.22 LDC bore clearance study print files

[subscriber]

Bruce,

If color and light are models for filtering, where some parts of the spectrum are reflected and some absorbed, would the resonance frequency of each expansion chamber in an airgun muffler predict which frequencies would be amplified and which absorbed? 

From a musical perspective, a one inch long air tube would have a pretty high resonant frequency, at one atmosphere.  Inside a muffler the pressure would start out at several atmosphere, then  decay to level still above an atmosphere when the air leaves the front bore.  Higher air pressure should raise the resonant frequency.  I suspect that for baffle bore lengths of 0.5 to 2" a lot of that will be in the ultrasound range.  If I am right, the question is what does that do to the other frequencies?

An expert on AGN forum states that all airgun muffler are doing is converting audible frequencies to inaudible ones, like a dog whistle.  I think there are designs where that principle is used and others not.  My toroidal baffle design that traps air by spinning it almost certainly shift the vibrations created up in frequency.  This, while Weihrauch's large, long three chamber designs offer plenty of expansion volume, and produce a low tone.  The latter have "hair curlers" with felt to absorb audible higher frequency sound.

As for my mental models, they are based on observation and an attempt to explain what is seen, based on scientific principles.  Only, I am not sure if I am locking on to the right principles.

An airgun muffler drops the bulk pressure of the air exiting the barrel muzzle, and blocks, filters and converts sound.   Other than contain noise, a muffler generate less sound than the uncorking of a bare muzzle, by dividing one large bang into several smaller ones. 

We know that higher muzzle pressure is louder, yet that is not all there is too it.  If a 30 FPE .22 PCP has a muzzle pressure of 300 PSI the task of making the sound comfortable would seem daunting, considering how loud a balloon pop is at only 3 PSI.  So, clearly air volume and the "size" of the "speaker cone" where it couples to the air also matter.

Perhaps it is as simple as calculating the energy in the air when it is released abruptly.  Basically pressure x volume.  So explaining how a balloon can be so loud, when its "muzzle pressure" is so low.  A bit like comparing the report from a 12 gauge bird shot load to a .223 centerfire rifle.  Both are very loud, but one is a small very sharp sound, compare to a larger less sharp sound.  Unless the size of the bore is also part of that equation, in this case, a smal VS larger speaker cone.  On the other hand, higher pressure gas will leave the muzzle more radially behind the projectile than low pressure gas.  One can see a ball of gas leave a gun muzzle directly behind the projectile; then that gas turns into more of a linear jet as the pressure drops.  Yes, part of that gas plume shape is due to the projectile being in the way of the gas. 

It would be great if you could model what happens inside an airgun muffler to lead to more effective designs that are smaller, lighter and cheaper to produce.  They should also not degrade grouping ability, nor shift it excessively from bare muzzle.   Sometimes such changes are due to a shift in barrel harmonics and can be restored via velocity tuning.  This can happen with any muffler, but is generally worse with long heavy mufflers; so shorter and lighter is less likely to cause harmonic problems.   All this assumes the bores are not so tight as to contact the pellet, and that air currents inside the muffler are not going to steer or buffet the projectile to the point of grouping degradation.  Usually, the air stripping function of mufflers improves groups, so that is what we like to see.

Other than what happens to the air inside a muffler, the outer casing also generates and transmits sound directly to the air.   That is a related field of study that is important.  Currently, wrapping the casing in something like a foam rubber sleeve is a known method to avoid adding casing noise to air noise.  The casing material certainly matters.  I was informed that some of my insert designs that were printed in slightly flexible TPU were quieter and had a less sharp sound than the same mufflers printed in PETG.

[WobblyHand]

Bruce,

If color and light are models for filtering, where some parts of the spectrum are reflected and some absorbed, would the resonance frequency of each expansion chamber in an airgun muffler predict which frequencies would be amplified and which absorbed? 

This is not an active system, it is passive.  Therefore there is no amplification.  However, each expansion chamber does have a resonant frequency range.  By appropriately staggering and weighting these resonant frequencies we can change the overall frequency response.  I would think, in general, the resonant frequencies are controlled by the cavity length of an expansion chamber, and we can weight the individual filters by judicious choices of the apertures.  So far this sound a lot like conventional filter design.

From a musical perspective, a one inch long air tube would have a pretty high resonant frequency, at one atmosphere.  Inside a muffler the pressure would start out at several atmosphere, then  decay to level still above an atmosphere when the air leaves the front bore.  Higher air pressure should raise the resonant frequency.  I suspect that for baffle bore lengths of 0.5 to 2" a lot of that will be in the ultrasound range.  If I am right, the question is what does that do to the other frequencies?

