Mathematics of the Transmission Line Speaker Enclosure:

Derivation, Simulations & Recommendations

This system features a Morel MW266 woofer in lightly stuffed TL  tuned to 27 Hz and a Morel CAT378 tweeter. 25Hz-20KHz range with very warm, rich bass, creamy midrange, extremely detailed high end and outstanding impulse response.

The Transmission Line is a seldom-used speaker design which is preferred by many audiophiles. Fans say TL’s sound less resonant, less boxy, more open. I’ve built several, one very recently. I don’t think TL’s are necessarily more “accurate” – they do have flaws. However they have good impulse response as they are in fact less resonant. I prefer the sound of a TL to a ported box.

In 1990, when I was a senior in college, I was frustrated that the kind of mathematical models that are available for acoustic suspension and bass reflex boxes were unavailable for Transmission Lines. I was taking an acoustics class, so for my semester project, decided to derive the math myself.

This is that paper, very slightly cleaned up and digitized, otherwise unchanged. I did not find this project to be at all easy. However when I was done I greatly enjoyed the satisfaction of knowing that I had started with basic physical properties like the mass of a speaker cone and the density of air and derived the entire response of a complex system. And that I had done something that nobody to my awareness at the time had done.

As you will see, TL’s are complicated. They can’t be reduced to simple differential equations like normal boxes can. In this paper, I ran all my simulations on a Hewlett-Packard 28S calculator. I would have loved to have been able to run them on a computer, but I didn’t have access to the right software.

I came to several conclusions that reinforce my personal experience from building transmission lines. I also explore closed TL’s, which have received very little attention in the literature.

I welcome anyone who wants to apply my models on a computer, math program or online calculator. Enjoy this paper.

 A Derivation and Analysis of the Transmission Line Speaker Enclosure

I posted this in Martin King’s Transmission Line Discussion Group and Martin responded:

Your paper brought back a lot of memories of my early TL studies. Your paper was well written and laid out the TL theory as it stood when I started in the mid 80’s and continued working with it up to the late 90’s as computers and measurement systems started to become more accessible to the speaker builders.  The math and theory set forth by Bradbury, based on Bailey’s tests data, was so elegant and attractive I spent years working out the details and programming the solution. I worked on computer codes written in Basic on an old PC XT clone and produced curves similar to yours, unfortunately I could never get the models to correlate with measured designs I found in magazine articles and AES papers or my own measurement data.
By the late 90’s and early 2000’s independent TL developers had built and tested TL speakers (in my case a cardboard tube with stuffing) that showed Bradbury and Bailey were wrong. The fibers did not move and the speed of sound was not reduced significantly. New computer models were developed that correlated well with test data and more builders starting making TL derived enclosures. The calculations were very accurate and designs were built with some confidence that they would work as predicted. The old school moving fiber theorists fought against the newer computer models on the discussion forums but as the years passed they have become less and less vocal and I believe now most speaker builders have moved past this incorrect theory.
If you are still interested in TLs, I think your computer model could be updated with a better fiber damping correlation and be pretty accurate for a constant cross-sectional area TL.

Martin is correct, my assumptions about absorption have been replaced by a better understanding. The best source I know of for modeling transmission lines is Martin’s site http://quarter-wave.com/. Martin offers a free MathCad computer model that accurately predicts system response.

Related:

The Ultimate 2-Way System

My product reviews at Parts Express (which detail significant aspects of some of my projects)

Archived Comments

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  • Faustin Baron says:

