On a 6kHz Helmholtz Resonator for the Sennheiser HD800

On a 6kHz Helmholtz Resonator for the Sennheiser HD800

Firstly, props to sorrodje for his work on the SuperDupont Resonator mod and demonstrating its positive effects, getting there by trial and error no less. Reading through the mod threads, there seems to be a bit of confusion over what manner of device the SDR is. It is in fact a Helmholtz resonator.

In going over various resources to better understand the mod and what Sennheiser may have done with the HD800S in terms of its “absorber”, it seemed to me at first blush that designing a Helmholtz resonator for the HD800 to mitigate its 6kHz peak boils down to a simple optimization problem. We have an equation that gives the dimensions of a Helmholtz resonator provided a desired resonance frequency, we just have to identify constraints and what to optimize, no? Unfortunately, uncertainty over how to account for some factors prevents easy application of the formula. I’ll rehash some of the info so we’re all on the same page, then go on to discuss some of the salient issues. These are basically my notes from the bit of research I’ve been doing on the subject. My hope is that this would be interesting for others and helpful in guiding further experimentation. Some links to start:

https://newt.phys.unsw.edu.au/jw/Helmholtz.html
https://en.wikibooks.org/wiki/Engin...control_with_self-tuning_Helmholtz_resonators
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/cavity.html
http://www.audioholics.com/room-acoustics/helmholtz-resonant-absorber


The Helmholtz Resonator

While Sennheiser hasn’t explicitly stated that a Helmholtz resonator is what has been implemented in the HD800S in what used to be the empty space at the center of the grill and ring radiator in the HD800, the absence of any structures other than the donut obstruction would strongly suggest this. Reason being there doesn’t seem to be another way of realizing an acoustic notch filter with what can been seen in the HD800S. Side note for Tyll: with an air permeable bottom, the device would no longer be a Helmholtz resonator if I'm reasoning this correctly. Imagine a spring mass system where the spring isn't anchored to anything. A Helmholtz resonator can be implemented in the form illustrated below on the left.



Of course, you’ll note the resemblance between the figure and the SuperDupont Resonator. Key features: 1) a sealed cavity of air connected to the outside via 2) a port in the form of a neck, though an aperture may be enough (will get to this later). The enclosed air is analogous to a spring mass system (air in neck corresponds to mass, air in cavity corresponds to spring) with a resonant frequency determined by the size of the two air masses. When this air is excited by sound at its resonant frequency, energy is transferred into the system and stored in the form of oscillation of the air masses. Here’s a video:



Since Sennheiser seems to have implemented a simple single port resonator with cylindrical neck and cavity, it makes sense for us to go with that for a DIY resonator mod. And of course sorrodje has already shown such a route to be effective. Fortunately, equations are readily available that describe the behaviour of a Helmholtz resonator given its geometry. Of primary interest is this one for the resonant frequency.



c = speed of sound in air = 343m/s @ room temp
A_n = cross-sectional area of neck
V_c = volume of cavity
L_n,eff = effective length of neck with end correction (https://en.wikipedia.org/wiki/End_correction)

I also like the illustration at the top of the HyperPhysics page that spells out the relationships in qualitative terms.



For the general equation to hold, the dimensions of the Helmholtz resonator needs to be smaller than the wavelength of the sound we’d like to affect. This is so that the volumes can be treated as lumped parameters. Also, the dimensions of the neck and cavity individually should be less than the quarter wavelength of said sound to minimize standing wave effects in the resonator. Take our case where we’d like to attenuate sound about 6kHz, this would mean our Helmholtz resonator should have dimensions smaller than 57mm, and neck/cavity should have dimensions smaller than 14mm. Since we’re fitting a Helmholtz resonator in an empty space with a diameter of 14mm and height of 7mm (thanks sorrodje for these figures posted in the mod thread), these conditions are satisfied.

For a cylindrical neck flanged (infinite baffled) at both ends, add twice 0.85 times its radius to get its effective length. For a Helmholtz resonator sized for the HD800, end correction can contribute significantly to the effective length of the neck, hence the comment earlier suggesting an aperture may be enough. Looking at Tyll’s close-up photo of the resonator in his HD800S review, it would appear that the port is very thin so it’s possible this may be the case.

