues are way beyond the groundwork I presented in the OP. Could we get back to that? Btw your 174 dB is effectively the sound pressure level at source ie somewhere inside the pipes which are typically 12+ mm diameter. There's a scaling factor of some thousands down to the sound pressure level at 1 metre. And I'm working with peak values not rms so 10 kPa at source will typically become oto 20 Pa at standard distance.

10 kPa, 8 K, 2.4 W, 120 dB seems to give a consistent picture as back-of-envelope calculations go. It's your 174 dB which is way out in left field. That's instant pulmonary embolism, burst lung zone.

I can never figure those things out. dB is for electricians. On a practical level, I've plugged a water manometer in the side of my mouth and registered about a metre water gauge when I'm playing a little more than reasonably loudly. If someone stuck their head within 1 metre of the business end, they'd be in some discomfort. That ties in with the 'threshold of pain' in the tables at 120 dB. I'm not sure whether the physics of sound intensity is frequency dependent either, but peoples perception of loudness most certainly is. As a rough guide, when I could play high and loud, I'd be pushing about a quarter litre of air per second into the instrument, so the PdV would be 10,000 * 0.00025 = 2.5 W. Your calculator gives 124 dB for 2.5 W/m2 so it seems to tie in. There will inevitably be some losses.

Depends on the frequencies and how close you are. Probably getting towards 120 dB at 1 metre.

A 10 kPa pressure amplitude swing around atmospheric should give a +/- 8 K swing around 300 K. That's a total variation of ~3% in sonic velocity. ie the peaks are travelling 10 m/s faster than the troughs. It's enough to seriously distort the waveform and raise questions about the usual sinusoid assumptions.

5.04 is integrated along 5.05, the line dr/dt = u + c, to give the forward wave. 5.06 is integrated along 5.07, the line dr/dt = u - c, for the return wave. Trumpets only exceptionally run above 10 kPa gauge and that's well within ideal gas range. PS You really DON'T want to hear acoustic waves with pressure amplitudes significantly higher than that.

Just about every work I've read on air acoustics (I've read a lot) starts by assuming constant sonic velocity. I sort of understand the reasons for this, but it simply isn't accurate. The peak of the pressure wave is hotter and the trough cooler due to compression/expansion and this does affect sonic velocity. I believe it's significant enough to be a major factor in eg. developing the characteristic timbres of wind instruments which nobody seems to have got a proper theoretical handle on yet. I've made a start on developing a mathematical system modelling a spherical wave accommodating a variable sonic velocity and attached a brief summary. I'd be most grateful if someone would give it a quick once over to see if I've made any blunders along the way. The pair of simultaneous ODEs I've come up with are beyond my skills to solve analytically, but they're quite amenable to numerical integration. Any hints from the more mathematically gifted would also be much appreciated. Spherical Adiabatic Acoustics.pdf

So while the dark matter does experience some gravitational attraction to the galactic disc, it just ploughs straight through it and out the other side without losing momentum whereas normal matter will inevitably dissipate heat due to friction during transit, and ultimately after a few transits will join the disc. Have I got that right?

I think the OP may fall more into the orbit of BS 5950 than BS 6399.

To get methanol levels down to those of London Gin, you must use multiple stages of fractionation such as the Coffey Still (modern separation technology could effectively eliminate methanol from the product quite simply). To enforce those levels across the drinks industry would effectively ban the use of pot stills. A lot of people would get upset about that.

I'd guess that method would give you a total stress loading on the structure as a whole, but how would it indicate how those stresses were distributed amongst the various sections?

Having rekindled my interest in the topic I dug out Clausius-Clapeyron for P* estimates (https://en.wikipedia.org/wiki/Clausius–Clapeyron_relation) and plugged it into Excel: A bit rough and ready (I've used fixed latent heats) but it ties in with my gut feel.

Just to clarify this last bit. We're not descended from dinosaurs. About 320 million years ago the ancestors of both humans and dinosaurs were to be found among a group of related small lizard-like creatures. One of these had made a few readjustments to its skull to improve the efficiency of its bite. It may have been a creature called Archaeothyris found in Nova Scotia, or a species very closely akin to this, and we (along with all other mammals) are their descendents. Dinosaurs and all other modern reptiles are descended from one of the species that had not developed this advanced bite.

I don't think it works quite like that. Firstly for an ideal liquid mixture (methanol-ethanol mix is close to ideal) the partial pressures of each at any given temperature follow the simplest form of Raoult's Law (at pressures below 10 x atmospheric at least) partial pressure of i (Pi) = partial pressure of pure i (Pi*) x mole fraction of i (xi) So if xi is small, the partial pressure is small even at the boiling point of the pure substance (in the case of the most volatile component). If you boil a methanol-ethanol mix, yes, there will be a significantly higher mole fraction of methanol in the vapour phase and xi will fall a bit but for a single stage operation, the idea that what's left behind has a safe methanol level has made many, many people "mad, blind and dead - or was it blind, mad and dead" as my school chemistry teacher put it. When you add water to the mix, things get a little more complicated. For both the methanol-water and ethanol-water systems, the alcohols are more attracted to themselves than to the water molecules. So these are non-ideal mixes and follow a modified version of Raoult's law that includes an activity coefficient ai. Pi = xi ai Pi* For these 'positive deviation' mixtures the activity coefficient >1.0 and offsets the effects of low mole fraction by increasing the alcohol's volatility. ai for methanol in its aqueous mix is typically around 2 but rises steeply in low concentrations to about 3.5. So a 50-50 methanol-water mix (xi = 0.5) will actually boil close to the boiling point of pure methanol. ai for ethanol in its aqueous mix is actually a bit higher, ~4 in low concentrations, which pushes its volatility even closer to that of methanol, making the separation even more challenging. Now obviously I have no practical experience of distilling strong spirits for personal consumption and strongly advise against anyone even considering such a reckless practice. But if say I was in urgent need of quality surgical spirit for medical purposes, I think I'd consider starting off the separation with a very slow simmer, well below boiling point and let the vapour rich air currents rise by thermal convection into my condenser.

For a more complete descrption of the above, refer to: https://www.youtube.com/watch?v=BsNoiFj3wlw&ab_channel=NurdRage

Couldn't the inherent conservatism of the design calculations be intentional? Particularly given the potential for multiple fatalities if the structure failed, a safety factor of 10x would not be unusual (as with elevators etc.)

Eat That Question - The Grand Wazoo - Zappa

No idea why, but Laurence Kuhn seems to have decided to feed me a bunch of backup material The meat in this one is possibly best summarised in https://en.wikipedia.org/wiki/Wheeler–Feynman_absorber_theory (basis of Davies' PhD studies)

Such vitriol.

£10,000 per year a 'marginally tiny amount' compared to the £3,000 per person per year spent by the Department of Health and Social Care in England? You didn't check the sums before that post, did you.