Well, if the chamber dimension is too small, it tends to cut off propagation for lower frequencies.  In waveguides, for frequencies below cut off, the lower frequencies decay at an exponential rate.  The lower the frequency, the faster it decays.  So small chambers are sort of a high pass filter, at least in microwaves, and I suspect this applies to acoustic propagation as the math is similar.  But it isn't that simple, because in the case of a moderator we have to let the projectile pass through...  And sound does leak through the hole.  So this complicates things a bit.

An expert on AGN forum states that all airgun muffler are doing is converting audible frequencies to inaudible ones, like a dog whistle.  I think there are designs where that principle is used and others not.  My toroidal baffle design that traps air by spinning it almost certainly shift the vibrations created up in frequency.  This, while Weihrauch's large, long three chamber designs offer plenty of expansion volume, and produce a low tone.  The latter have "hair curlers" with felt to absorb audible higher frequency sound.

I don't know about experts on forums, I tend to take a lot of things said with a grain of salt.  Heard plenty of blowhards in my career, not to mention on forums.  But following up on waveguides below cut off, this tends to support that the filters may be reducing sound at lower frequencies.  It's unclear there is any up conversion in frequency at all, I don't remember the spectral response with and without a moderator.  If there are longer chambers, then more low frequencies will pass.  Felt will attenuate high frequencies in any of the designs.
[/quote]

An airgun muffler drops the bulk pressure of the air exiting the barrel muzzle, and blocks, filters and converts sound.   Other than contain noise, a muffler generate less sound than the uncorking of a bare muzzle, by dividing one large bang into several smaller ones. 

Not sure I agree with converting sound, but I am in agreement with the gist of the rest.

It would be great if you could model what happens inside an airgun muffler to lead to more effective designs that are smaller, lighter and cheaper to produce.  They should also not degrade grouping ability, nor shift it excessively from bare muzzle.   Sometimes such changes are due to a shift in barrel harmonics and can be restored via velocity tuning.  This can happen with any muffler, but is generally worse with long heavy mufflers; so shorter and lighter is less likely to cause harmonic problems.   All this assumes the bores are not so tight as to contact the pellet, and that air currents inside the muffler are not going to steer or buffet the projectile to the point of grouping degradation.  Usually, the air stripping function of mufflers improves groups, so that is what we like to see.

Other than what happens to the air inside a muffler, the outer casing also generates and transmits sound directly to the air.   That is a related field of study that is important.  Currently, wrapping the casing in something like a foam rubber sleeve is a known method to avoid adding casing noise to air noise.  The casing material certainly matters.  I was informed that some of my insert designs that were printed in slightly flexible TPU were quieter and had a less sharp sound than the same mufflers printed in PETG.

Not going to mislead you, this is one heck of a stretch for me.  I don't know a thing about computational fluid dynamics.  But I know some things that might be transferable.  At the moment, I am just starting out with the tool and don't know how to use it yet.  I suspect I will have to watch about 10 tutorial videos before doing much.  And using open source tools always has the risk that it simply won't or can't do what is needed.  That being said, I will start watching the vids and hope to be able to use the tool in some basic fashion.  I self taught myself FEA, so maybe CFD won't be so hard.

Totally agree that the materials matter.  They will transfer the internal acoustic energy to the outside world by vibrating as a cylindrical speaker.  It may not be an efficient speaker, but it will make sound.  It makes sense that a material that is flexible and in a sense lossier would help.  Increasing thickness or material type would help.  If the material were denser, it would shift the resonant frequency higher.  But for that matter, so would adding stiffening ribs.  As I recall, you sometimes add them in.  They will shift resonant frequency higher.

[WobblyHand]

Regarding frequency translation.  This image https://www.airgunnation.com/attachments/with-jpg.394810/ was used to claim frequency translation of a particular moderator design.  The claim is technically not true.  What it does show is the moderator is a passive device, otherwise know as a passive acoustic filter.  In no frequency bin is there amplification for the moderated response as opposed to the corresponding unmoderated frequency bin input.  This makes sense as the moderator (the filter) is un-powered by either mechanical, electrical, or pneumatic inputs.  A passive filter device cannot amplify frequencies, it only can attenuate them.  Now the author of the post seems to have a LOT of practical experience in the subject area, but any claims of acoustic amplification cannot be true due to conservation of energy.