Digging In Deeper

Knowing the above, it seems like we can use a spreadsheet to easily calculate dimensions of the 6kHz Helmholtz resonator that we want (h_n + h_c should equal 7mm, d_n < d_c ≤ 14mm). One example below.



So what are the optimal dimensions to maximize the efficacy of a Helmholtz resonator given our space constraints? My understanding is, intuitively, we’d like to maximize the cross-sectional area of the neck to capture as much sound wave pressure as possible and also the enclosed volume to maximize the amount of air that’s absorbing energy at the resonant frequency. I’m not certain that’s all there is to it, but again looking at the resonator in the HD800S, it’s possible that this is a reasonable approach. Partway through the wikibooks page, there’s a section on the effect of resonator volume on sound attenuation that’s relevant. Excel has a Solver plugin that solves optimization problems which I tried. Unfortunately, it tends to converge to the solution where both r_n and r_c equal 7mm and we no longer have a Helmholtz resonator. Seems like we’ll have to impose other constraints so we have a valid solution?



Now let’s step aside and examine the SDR for a second. Plugging in the SDR dimensions (d_n=6mm, h_n=2mm, d_c=10mm, h_c=5mm) into the equation gives 5.5kHz as the predicted resonant frequency. If we were to increase the diameter of the creatology foam ring (neck) by 1mm to 7mm without changing anything else with the SDR, the predicted resonant frequency would be 6kHz. I understand that sorrodje has already tried this since the diameter of the port in the HD800S absorber is also 7mm, but it didn’t result in any measurable differences. This most likely is because the present formula doesn’t accurately represent the Helmholtz resonator at the dimensions we’re working at.

One possible problem you may have noticed is that the cavity end of the neck in our desired Helmholtz resonator would be more of a step than a flange. Another possible problem is that the cavity in our Helmholtz resonator is relatively small while I suspect the end correction coefficients found in the earlier links are for pipes opening into (effectively) open air. Yet another possible problem is that the coefficients may not be constant wrt frequency. It turns out there may not be an accurate end correction coefficient number readily available for the cavity end of the neck in our desired Helmholtz resonator. Paper (a) linked in the Further Reading section below describes some relevant investigation. Interestingly from the literature I’ve been glossing, most of these end correction coefficients are determined by experimentation. I think we may have to hereby scrap the notion of designing our desired Helmholtz resonator based on the formula. At present, we can use the numbers we get from it as rough indicators of whether certain dimensions are suitable starting points for our own experimentation and that’s about it.

Another design uncertainty is optimal damping. Further down the wikibooks page is a section on the effect of resonator damping on sound attenuation that’s relevant. The Helmholtz resonator is inherently damped so its oscillations die out unlike in the spring mass system. Damping occurs due to losses from reradiation out of the port, viscous losses from the air vibrating in the neck, and thermal losses from the air expanding and contracting in the cavity. The HyperPhysics page claims a Helmholtz resonator is “highly damped” but I haven’t found any quantitative examples that would give a sense of what this means. Apparently, losses due to reradiation dominates such that the other two can be neglected. This means we shouldn’t have to worry about the materials used to make our Helmholtz resonator and can use whatever is easiest to work with. Further, the driver screen and dust cover most likely contribute significantly to damping so ultimately I should think we need not worry about tweaking damping. Though Tyll did note in his mod video (3:48) that the screen may be different in the HD800S so Sennheiser may have done some tweaking in this regard.



Papers (b) and (c) linked below go into further detail.

Next Steps

Try replicating more accurately the HD800S absorber and see what that gets us. This involves sealing off the outer end of the center empty space and sticking a donut of x thickness with a 7mm diameter hole in the empty space to the underside of the screen. Optimal thickness to be determined by experimentation.