    Just finished reading your “Mathematics of the Transmission Line Speaker Enclosure”. I didn’t go through the math, stayed with your thoughts and conclusions. Interestingly, when Martin first started posting his transmission line information we started to email each other. After much time had passed I asked him how he became interested in transmission lines. He replied after hearing and being impressed with a pair at an Albany State professors home. I turns out that the transmission lines were a pair that I had built and sold to the professor and that, in fact, Martin lived near me. Many years before I had been building transmission lines from Sonotube and passed that information along to Martin. We were already using Sonotube for our experiments about 10 years before, with lines as long as 14 ft. Martin had asked me if I thought or observed the fibers moving and I did as I think you described. What is critical is when using wool as a stuffing that it is well teased, using a hacksaw blade. At that time my thinking was to build the line as long as was practical while focusing on the resonant frequency. This always worked well and the lines were always turned out well. Eleven feet seemed to be ideal for the Focal drivers that I was using. I built much shorter lines that others were happy with, but the longer lines I thought always performed better. Years later Martin designed a line for me based on another set of Focal drivers and I was not happy with the performance compared to magic of my longer lines. Before emailing with Martin I had built an 11 ft line with compound loaded drivers, the drivers were facing each other, and that was clearly superior to the single driver in the same line. At that point I was only using this in a subwoofer application. Since then I have built a few full range systems. With those speakers I use sealed box woofers to capture the bottom end, in particular the low end of the old Celestion SL6s. While the bass/midrange drivers in the Celestions are small, the bass extension is quite noticeable compared to running the Celestions full range. Perhaps comparable to the idea from Magnepan where they market a bass driver to go with their full range speakers rather than a subwoofer. Incidentally, years ago when Jim Winey from magnepan was contemplating building a tonearm he and I discussed it over the phone and he incorporated my ideas for the counterweight when the arm went into production. My observation is that using a woofer/midrange on the bottom gives a better defined and cleaner low end. My experience is that I can’t mate an open ended midrange transmission line to an open ended bottom. IMF with their Monitor and Studio speaker had their midrange in a closed end transmission line and the bottom in an open ended transmission line. As I recall, IMF used some fiberglass in their line, but weaved it in strips through dowels, essentially extending the length of the line for certain frequencies. If I use the Celestions (sealed cabinet) full range with a transmission line bottom it works out as well, provided that I don’t cross over too high. Thanks… Faustin

    Reply
    • Perry says:

      Thanks for your note. Martin has made a tremendous contribution and I LOVE his work. Can you pass the following comment to him, since I don’t know him personally? I suggest he transfer his transmission line math program into an entirely web-based application and make it a membership site that costs $9.95 per month – and allow people who use it to use the designs commercially. I was not willing to use it because in order to do the whole TL math program thing I had to use a Windows machine. I hate Windows. Plus I would have had to learn the math program too. I think it would be a small but viable business, it would make him a few thousand dollars a month. My 2 cents as a very experienced marketer.

      Reply
  • William Duncan says:

    Dear Perry, Looking to make some speakers for our school – see url and came across your article on the two system using the BG RD 75 planars. Do you have any more details. Very nice impulse response – I thought line arrays were supposed to suffer from non-linear impulse responses? See Design Guide for Practical Near Field Line Arrays by Griffin…

    Love to hear more,

    Thanks

    William

    Reply
  • PAOLO PETRINI says:

    Hallo, where could I buy the B&G RD 75 ribbons ????, Parts express does not sell them anymore, I have been considering for 8 years to make a two way elec x overed two amped system—thanks

    Paolo

    Reply
    • Perry says:

      Contact the manufacturer?

      Reply
  • David Michno says:

    Hi,
    I worked with David Graebener as a mechanical engineer for BG back in the day and developed that 75″ driver with him. BG is now out of business. I may know of a small supply of the 50″ and possibly a couple 75″ drivers plus many various sizes of neo planars. I am not a bussiness and this comment is meant not to be self promotional so not sure how you are to contact me if interested. After working for 3 planar driver manufactures I have a number of planar drivers/knowledge hanging around doing nothing.

    Reply

16 Comments on “Transmission Line Speaker Calculation”

  1. Hello all
    I have been involved with speaker building and repairs for a very long time, say 50 years or so. Yes like many of us it started with salvaged TV consoles and such, the speakers pulled out and played with. But later in life I started the base reflex style using the manufactures specs like the Fs value, cabinet volume, port diameter and a chart of general specified lengths. Well it worked well. Then yes the TRANSMISSION LINE type. The tunnel size first 25% was 1.5 cone area as well as the last 25%, and the middle was the same as the cone area. The length was 3360″ (general constant used), divided by the Fs of the woofer = 1/4 the wave length of the Fs. Padding all accept the last 25% exit path.

    Yes this worked for musical instrument and DJ type speakers. At least I noticed as did all my friends, the difference in the smooth sound.

    I’m still playing today with it in my retired years, rebuilding, repairing old vintage speakers in good shape when ever I find them. Modern day technology has changed the game quite a bit and so I rely on the manufacturer to fill in the gaps.