One method of designing microwave filters is called the high Z - low Z method, also known as stepped impedance.  It consist of high impedance and low impedance structures, be they transmission lines, or waveguide cavities.  If this were to be directly translated into acoustics it might look like cavities of some length and different diameters perhaps with apertures.  For a moderator to be practical this isn't helpful.  No one wants to tote around a 4" diameter moderator.  But, like in waveguides and other structures, we can do some magic to make it appear like it is wider.  It is sort of a virtual wideness.  As a matter of fact, it would seem to me, that angled cones and tesla like structures behave exactly like virtual widening of the cavities, allowing a more compact design.  There are other design techniques that would work as well, I don't want to bore you on basic filter design or microwave filter design.  There's also coupled resonator design methodology.  BTW, this is stuff that I learned around 1985 or so. 

So, if I were to continue the thought, perhaps the desired frequency response could be synthesized in an unfolded design at first and validated.  Perhaps then the folded design would be introduced.  There can be a blurring between the two, as obviously the unfolded design is undesirable.  But perhaps one could see if the general approach yielded any interesting results.

So, what exactly is the design goal?  Mouse fart quiet is not a technical filter design parameter!  Is there a filter response or rough attenuation that is desired?  Because that is how filters are designed.   Are we looking for a low pass, high pass, or bandpass type response?  Is 20 dB attenuation sufficient?  (Not aware of any that are that good.  Is there a moderator with 20 dB attenuation?)  What is a desirable cut off frequency?  In the beginning, we can just use WAG's for a starting point.  Anyways, this is interesting, at least to me.

Couldn't find my copy of Matthaei, Young and Jones, but found a link on the internet archive.  Not light reading...

[subscriber]

Bruce,

With regard to the chart you posted above; some of the charts from the same source; with and without moderators seem more convincing when it comes to the frequency shift claim.  It is not my claim, so I am not going to try to defend it, but I assume you noticed the scales are not the same on the Y-axis?  The charts that seem more convincing for frequency shifting are produced using very small volume mufflers for the power level of the shot.  Anyway, what you are looking to do does not hinge on such "shape shifting".

I am not usually impressed by a muffler producing less than a 10 dB drop, compared to bare muzzle, but that statement should be seen in context.  Reducing 120 dB to 110 is relatively easy.  A greater than 20 dB drop from bare muzzle is routing with PBs, but they go from eardrum busting to barely hearing safe.  So, an impressive drop, but still much louder than considered acceptable for airguns.

For backyard friendly airguns, I have seen very few moderators drop below 70 dB, and then the absolute accuracy of measurement comes into question.  At such a low dB readings, noise from the trap can easily dominate.   So, a drop from 90 to 75 dB seems possible.  But 80 to 65 dB seems very unlikely.

In other words, while we look at the drop in dB, we should try to define how quiet, is quiet enough.  If the reading is between 70 and 80 dB it is probably low enough.  Even around  90 dB for the moderated sound, the nature of the sound is often more important. 

People like to argue numbers, because sound quality is too subjective.  Even with only a 4 dB reduction, the nature of the sound can make all the difference:

At the video below; the first muffler on the Huben pistol is only 2.75" long.  If I remember correctly, it only offers a 4 dB drop over bare muzzle.  But compare the nature of the sound to the bare muzzle at the end of the video.  The latter has an echo, with a definite "gunshot" signature, while the short muffler produces a loud, but much more pleasant sound.   The second muffler in the video is longer than the first, yet it sounds harsher.  I can't explain that, and I designed both mufflers.



So, asking for muffler dB or dB reduction specs is actually difficult, because they matter in context with the nature of the sound, to me.  Despite many people caring a whole lot about one muffler winning by 2 dB over another.  Usually, it is because they are selling the quieter one.

I started this thread about muffler bore clearance because I wanted evidence to support what a few of us know already:  Making the bore a littler larger loses very little in performance; and it may actually result in a more pleasing sound.  Maybe even a little quieter in some cases - as discussed a few posts up.  If the terrible loss in performance was real, should we open the baffle bores by 1 mm, then no one would suggest shooting a .117 from a .22 muffler; or a .22 from a .25 muffler.  Clearly, that is a very common practice.

[WobblyHand]

@subscriber "but I assume you noticed the scales are not the same on the Y-axis?"

Indeed I did.  Which is exactly the reason I asserted that the moderator is a passive device and incapable of any amplification, only attenuation.  In all cases, the moderated frequency, bin by bin, is less than the un-moderated frequency bin.  The moderator attenuates the input by its transfer function, which is the plot of the frequency response vs frequency.  The scale of the two plots is different, which fools us into thinking there is amplification.  There isn't any amplification, which is line with conservation of energy. 

