Acoustic Waves in Air with Variable Sonic Velocity

[sethoflagos]

57 minutes ago, Mordred said:

I would like a little clarity on the goal in terms of acoustics.

Sometimes so would I. I've worked hard all my life and set aside many topics that interested me because I could spare the time. Now that I have the time, I really would like to produce something academically worthwhile before I kick the bucket.

Anyway, some weak field results hot off the presses

[sethoflagos]

Shucks, missed the cut-off. r within the trig functions should read r-r₀ of course.

[Mordred]

1 hour ago, sethoflagos said:

Sometimes so would I. I've worked hard all my life and set aside many topics that interested me because I could spare the time. Now that I have the time, I really would like to produce something academically worthwhile before I kick the bucket.

 

Well I can understand that, if that's the goal then I would recommend including a planar wave examination and a comparison between the two. 

 here is an example which should help. Though primarily it simply explains that there is a difference between the two but the article is lacking in what those differences entail in acoustics.

https://ccrma.stanford.edu/~jay/subpages/Lectures/Lecture1-Acoustics.pdf

I didn't spot any mistakes in the first article of equations you applied however still looking it over as I have time

Edited January 8, 2023 by Mordred

[sethoflagos]

29 minutes ago, Mordred said:

Well I can understand that, if that's the goal then I would recommend including a planar wave examination and a comparison between the two. 

Well there's the rub. As r approaches infinity the wavefront tends toward planar. Essentially I simply lose the 3rd term (2cu/r) from the ODEs and I have a planar wave. It's a little trick I found to study standing waves in low angle conical to concentric tubing.

Btw I really appreciate the time you're spending on this. I know you've many other matters to attend to.

[Mordred]

yes that would make sense in near field and far field examinations I've come across though acoustics isn't something I often look into. The article I posted hints at that as well if I recall. No problem on trying to supply some direction etc its always nice to see a speculation posting that includes the related mathematics

Edited January 8, 2023 by Mordred

[sethoflagos]

7 hours ago, studiot said:

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

Good point but not insurmountable I think. The really scary challenge is that the sound output propagates back up the instrument to the mouthpiece where it acts as a high gain servo-assist on opening and closing the embouchure in sympathy with the output waveform. The output largely determines the input so there is absolutely nowhere to hang a set of initial boundary conditions unless you start from silence and integrate through all the note establishment phase. 

[sethoflagos]

Amendments to the previous update plus preliminary approximations to all relevant fields. I really could do with someone (@Mordred) to put my use of Eqn. 06.05 under the microscope. The results that follow look credible at least as approximations, but ....

 

[Mordred]

let me guess your adding two waves of opposite phase and seeing c_o disappear ? still going through your math but I'm glad to see you got the displacement phase lag that showed in the reference I posted though the graph in the article isn't very clear.  

here is a decent article to use for comparision

https://hal.archives-ouvertes.fr/hal-03188302/document

Edited January 10, 2023 by Mordred

[sethoflagos]

Thanks for the reference - that one's a keeper and gone straight into my acoustics library.

I'm just focussing on the forward wave for now as that generates forms that are easily compared with standard reference material.

If I include the reverse wave on an equal footing, I get a standing wave of sin(wt)cos(w(r-r₀)/c) form. I'm not 100% sure why. Perhaps this is where I start having to take a serious look at impedance.

[Mordred]

I would highly recommend you look into independence in particular for planar and spherical waves.

 You should also be able to match up to the relevant logarithmic functions as they apply to acoustics. If your not it indicative of an error or some missed factor.

 Once you get into more than one wave you will definitely need to factor in your wave equations primarily in amplitude and phase angles. In essence constructive and destructive interference factors.

Edited January 10, 2023 by Mordred

[sethoflagos]

1 hour ago, Mordred said:

I would highly recommend you look into independence in particular for planar and spherical waves.

 You should also be able to match up to the relevant logarithmic functions as they apply to acoustics. If your not it indicative of an error or some missed factor.

 Once you get into more than one wave you will definitely need to factor in your wave equations primarily in amplitude and phase angles. In essence constructive and destructive interference factors.

I was able to download the full text of Hansen's "Fundamentals of Acoustics" 4th Ed.

It's written in a language I can follow and I've attached a sample chapter. Is this likely to guide me in the direction you're recommending?

Fundamentals of Acoustics 4ed-Cap5.pdf

[Mordred]

yes this is precisely the direction I wanted you to take, these equations are ideally suited. Good find key note  on equation 5.5.4 in regards to specific acoustic impedance involving a linear, lossless wave equation. 

Edited January 10, 2023 by Mordred

[studiot]

Perhaps you can get a hold of this older degree level book.