    You all take care and happy building

    1. Thanks for your post.

      Hey you know what I’m looking for? A 1982 McGee Radio catalog that someone would be willing to part with. Let me know if you know where something like that can be found.

  2. I had a good friend of mine introduce me to the concept of a transmission line speaker back around 1977. According to my friend, he had built a set of transmission lines speakers using 1-10” KLF speaker, 1-6” mid-range and 2- 3” ring dome tweeters. I recall the speaker box dimensions approximately 42” (H), 18” (D), 14” (W). I don’t have the exact baffle design notes, but I think the transmission line length was approximately 115”. The cool thing is that his father had access to RCA labs acoustics chamber and my friend had the opportunity to test his speakers. The test results were very impassive, great frequency response and extremely efficient regarding power vs. sound output. The only thing that was intimidating is that his father had a massive wood shop at home and was able to design very complex wood projects. In other words, to put the transmission line speakers together, the error factor in the box was no more than 64th inch per the entire length of the box. The wood working precision was extremely difficult to build and very time consuming at best. I believe he took 4 Months to build working every day at least 3 hours on average.
    During the mid to late seventies, I was very Blessed to have a friend whose father was a skilled wood worker craftsman and made custom cabinets and furniture and had over 40 years of experience. I was able to spend at least 4 to 5 years working and learning woodworking skills from his father.
    Back in 1978, I decided to build my own set of transmission line speakers and tried to replicate my friends design. It took me 5 Months to build my speakers in my father’s garage. I unfortunately had no access to a sound lab to test my speakers. One good factor was that I was able to design my crossover networks with some great equipment at Tellabs in Lisle using LRC bridge and a great spectrum analyzer plus other equipment. So, after several weeks of testing, I had finally built my transmission line speakers and they sounded great. One day a friend of mine wanted to see how loud my speakers would sound. I had an amp with a power meter and we adjusted the sound until it was so loud, we could not scream at each other. The power meter measured 2 Watts and the room size was 30’ by 18’. The bad news, was that the speakers were so efficient that I had blown a plaster statue that hung on the back wall in the room and cracked the plaster walls behind the speakers. Yes, the speakers were so loud that a UPS truck stopped by to tell me that he heard my speakers two blocks down the road and had to stop by and ask me what the heck was going on. He said the sound from of the music was like a live concert. The bottom line, are transmission line speakers efficient “yes”, are they hard to build “yes”, and I would recommend them to anyone who has the time and the skills to make them. BTW, I have perfected a new woodworking method to help reduce the time to make each transmission line speaker. 40 years later, I am going to test these speakers with new woofers and some new test equipment and pass this onto my son. Maybe he will learn how to make his own transmission line speakers and enjoy the speakers like I have and hope other will do the same?

  3. Perry how can I build one for trunk of Honda civic for one single 8″ l7 solobaric. I built sweet spot ported box that sounds like 2 15s. But want more boom for a single sub

  4. Hello,
    I am trying to set your formulae in the article ‘derivation and analysing the transmission line’ into an Excel program.
    To see what happens with Zb when I change for example the Fs, the Sd or Sl etc.
    My question is about the mechanical impedance of a piston formula (14).
    The S in this formula is not explained in the article so I think this the cross section area in m^2 of the pipe. Is this correct?

  5. Perry,
    I really enjoyed reading about your RD 75 two way project. However, I am unclear on one very vital aspect – did you use the RD75 in monopole or dipole orientation? The monolith appearance would suggest dipole baffled orientation as most monopolar applications minimize front baffle width and maximize depth for back wave absorption. Care to discuss and/or illustrate/diagram further the RD75 housing? If it is in fact a dipolar orientation, as you know the baffle dimensions and geometry GREATLY affects performance and frequency response

    Would also like to hear more about the DSP 2×8 digital crossover and the improvements it offers.

    1. Scott,

      It’s monopole with a sealed back. If I had to do over again AND had a bigger room, I’d use dipole. As it is now, there’s not much to say about it. It’s just a box. Tom Perazella, who’s a friend of mine, has written about his dipole design in AudioXpress / Speaker Builder magazine.

      The MiniDSP is fantastic. The D-A converters are MUCH better than the Behringer DCX2496, in fact my brother said it sounded like a completely different system. So much for the myth that all D-As are alike. The Behringers really are not audiophile quality. They are good enough for sound reinforcement though.