Had the plots been the same offset and scale, this would have been obvious.  If I had access to the point data, for both plots, I could plot it and show you.  The moderated output power for each frequency is less than or equal to the input un-moderated power for the corresponding frequency.  It has to be, it is a passive system!  Actually, if I had those points, I could plot the transfer function of the moderator.  It is just the difference (in dBs) between each and every frequency bin, plotted vs frequency.  The transfer function would be very interesting indeed.

BTW, had there been ANY amplification, it is only due to the statistical variation between the shots.  We all know shots vary.  If one did 10K trials of each, and they were averaged correctly (not in dB's) it would be extremely unlikely so see any apparent amplification.  Had you truly seen amplification, you would have invented a "perpetual motion" machine, which to my knowledge is not possible with any passive system.

[WhatUPSbox?]

From the chart posted, it looks like the moderator was most effective below 5kHz with the upper frequencies relatively unaffected. Not sure what a frequency shift would be. This looks like bandpass.

[WobblyHand]

I am not usually impressed by a muffler producing less than a 10 dB drop, compared to bare muzzle, but that statement should be seen in context.  Reducing 120 dB to 110 is relatively easy.  A greater than 20 dB drop from bare muzzle is routing with PBs, but they go from eardrum busting to barely hearing safe.  So, an impressive drop, but still much louder than considered acceptable for airguns.

For backyard friendly airguns, I have seen very few moderators drop below 70 dB, and then the absolute accuracy of measurement comes into question.  At such a low dB readings, noise from the trap can easily dominate.   So, a drop from 90 to 75 dB seems possible.  But 80 to 65 dB seems very unlikely.

In other words, while we look at the drop in dB, we should try to define how quiet, is quiet enough.  If the reading is between 70 and 80 dB it is probably low enough.  Even around  90 dB for the moderated sound, the nature of the sound is often more important. 

People like to argue numbers, because sound quality is too subjective.  Even with only a 4 dB reduction, the nature of the sound can make all the difference:

The language of filter design is desired transfer function.  Above you indicate some numbers like 70dB, which on the surface isn't yet meaningful to me.  Relative to what?  What was the bare muzzle reading?  Is this a single number, rather than a spectral plot?

At the video below; the first muffler on the Huben pistol is only 2.75" long.  If I remember correctly, it only offers a 4 dB drop over bare muzzle.  But compare the nature of the sound to the bare muzzle at the end of the video.  The latter has an echo, with a definite "gunshot" signature, while the short muffler produces a loud, but much more pleasant sound.   The second muffler in the video is longer than the first, yet it sounds harsher.  I can't explain that, and I designed both mufflers.

So, asking for muffler dB or dB reduction specs is actually difficult, because they matter in context with the nature of the sound, to me.  Despite many people caring a whole lot about one muffler winning by 2 dB over another.  Usually, it is because they are selling the quieter one.

I started this thread about muffler bore clearance because I wanted evidence to support what a few of us know already:  Making the bore a littler larger loses very little in performance; and it may actually result in a more pleasing sound.  Maybe even a little quieter in some cases - as discussed a few posts up.  If the terrible loss in performance was real, should we open the baffle bores by 1 mm, then no one would suggest shooting a .117 from a .22 muffler; or a .22 from a .25 muffler.  Clearly, that is a very common practice.

Perhaps if we could get a spectral plots of the pleasing vs non pleasing moderators AND the un-moderated input, we actually might be able to determine what the differences in transfer function might be.  Without any quantitative measurements we are reduced to waving out arms about.  There has to be something, some characteristic that makes something better sounding, right?  I'm still maintaining that a scientific/engineering approach can get us closer to the goal.

From a point of view of changing the clearances, I have no issue with what you are saying.  Actually I think the aperture size could have a very strong influence as it changes the acoustic coupling into the next chamber.  That means it changes the characteristics of the transfer function - which could make things sound better or worse.  In my opinion, there is no reason the aperture has to be either uniform, or tapered, it could have a pattern so the transfer function is more pleasing.  Besides the obvious limit of allowing the projectile to get through unimpeded, it could be larger than necessary for passage, and make the sonic qualities more favorable.

I have no idea if this is true, but perhaps more pleasing is simply a lower cut off frequency for the low pass filter.  I also think you are right, a couple of dB doesn't matter that much, IF the filter has a more pleasing shape (transfer function).  Although it wouldn't be bad if it had higher attenuation, and be pleasing.

