Acoustic Waves in Air with Variable Sonic Velocity

[sethoflagos]

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

Edited January 6, 2023 by sethoflagos
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[Mordred]

I'll look at it in more detail later on but on a quick glance I'm impressed good job so far

[studiot]

Well so far I've got as far as printing the pdf out and having a swift butchers.

It's really good that you have declared (hopefully all) your variables at outset. +1

It's also interesting to see a chemical engineer using the ideal gas equation rather than z or w factors.

Anyway I'm not sure which 2 equations you want to solve for, presumably the characteristic equations at the end 5.06 and 5.05 or 5.07 ?

Edited January 6, 2023 by studiot

[sethoflagos]

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. 

[sethoflagos]

[swansont]

3 hours ago, sethoflagos said:

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.

There isn’t all that much energy in sound waves. How much of a temperature variation are you claiming?

[sethoflagos]

26 minutes ago, swansont said:

There isn’t all that much energy in sound waves. How much of a temperature variation are you claiming?

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.

Edited January 6, 2023 by sethoflagos
Clarification

[swansont]

31 minutes ago, sethoflagos said:

A 10 kPa pressure amplitude swing around atmospheric should give a +/- 8 K swing around 300 K.  

How many dB does that represent?

[sethoflagos]

Just now, swansont said:

How many dB does that represent?

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

[swansont]

5 minutes ago, sethoflagos said:

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

I’m getting just under 174 dB. (not sure why it depends on the frequency) with 20 microPa as the reference pressure

https://www.omnicalculator.com/physics/db

https://en.m.wikipedia.org/wiki/Sound_pressure

[sethoflagos]

18 minutes ago, swansont said:

I’m getting just under 174 dB. (not sure why it depends on the frequency) with 20 microPa as the reference pressure

https://www.omnicalculator.com/physics/db

https://en.m.wikipedia.org/wiki/Sound_pressure

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.

Edited January 6, 2023 by sethoflagos

[swansont]

But if it’s 174 dB for your 8 degrees, and the actual sound level is 124 dB, then the pressure amplitude is below 100 Pa, with a corresponding drop in temperature. < 0.08K doesn’t seem like it’s a big deal, which is likely why it’s ignored.

[sethoflagos]

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. 

[swansont]

35 minutes ago, sethoflagos said:

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. 

How did you get from 2.5W to 2.5W/m^2?

Being able to generate 10kPa of pressure for airflow does not mean that’s the sound amplitude

[sethoflagos]

3 hours ago, swansont said:

How did you get from 2.5W to 2.5W/m^2?

Being able to generate 10kPa of pressure for airflow does not mean that’s the sound amplitude

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.

Edited January 7, 2023 by sethoflagos
Clarification

[swansont]

8 hours ago, sethoflagos said:

Being 1 metre downwind of a trumpet bell will put you in the middle of a soundfield oto 1 m².

So at some other point the value is different. If you measure it at the bell, it’s a smaller area, and thus a larger pressure value

Your analysis and conversion seems ad-hoc. It wouldn’t work for a point source. An actual bell, for example.

8 hours ago, sethoflagos said:

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.

Citation, please

Quote

And I'm working with peak values not rms so 10 kPa at source will typically become oto 20 Pa at standard distance.

So the temperature effect at “standard distance” will be much smaller.

[sethoflagos]

3 hours ago, swansont said:

Citation, please

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?

[swansont]

9 minutes ago, sethoflagos said:

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?

Speculations need to be backed up by evidence.

I’ve been trying to get you to show that there is some solid foundation for the idea. You claimed that “The peak of the pressure wave is hotter and the trough cooler due to compression/expansion and this does affect sonic velocity.” but this seems to not be much of an effect for normal sound levels. There’s nothing speculative about establishing this. The pressure amplitude of the sound generated is not directly from blowing, but from vibration, be it a reed or one’s lips, and trying to equate the two is erroneous.

If you want to analyze the temperature effects from really loud sounds, that’s fine, but you need to acknowledge when the analysis applies and when it doesn’t. Constant speed of sound looks to be a really good approximation for most cases.

[sethoflagos]

1 hour ago, swansont said:

Speculations need to be backed up by evidence.

I’ve been trying to get you to show that there is some solid foundation for the idea. You claimed that “The peak of the pressure wave is hotter and the trough cooler due to compression/expansion and this does affect sonic velocity.” but this seems to not be much of an effect for normal sound levels. There’s nothing speculative about establishing this.

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.

1 hour ago, swansont said:

The pressure amplitude of the sound generated is not directly from blowing, but from vibration, be it a reed or one’s lips, and trying to equate the two is erroneous.

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.

1 hour ago, swansont said:

Constant speed of sound looks to be a really good approximation for most cases.

Newtonian gravity is a far better approximation for anything you or I are likely to experience first hand.

1 hour ago, sethoflagos said:

Do you have anything to say on the meat of the OP or are you happy just sniping at the peripheral stuff?

I guess this has been answered by default.

[swansont]

6 minutes ago, sethoflagos said:

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.

So your estimation of 8 degrees was not based on any well-established science?

 

6 minutes ago, sethoflagos said:

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.

Was your statement based on any science or measurement, or did you just make it up?

 

[sethoflagos]

1 hour ago, swansont said:

So your estimation of 8 degrees was not based on any well-established science?

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.

 

 

Edited January 7, 2023 by sethoflagos

[swansont]

1 hour ago, sethoflagos said:

It was based on Eqn. 02.06 given in the document posted above complete with a full derivation from thermodynamic fundamentals.

Speed of sound wasn’t one of the given variables. Pressure was. 

 

1 hour ago, sethoflagos said:

Suggest you try reading it before you make any further inflammatory statements.

Inflammatory? You’re the one who said “these questions cannot be quantified with authority,” not me. 

[studiot]

Gentlemen can you please give others the time to go through the 9 pages of maths posted before closing this ?

I am been rather busy with other things this w/e but would comment further as follows.

 

The NS equation or equations is a single 3D vector PDE or 3 separate scalar ODEs.

That is not enough by itself to solve since it only gives 3 equations for the 5 variables involved.

To solve it we require to introduce two further equations.

One way is to use the continuity equation and a gas law equation of the form density = a function of pressure and temperature.

This I think on first reading is Seth's method.

It may be possible to reduce the number of variables by specifying spherical symmetry.

But a trumpet symmetry is decidedly non symmetiric. Much of the sound energy is focused.

 

I do believe that @MigL is a NS equation specialist and would welcome his comments as well as whatever @Mordred comes up with.

 

[sethoflagos]

50 minutes ago, studiot said:

But a trumpet symmetry is decidedly non symmetiric. Much of the sound energy is focused.

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. 

 

[Mordred]

I would like a little clarity on the goal in terms of acoustics. I fully agree that there is distinctions in the behavior of plane waves and spherical waves  in terms of the specific acoustic impedence. This is known, the specific acoustic impedence is the relations the calculator that Swansont  linked above applies.

 For plane waves the Z as it's commonly denoted stays relatively constant. However in the spherical wave you will have a 1/r relation.

This affects the sound intensity. 

Is our goal specifically reverberation time as per Sabines equation ?

 

 

 

Edited January 8, 2023 by Mordred