      You can find all kinds of info on MiniDSP units; I just arranged the various filters to achieve the result I wanted.

      They’re really great units and you won’t ever want to go back to passive xovers after you’ve used them. Oh by the way the 2×8 does not actually have its own D-A and A-D. I used a Maverick Audio D-A converter with that, and was very happy with it. Later I switched to the MiniDSP 2x4HD which has its own D-A and those sound superb. Those are the ones I would recommend for a 2-way. $200 ea. They also have higher output than the original 2×4 “standard” unit which turns out to be pretty important oftentimes. The standard unit doesn’t have enough output to drive a lot of power amps all the way to clipping.

      Oh also the MinDSP 2×4 HD has capability for FIR (Finite Infinite Response) filters which are a whole different ballgame. Harder to use and has some limitations – can’t go super low in frequency given the number of FIR taps on the 2x4HD. I use the Eclipse Audio FIR filter designer. FIR is for advanced designers. Don’t bother with it if you’re new to DSP. Standard DSP is “IIR – Infinite Impulse Response.” That is built into the standard filters of the MiniDSP 2x4HD.

      I stream music lossless via Apple TV. The Apple TV unit has a fiber output and the MiniDSP has a fiber input. So it’s full digital from source all the way through the DSP unit.

  6. Hello, my name is Bill Wyko, owner of Audio 2000 in Tucson AZ. We will be starting a build in the next few weeks to attempt a world record spl in the 180’s Please see my website http://www.tucsoncaraudio.net to see I’m not just dreaming. I have 30 years in car audio. I’m interested in the concept of the TL enclosure to achieve our goals. We’ve been given the budget and will have in excess of 150,000 watts in amplifier power. My question is simple, should I pursue a TL enclosure to achieve these numbers?

    1. You should join Martin King’s Transmission Line discussion board on Yahoo Groups to discuss this. I would expect that a TL has its merits here.

  7. 1 more thing, i am a retired SPL compeditor 1998 to 2003 and have achieved 171.5 DB @ a world finals in Kansas city Missouri in 2003 documented… Again. Help please. Tim

  8. I am building a t line with 4 10″ drivers in a subaru forester wagon and have more then enough space. My questions are a) how many lines are needed. B) does the lines have to be × 4 the size. C) is round or 45 degree @ line turns make a difference. D) is egg carton foam better then stuffing. E) why is it that after 30 hours searching for thease answers on all the t-line websites their are nothing but theory’s and trial an error and no solid design’s that can be used. Help, please.

    1. Timothy,

      I think you’ll get good answers on the Quarter Wave discussion board. Sorry, I can’t spare the time to dispense speaker advice here.

  9. I just built my first transmission line home theater subwoofer with my 7 year old son. We built and installed between studs and flush mounted in the lower half of the wall where the pine Wayne’s coating is and we utilized the Wayne’s coat over the top and it allowed us to make a nice speaker screen that snapped right in. Basically about 3.2 cubic feet with a 10″ 8 ohm sub driven at 480 watts bridged at 4 ohms. We have approximately 45″ of transmission line yet a the base we installed a 3″ x 11″ PVC tube so it doesn’t have the full slot. It actually works quite well, very clean, better than any polk audio active sub I’ve owned. I didn’t however stuff it with anything, although I did leave an access cover unglued in case I needed to make further adjustments to our new project which seems to be the case. Honestly this has been one of the more difficult things I’ve done over the years yet something I want to do again for the front half of the theater room as this is installed in rear corner. No unusual booming or chuffing. Very little air passes through the port tube though from what I’ve gathered here there shouldn’t be any just sound waves so stuffing is what I’m guessing we should do. I’ve heard to use standard polyester pillow stuffing from joann fabrics and that will work fantastic according to others. Any thoughts? I also didn’t bevel my corners or 45 inside corners. Is that really an issue here in this case. Am I missing something I can do better, add more transient plates? Does it matter distance between plates? Mine are currently 2.75″ of air between each plate. The cabinet is 26.5″ long x 15″ wide (between studs) and 16″ deep out in garage and outside of cabinet is wrapped in carpet padding. At it’s loudest level I cannot hear bass in garage so it’s apparently sealed well. Let me know if anything I should do to tune it better than it is. Thank you in advance your clearly the genius here and inspiring to me to read.

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