[WobblyHand]

From the chart posted, it looks like the moderator was most effective below 5kHz with the upper frequencies relatively unaffected. Not sure what a frequency shift would be. This looks like bandpass.

I guess a basic moderator structure would be a high pass, simply because of the hole through it!  Didn't look at it carefully enough to plot any differences.
Can't imagine there is a frequency shift, it's not an active device.  The transfer function is probably lumpy, simply because it wasn't designed as a filter.

Could be a bandpass, though, but I don't think it is likely.  I would bet this filter is periodic in a sense and lets through ultrasonic frequencies.  In microwave structures like this, the frequency response repeats about some magic frequency related to the dimensions of the structure.  So there would be periodic stop and pass bands.  Most of the time this doesn't matter, since those frequencies are not of interest, or in this case, we humans simply can't hear them.

[WobblyHand]

Ok.  Math time.  If I properly understand the analogy, we can compute the cut off frequency of a circular wave guide as:

f_c = ( 1.8412 * c_s) / ( 2 * pi * r)  where c_s = speed of sound, and r is the radius of the inside of the waveguide.

Since the speed of sound is 343 m/sec, I will use the radius in m. 

Lets take as an example, ID = 25mm, so r = 12.5mm or 12.5x10^-3 m
I get f_c = 8040 Hz.  So the tube cuts off frequencies below 8040 Hz.  Basically there should be sort of a rough ramp, with the maximum attenuation at DC, up to 8040 Hz, where the sound is unattenuated by the tube diameter.  I will have to get back to you all on this, to let you know the slope of the curve.  I want to say 20dB/decade, but actually I don't know.  The info I seek is buried in an old college text book which I have yet to find.

The larger the diameter, the lower the cut off frequency.  Thus narrow diameter moderators "should" have less low frequencies, if we prevent the shell from vibrating.  Larger diameter moderators should pass more lower frequencies.  I am pretty sure this passes the reality test so far.  Small apertures will tend to pass higher frequencies, but fortunately, they are limited in their length, so some low frequencies do get through.

I am going to have to go pound the books for a while, I'm not smart enough to do any of this on the fly...

But I will say, the tube itself is a high pass filter, as it has a low frequency cut off.  This means the low frequencies are being selectively attenuated, but not high frequencies.

[WobblyHand]

Curiously, if the tube diameter = 12mm, the cut off frequency is 16.7 KHz.  Which is nearly out of hearing range.  Now that's an interesting thought!  Just a simple straight tube. 

I need to figure out the attenuation per unit length, might not be enough to be reasonable.  I don't think it will work, but hey, its dead simple to try.  Slowly getting there.

[TorqueMaster]

We've been doing it wrong all along.  I just tried a 1/2" copper pipe, and I couldn't even hear the hammer drop!

Of course, I'm kidding, I actually appreciate your trying to use established methods and apply them to our audio issue.  The closest thing I've dealt with are bass ports on loudspeakers -- "tuned" to allow more bass through than a sealed box would.  Not sure that is applicable.

I have a question when you say there is no amplification -- because these are only passive devices ---   If I blow through a soda straw, it is low amplitude white noise, more or less.  If I blow the same into whistle, or a flute, I surely get an amplitude increase in one of the "frequency bins."   Overall energy must be conserved, so must be other frequencies getting attenuated as well.  Is an unmoving hunk of metal or plastic (the whistle) "active"?

[WobblyHand]

We've been doing it wrong all along.  I just tried a 1/2" copper pipe, and I couldn't even hear the hammer drop!

Of course, I'm kidding, I actually appreciate your trying to use established methods and apply them to our audio issue.  The closest thing I've dealt with are bass ports on loudspeakers -- "tuned" to allow more bass through than a sealed box would.  Not sure that is applicable.

I have a question when you say there is no amplification -- because these are only passive devices ---   If I blow through a soda straw, it is low amplitude white noise, more or less.  If I blow the same into whistle, or a flute, I surely get an amplitude increase in one of the "frequency bins."   Overall energy must be conserved, so must be other frequencies getting attenuated as well.  Is an unmoving hunk of metal or plastic (the whistle) "active"?

Bass ports are applicable, I'd think.  As for doodling around with this, like you I have an interest in this (and a bunch of other weird stuff).