Following from there, according to paper (a), we can assume the end correction coefficient for the cavity end of the neck in our desired Helmholtz resonator is less than 0.85, probably close to 0.55 if I’m interpreting section II.C correctly. Knowing this, we can go about pinpointing the proper coefficient number. This can be done by varying the diameter of the donut hole from 7mm to say 13mm in say 1mm increments and finding the optimal thickness for each diameter, then doing some calculations. Of course in the process of doing this, an optimal Helmholtz resonator would be found. I have my doubts about whether such a resonator would be much more effective than the SDR however.

Further Reading:

(a) http://www.jaist.ac.jp/~jdang/DangLabHomePage/JAS01075.pdf
(b) http://www.acoustics.asn.au/conference_proceedings/ICSV14/papers/p406.pdf
(c) http://www.mypolyuweb.hk/mmlcheng/assets/pdf/081.pdf
(d) http://www.acoustics.asn.au/conference_proceedings/INTERNOISE2014/papers/p193.pdf

 

Holy shit, this is awesome. Thank you so much for this.
A lot of what you said is also what I noticed in my measurements (closed HD800 hole resonates at 12.3khz, SDR probably at 5.5-6khz, dust screen (and probably dust cover even more so) damping the effectiveness of the resonator, ...)

 

I've made a couple of additions to the part about the resonant frequency formula for completeness.

 

Great work! I would just like to briefly point out that these types of resonator calculations were done (at least by myself and I am pretty sure @sorrodje too) in the process of going through different iterations in the SDR thread. I have (and again I'm sure SD has as well) gone through all of the suggested geometries that you have listed. I am not at all saying that we have exhausted all of the possibilities though. I would definitely still like to try more materials that have different absorption/reflection properties. There was a limit to the geometries I have tried as well due to the tools and materials I have at hand. Also if there are any of these geometries or designs which you would like to see some measurements or comparisons for simply let me know and I would be happy to provide them. Thanks for going into all of this research. The end correction factors and flange type factors are one area that I have not looked into yet.

 

Very nice write-up!

Some questions which I've been asking myself:

1) Is there any explanation you can think of as to why the HD800S has more second-order bass harmonics (distortion) than the HD800 and HD800 modded?

(See Tyll's measurements: http://www.innerfidelity.com/conten...uland-french-diy-response#vjBsPSCiHyXrbUzQ.97)

2) If one took a regular plastic sticker, cut it to be round, punched a hole in with the right size and stuck it over the center of a HD800 driver would it adequately imitate the HD800S? Is that really all there is to it? The material doesn't really matter that much but the size has to be right?

Once again, nice write-up!

 

Corrected the side note for Tyll. The original quoted below applies to an air impermeable but flexible bottom, not as written.

Thanks all for the feedback, glad people are finding this useful.

Yes, I assumed you and sorrodje would have done some of these calculations. It's unfortunate that the formula most likely doesn't give accurate predictions due to unsuitable end correction coefficients.

Have you already tried not lining the cavity with anything so that it has a maximum diameter, ie 14mm, thus volume, then varying the thickness of the donut with 7mm diameter hole (first part of Next Steps)? I'm thinking cork would be a suitable material to use since it's easy to vary the thickness very finely. I really need to buy back that HD800 I sold when news of the HD800S broke so I can try some of this.

And I do like how the current SDR is constructed as it makes it easy to pop it in and out of the hole. This necessitates the felt ring cavity wall though so it's a trade-off, assuming increasing resonator volume does in fact make the resonator more effective.

1) No idea, but I think Tyll's right in saying Sennheiser did this purposely to increase perceived bass. Should be possible to tweak the construction of the driver so that it produces more 2nd order harmonic distortion.

2) The thickness is critical as well, so you may try layering stickers on top of each other until you get the best result to your ears. And of course you need to seal off the outer end of the center empty space.

 

Double posting to add a separate thought. Has measurements been done for a HD800 with center hole completely blocked on the inside end? This is essentially what we're doing with these mods after all.