The Students Physics VolI

Acoustics

Alexander Wood

Blackie.

 

Wood goes into great detail about the subject including lots of experimental material.
He treats many musical instruments including brass in general and trumpets in particular and offers measurements on them.

 

Here are some pages from the more basic section involving the velocity of sound, both measured and linked to theory includng corrected gas laws.

[sethoflagos]

Many thanks! 

Since I got no alert, I've not checked this thread for a couple of days so I missed your post. Perhaps I should press 'Follow' on my own threads!

[studiot]

14 minutes ago, sethoflagos said:

Many thanks! 

Since I got no alert, I've not checked this thread for a couple of days so I missed your post. Perhaps I should press 'Follow' on my own threads!

As far as I know 'follow' comes up automatically  ie the box is ticked  when I start a thread.

Perhaps someone with more knowledge of the system might advise you on this.

[iNow]

5 minutes ago, studiot said:

As far as I know 'follow' comes up automatically  ie the box is ticked  when I start a thread.

Perhaps someone with more knowledge of the system might advise you on this.

It likely does, but that behavior is governed by your personal notifications settings under user profile (i.e. because you chose for it to be that way).

 

https://www.scienceforums.net/notifications/options/

 

Edited January 11, 2023 by iNow

[studiot]

@sethoflagos

You might find this book of interest.

It is due to be published in early 2023 and a part of the Cambridge University 'Student's Handbooks' series in applied maths and physics.

A student's guide to the Navier Stokes Equations

Justin E Garvin

Cambridge University Press.

[sethoflagos]

1 hour ago, studiot said:

@sethoflagos

You might find this book of interest.

It is due to be published in early 2023 and a part of the Cambridge University 'Student's Handbooks' series in applied maths and physics.

A student's guide to the Navier Stokes Equations

Justin E Garvin

Cambridge University Press.

No offence taken, but do you really think my handling of the N-S equations is sub-graduate level? 

What I was primarily looking for was some help with the differential equations.

On 1/6/2023 at 7:56 PM, sethoflagos said:

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.

Essentially, if I can't get Laplace transforms to work I'm stumped on the analytical solutions. I've had more success with numerical methods on the simplest cases, but anything beyond simple (eg the slightest hint of rotational flow) basically needs a Cray with the methodology I've been using.  

Edited January 20, 2023 by sethoflagos
sp

[Mordred]

6 hours ago, studiot said:

@sethoflagos

You might find this book of interest.

It is due to be published in early 2023 and a part of the Cambridge University 'Student's Handbooks' series in applied maths and physics.

A student's guide to the Navier Stokes Equations

Justin E Garvin

Cambridge University Press.

Thanks for the info as I tend to collect good literature I may pick up a copy

[studiot]

12 hours ago, sethoflagos said:

No offence taken, but do you really think my handling of the N-S equations is sub-graduate level? 

I was suprised at this response from an intelligent, well educated person of significant experience.

7 hours ago, Mordred said:

Thanks for the info as I tend to collect good literature I may pick up a copy

I haven't seen this book (yet?) so I would be interested in your assessment if you do.

The Cambridge Student Guides series are not formal texts or treatises but specially written to provide a depth of understanding not realised in normal textbooks.
They are written by modern experts from around the world and some of the one I have show nex material or approaches not seen elsewhere.

As such they contain material whch may be of undergrad level in some courses and postgrad in others depending upon the main coverage of any particular course.

The single topic volumes such as Lagrangians and Hamiltonians; Entropy; and Vectors and Tensor are particularly well presented in my opinion.

It would be a brave man indeed who cannot learn something from each or any one.

 

[sethoflagos]

41 minutes ago, studiot said:

I was suprised at this response from an intelligent, well educated person of significant experience.

Sorry. Caught me on a bad day. 

[studiot]

1 hour ago, sethoflagos said:

Sorry. Caught me on a bad day. 

No worries.

[Mordred]

4 hours ago, studiot said:

 

I haven't seen this book (yet?) so I would be interested in your assessment if you do.

The Cambridge Student Guides series are not formal texts or treatises but specially written to provide a depth of understanding not realised in normal textbooks.
They are written by modern experts from around the world and some of the one I have show nex material or approaches not seen elsewhere.

As such they contain material whch may be of undergrad level in some courses and postgrad in others depending upon the main coverage of any particular course.

The single topic volumes such as Lagrangians and Hamiltonians; Entropy; and Vectors and Tensor are particularly well presented in my opinion.

It would be a brave man indeed who cannot learn something from each or any one.

 

 

I have read numerous Cambridge materials as well. Some of the better examples of various math treatments in numerous fields such as the ones you mentioned and others are contained within them.  I have always found them handy