Whistles are interesting.  Actually, I know little about them.  I just went to Wikipedia and a lot of what's there isn't all that digestible.  Perhaps what you are saying is true, I'm not sure at this point.  It seems they rely on small apertures and chaotic flow.  These apertures or cavities will have high resonant frequencies.  But I am not sure if there is amplification.  If you blow weakly into a whistle it doesn't work that well, if I remember correctly. 

Now it is possible, if the cavity has high enough Q (quality factor) large fields (or sounds) can build up.  But they are still passive devices, but they were pumped over time.  Obviously can't get more total power than you put in.  However, if due to a high Q cavity, you can pump it with energy from the frequencies around it, it gives the appearance of amplification.  Saying it a different way, if you collapse the energy of a bunch of frequencies into one, yes the amplitude will be greater in that bin, but less elsewhere.  So perhaps for a whistle mechanism one could do some spectral management.  If the whistle is ultrasonic, and we were able to efficiently pump or excite it, then I'd think it was possible for energy to be transferred from audible frequencies to the ultrasonic.  I don't know if the process is efficient or not.  If not a whistle, I think I'd state at no time can there be an increase in spectral power between input and output.  Maybe the whistle is non-linear, which can act as a mixer.  But this is total speculation on my part, I just don't know.  Mixers can translate energy from one frequency to another. 

For a passive system, which includes moderators and whistles, the total summation of all the power contained in all the bins, has to be less than the total summation of power contained in all the input frequency bins. 

Yeah, I wish we were doing it all wrong.   Lots of people have worked on this, over, what a century now.  It doesn't help that details on this technology are not widely available.  So it's hard to just look stuff up.  Yes you can read some patents, but I mean engineering papers on the subject.

So we are left with copying original designs from last century, diddling around with variations, empirically trying stuff, or making feeble attempts to jump out of the box.  I'm only jumping out of the box because it seemed that maybe I could transfer some knowledge from a different field.  Maybe it's the wrong thing to do, or it won't lead anywhere.  But perhaps it is worth a try. 

[subscriber]

Bruce,

I was not very clear with all the dB numbers I offered:  The "70 dB" was not relative to anything, it was absolute.  So not a reduction of 70 dB, but the loudness of a very quiet muffler equipped airgun report.  Was that obtained by standard PB measurement method, 1 meter to the side of the muzzle, 1.5 meter off the ground?  Almost certainly not.

So, rather than provide dB numbers and then argue about the calibration of the measurement system, it helps to shoot a number of different mufflers side by side, and compare them to each other and the bare muzzle. 

As I am not the one doing the testing, I cannot control the set-up; only suggest it.  Most important to me is avoiding sound levels that saturate the meter, or are too close to its rate full scale capability to be "accurate".  So, when I suggest standing the meter off by 5 meters to avoid meter limitations, the absolute readings are going to be a lot lower, than if they were measured at 1 m with a proper meter.

I am playing catch-up reading the rest of this thread, so I wanted to clear up the obvious confusion I created above, first.

[WobblyHand]

This might not work, because I may have over looked something.  That's always the case for thinking out loud in public. 

A round waveguide pipe is a high pass filter, whose cut off frequency is determined by some geometric constants, the speed of sound, and the radius.  I have given the equation previously.  You can test out this idea, by holding a 6" long tube to your ear, say an LDC.  A narrower one will sound different that a wider one.  Both will sound a lot like noise, but the pitch of the larger diameter one will have a lower tone.  This is because the cut off frequency of the larger diameter tube is lower in frequency.  This means you will hear lower tones or lower frequency noise.

The power transferred is dependent on frequency.  Actually, if we plot the attenuation as a function of the log of the frequency, we get a straight line, that has a slope of 20dB/decade.  The attenuation goes down as the frequency goes up.  At the cut off frequency, the attenuation is 3 dB.  After that, the attenuation asymptotically approaches 0.

Now, what happens if we open up the bore diameter to a diameter d1 that is larger than the bore diameter, for a length L1?  We get a low pass filter response for that section of the filter.  What does a low pass filter do?  It lets through low frequencies up to its cut off frequency and attenuates frequencies higher than the cut off.  It also has a slope of -20dB per decade, the opposite of the high pass filter.

If we cascade these filters, the composite response is just the addition of the two transfer functions (in dB's).  We now have a bandpass filter.  Pretty much like Stan said.  Now I found some equations, that I still need to prove to myself are "good enough", but I put together a program that plots the expected response.