 

Thank you for posting. I think this explains the acoustics of the situation very simply and thoroughly. This is how members gain "Friend" status

 

Yes I have tried putting no sort of lining inside the resonator with only a 6mm hole in foam on the top and then a solid piece of foam on the back side of the driver.
Green = lined with felt, Blue = unlined cavity


The green line does not represent either the official SD mod or my final version of it and both the official version and my version achieve slightly better results due primarily to a different type of felt used. Still the blue line pictured does have a slightly lower SPL at the peak than even my current version or the official SD mod (only very slight but it is a repeatable result). However when I was listening to it I did not like it quite as much and found it not as effective. I was wondering if it had to do with the smoothness until I looked at the CSD plots.

Felt lined CSD


Unlined CSD


Aha! Without the interior lining to dampen the cavity there is still a resonance which can escape. Also it is important to remember that felt is partially acoustically transparent so it does not necessarily reduce the volume of the cavity in quite like you might initially think.

I have not played around with the thickness of the opening or "donut" too much and this would certainly be an area for further discovery. However based on the above measurements as well as others I can tell you that at least based on what I have tried so far both the materials used and the interior damping are quite important.

 

i'm still baffled why it took a super giant headphone company years to figure out on simple idea scheme like this....i would assume it would've took plain someone two seconds to figure it out there. guess not.... i don't get all the math written but, i understand.. i would guess simple acoustic imagination would been simple solution. really do appreciate the write up though. it's interesting.

 

I was under the impression that the increased bass distortion was exclusive to the resonator in the HD800S, based on the fact that the SDR did no such thing to the distortion in the HD800. I do remember reading somewhere about speculation as to how it was achieved but I can't remember where nor how off the top of my head.

Well that would be assuming that they began working on it right away. Plus they would have looked at other options before finalizing the decision to go with a resonator in the centre hole of the driver, though it seems like such an obvious answer in hindsight.

 

Who says it took them that long? It's only an assumption that they were working away at it for years. I don't think they saw it as a "problem that needed solving". The engineers no doubt had the skills, maybe someone just needed to ask them? A trouble with big companies is the number of organisational layers between management, marketing and engineering.

 

Asymmetrical LF displacement, perhaps?

 

Do you mean the bottom of the hole blocked off?
Yes, you get a big dip at 10khz or higher. Searching for those atm.

EDIT: I can only find these right now:

I assume this is with the hole covered from the outside, instead of on the inside. I thought the dip would be quite a bit higher in frequency. You mentioned it being at 12.3khz, which is much more like what I expected. Might have to redo it with the hole covered on the inside.

 

You know what, I've just removed the remark about the center hole resonance as it's likely erroneous. That calculation was done without end correction (oh the irony) and measurements don't bear out.

Hmm, I knew I wasn't being clear enough there. What I meant was the effective starting condition when implementing any resonator mods is not a stock HD800 but this below:

So it would make sense to use measurements of the HD800 set up like this as reference instead of existing measurements. My last statement in that post is just alluding to the fact that at frequencies other than near/at the Helmholtz resonance, the system is effectively as shown above.

About your DIY resonator, I really have no idea. Perhaps the spaces between the filled straws have a closed pipe resonance near 6kHz and is being damped by the foam?

And the thickness of the foam ring is 2mm per the SDR correct?

Using the (inaccurate) formula to calculate for the resonant frequency does place it at about 4kHz so it's not surprising you're getting those measurements.


Do try playing around with the thickness in conjunction with the hole diameter of the foam ring since it makes up the neck of the Helmholtz resonator and adjustments to its dimensions will significantly change the HR's resonance characteristics as I've tried to convey in the OP.

Some ideas:



About materials, I have little experience and reading on the subject so will defer to what you've all observed in your experimentation.

Unlikely to be due to the resonator, Sennheiser must have done something to the magnet/voice coil/diaphragm, most likely the voice coil.

 

Knew this would be the case. I don't think I have the measurements of a completely blocked hole anymore. I might have to take the HD800 out of its box again after not using it for 2 days. I also think that the metal mesh is very permeable at such high frequencies, so that we can probably think of it as an open hole (at these frequencies).

 

Thanks, looks about right after accounting for end correction, ballpark at least.

Cheers, that would be helpful indeed.