Here are two graphs.  I'm not saying this is realizable or practical at the moment, but it is something that at least remotely makes sense to me.  If we have a tube with an OD of say 30mm and have a bore through it with an ID of 12mm, the HPF (high pass filter) cut off is 16.8KHz.  If we then carve out a section in the middle of the moderator and open up the diameter to 25mm for 65 mm, the low pass filter (LPF) that is formed has a cut off frequency of about 500Hz.  If we combine the two filters the minimum attenuation is expected to be 31 dB.  I don't think there is any harm in putting the LPF in the center of the LDC, with the narrow bore tube leading to it.

If we do the same experiment, but instead we have a bore of 8mm, rather then 12mm and changing nothing else, the cut off frequencies for both filters change, they get further apart.  Guess what, we get 41 dB of attenuation!  At the very least it seems promising.  Anyways, that's my days work on this subject.

[WobblyHand]

Bruce,

I was not very clear with all the dB numbers I offered:  The "70 dB" was not relative to anything, it was absolute.  So not a reduction of 70 dB, but the loudness of a very quiet muffler equipped airgun report.  Was that obtained by standard PB measurement method, 1 meter to the side of the muzzle, 1.5 meter off the ground?  Almost certainly not.

So, rather than provide dB numbers and then argue about the calibration of the measurement system, it helps to shoot a number of different mufflers side by side, and compare them to each other and the bare muzzle. 

As I am not the one doing the testing, I cannot control the set-up; only suggest it.  Most important to me is avoiding sound levels that saturate the meter, or are too close to its rate full scale capability to be "accurate".  So, when I suggest standing the meter off by 5 meters to avoid meter limitations, the absolute readings are going to be a lot lower, than if they were measured at 1 m with a proper meter.

I am playing catch-up reading the rest of this thread, so I wanted to clear up the obvious confusion I created above, first.

I pretty much knew the moderator didn't offer 70dB attenuation, at least not in the pass band.  This is why I asked, what was the unmoderated value.  From those two values we can determine attenuation.  And I agree with most of your perspective on testing.  Can't saturate the meter!  Anyways, I made a little progress on my ideas, as shown in the previous post.  At least there is some kind of indicator that sort of points in the direction to head.

[TorqueMaster]

Ok, you've piqued my curiosity.  What you've described is braindead easy to implement.
OD 30mm
Chamber ID 25mm, Chamber length 65mm
Bore ID 12mm
Are the lengths of the bore ahead and behind the chamber important or will any length work?   I just used 50 and 50 with overall length 165.

Do the chamber ends need to be exactly as shown, flat, perpendicular to the bore?  Does the exterior need to be cylindical, or would it work just the same if these cavities were inside a long square block, for instance?  My reason for asking is to simplify or overcome 3D printing limitations.

[subscriber]

Bruce,

A number of your statements have prompted answers and ideas; pasted and addressed below:

Perhaps if we could get a spectral plots of the pleasing vs non pleasing moderators AND the un-moderated input, we actually might be able to determine what the differences in transfer function might be.

This is a great idea.  Bob (TorqueMaster) has several spectral analyses of different muffler sounds.  It might take him some digging to find good examples.  From my perspective, a pleasing sound produces not just a lower amplitude full spectrum magnitude/time chart, but it starts with a taper, rather than at peak amplitude.

In my opinion, there is no reason the aperture has to be either uniform, or tapered, it could have a pattern so the transfer function is more pleasing.

If we look at the exhaust cones on the latest passenger jet engines, that V-notched shape may be inspirational.  That shape apparently reduces noise pollution considerably - by producing a less abrupt shockwave, rather than filtering out noise. See attached images. 

I will try to conjure something meaningful that incorporates the V-notch concept, that is also 3D printable.  Speaking of printable, flat baffles are tricky to print, so I use cones in my designs.  I am convinced of the effectiveness of progressively slanting cones, in managing air flow as the bulk pressure decays, based on lots of test results - much of which Bob TM produced; followed up by happy customers buying prints of my designs, many as printed by Bob.  Many printed by interested parties for their own use. 

I have it on good authority that TKO mufflers use flat baffles, to very good effect.  So there is at least one counterpoint to using cones.  When it comes to "air stripping" turbulent air from the pellet path, cones are deemed superior to flat baffles.  I presume this "common knowledge" is backed by test results and optimization.

Can't imagine there is a frequency shift, it's not an active device.

If I bang on a drum (skin), twice per second, what is the frequency of the sound coming off the drum?  If it were 2 Hz, then a small drum would not sound different from a large one; yet they clearly do. 

My point is that a muffler does not just filter the sound of a bare airgun muzzle blast; a different sound is generated inside the muffler, by each abrupt expansion event that occurs in successive chambers, starting with the first chamber.  All of those thumps are applied to the air inside, making it ring at its natural frequency.  Ditto for the structure of the casing.  Some of that casing vibration is re-applied to air in surrounding chambers.  The baffle bore communicates between chambers potentially exciting or damping the other chambers, by virtue of the degree of phase shift between them.

Even if all the chambers are identical, the fact that the bulk pressure decays from inlet to outlet means the natural frequency of each chamber is lower than the one before it.  The natural frequency of all chambers  drops as time passes, due to the drop of the bulk air pressure inside over somewhere around perhaps 1/3 second; with perhaps the first one or two 1/10 of a second being the most significant from a sound signature perspective.

Muffler that drive deliberate turbulence to slow down the exit of air from the front end, probably produce a lot of white noise, or hiss.  So the model for what happens when air has pressure waves bouncing between walls, may not apply to high turbulence flow device.  Or not apply to the same extent.

[subscriber]

Starting with the 6.5 mm bore muffler attached to the OP, I did the simplest design modification I could think of, to emulate the jet engine V-notches seen at the turbine and bypass fan exhausts, in the images above.

The new V-notched baffle bores span the (as printed) ID of 6.5 mm at the sharp tips, reaching 8.5 mm ID that the rounded notch "roots".  So, spanning the total 2 mm baffle bore diameter range of the three mufflers in the OP, but having a non-circular exit.

There are 9 notches at each baffle bore, so that none of the facets are parallel - as would be the case for an even number of facets.  See images below.

There are two variants to this design.  V1 has the same 1.2 mm wide flat annular ring on the leading baffle face, as the mufflers in the OP.  V2 extends the baffle forwards to a sharp edge (to a point, after the notches are cut). 

V2's baffle bore leading edge shape will be limited by the slicer and printer, in terms of how sharp those points will be, but the flat surface of V1 is in the same area is avoided.  The leading edge of a baffle acts as the exit of the previous baffle chamber, so while the scale of these jagged details is small, the effect may or may not be significant.

I predict that V1 and V2 will produce more hiss than the original set of three mufflers, but will not outperform them in dB or sound quality.  If, on the other hand the V-notched baffles provide a signal that suggests an improvement, then trying to optimize that will be worth the effort.  Up till now, the closest I have come to aircraft engine nozzle notches are V-structures I use as permanent printing supports for overhanging aspects of a design - see last image for that.

The STL files for both V1 and V2 are attached in one ZIP file, below.

[WobblyHand]

Ok, you've piqued my curiosity.  What you've described is braindead easy to implement.
OD 30mm
Chamber ID 25mm, Chamber length 65mm
Bore ID 12mm
Are the lengths of the bore ahead and behind the chamber important or will any length work?   I just used 50 and 50 with overall length 165.

Do the chamber ends need to be exactly as shown, flat, perpendicular to the bore?  Does the exterior need to be cylindical, or would it work just the same if these cavities were inside a long square block, for instance?  My reason for asking is to simplify or overcome 3D printing limitations.

What you have shown is the idea I modeled.  As I mentioned, it's not clear that I have done all my homework on the correctness of my sources, ie I found some stuff on the Internet. 

I don't think one has to have square edges exactly.  Perturbation theory would indicate that minor chamfered or filleted edges wouldn't change the answer much.  Unfortunately, at the moment I don't know how to model that analytically.  The only way would be to simulate it numerically, and I'm not yet up to speed there.

The only caveat I can think of is that the main tube impedance is not matched to the LPF.  That would take some work to verify.

I was going to print something quite similar, since it's dead simple.  Ran out of time yesterday doing research, analysis and chasing down math errors.

[WobblyHand]

Ok, you've piqued my curiosity.  What you've described is braindead easy to implement.
OD 30mm
Chamber ID 25mm, Chamber length 65mm
Bore ID 12mm
Are the lengths of the bore ahead and behind the chamber important or will any length work?   I just used 50 and 50 with overall length 165.

Do the chamber ends need to be exactly as shown, flat, perpendicular to the bore?  Does the exterior need to be cylindical, or would it work just the same if these cavities were inside a long square block, for instance?  My reason for asking is to simplify or overcome 3D printing limitations.

I neglected to answer one of your questions.  The derivation only applies to circular cylinders, not square or rectangular. 

Something else that I ignored for this is the possibility of multiple propagating modes in circular guides.  Might need to add some features for mode control, but hey it's rather early in the design cycle for me.  I thought it was exciting to start to see a design moving in the right direction.  It's expected that some iteration may be required.