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Webster and the Horn Equation

Arthur Gordon Webester

Arthur Gordon Webster (image: Wikipedia)

In July this year (2019), it is 100 years since Arthur Gordon Webster published his seminal paper "Acoustical Impedance and the Theory of Horns and of the Phonograph". Who was Webster, and what significance had his paper?

Arthur Gordon Webster was born in 1863. In 1885, he graduated from Harward College, and in the years from 1886 to 1890 he studied under Herman von Helmholtz for his Ph.D. From 1892 he was head of the Physical Laboratories at Clark University, and was appointed full professor in 1900. In 1899 he helped founding the Americal Physical Society, and became its president in 1903.

Webster was a proficient mathematician and a competent experimentalist, and spoke several languages fluently. His research was centred around acoustics and mechanics, and he invented the phonometer, an instrument to measure absolute sound intensity.

Webster committed suicide in 1923 following rumours that his physics department would be closed by the new president at Clark University.

His paper on horn theory was published in 1919, but it was actually originally read at a meeting of the American Physical Society at Philadelphia in December, 1914. It is not the first paper dealing with the horn equation, which was treated by Euler, Bernoulli and Lagrange in the 18th century, as described by Eisner (Edward Eisner, "Complete Solutions of the Webster Horn Equation", J. Acoust. Soc. Am. 41, 4 (2) (1966), pp. 1126–1146.). So why has Webster’s name been associated with it?

There are several reasons for this. First of all, it was a relatively accessible analysis of horns, presented at a time when it was needed. The old papers analysing sound in expanding tubes were not readily accessible, and Lord Rayleigh’s “Theory of Sound” was out of print. Second, Webster introduced the concept of acoustic impedance (although with a slightly different definition that what was later adopted), which tied in nicely with the impedance concept used by electrical engineers to analyse complex circuits. Thirdly, he showed how the equations for various types of finite horns could be developed, and in this way, the effect of the conditions at the mouth could be taken into account.

Why was it important? Up to this point, gramophone recording and reproduction had developed mainly through empirical methods, and radio broadcasting was in its infancy. With the vacuum tubes available at the time, output power was very low. To free the listener from using headphones an efficient loudspeaker (using a horn) was needed. The engineers needed a way to estimate and optimise the performance of the horn. As stated by professor V. Karapetoff in 1924, “This problem of horns is a 'house-on-fire' problem, in the sense that loud speakers are now being manufactured by the thousand, and while they are being manufactured and sold, we are trying to find out their fundamental theory.” Webster’s analysis made possible to find guidelines to use in the early stages of design, although the math was complex enough that for most cases rules of thumb based on equations that could be calculated by hand had to be employed.

The paper does not appear to have been widely recognized at first. The first citation seems to be one by Kennelly in 1923. But it was no doubt studied by engineers involved in loudspeaker design. And by the publication of the paper “The function and Design of Horns for Loud Speakers” by Hanna and Slepian in 1924, a much wider community was introduced to Webster and the horn equation. It is perhaps less widely known that at the same time, Paul B. Flanders and Donald A. Quarles at Western Electric Engineering Department, later Bell Labs, also studied horns and derived equations for predicting their performance. Even the performance of an entire system of driver, throat chamber and horn. But this work was only published in internal memos, and not publicly.

But the significance of Webster’s paper at the time was great. It was now possible to calculate the performance of horn loudspeakers, and one early user of this method (outside the Bell Labs) was Harry F. Olson of RCA, who published his findings in several papers. He and others studied the effects of horn flare, horn length and mouth size.

At this time, about 1925-1940, it appears to have been widely known that the equation was an approximation. As W. M. Hall puts it 1932, “The use of such approximate methods in the theory is justified only by the approximate nature of the results it is desired to obtain.” Experimenters often commented that the discrepancies between theory and experimental results were due to the approximate nature of the horn equation. This knowledge seems to have been more or less lost after the publication of Olson’s “Elements of acoustical engineering”, and Beranek’s “Acoustics”. Neither of these books mention the approximations in deriving the horn equation, and judging from published literature, it took several decades before it again became common knowledge among audio engineers.

Today, the main function of horns in pro audio is to provide directivity control. Compression drivers are crossed over at much higher frequencies than was common in the decades after Webster’s paper was published, and the higher power handling and cheap electrical power makes the acoustic load less important than it once was. And the only thing the horn equation really can do, is to predict the acoustic load, and the “transformer properties” of the horn. It is one-dimensional, and cannot predict directivity, which is a 3-dimensional quantity. For this we need more advanced methods, like FEM, BEM or MMM.

But for bass horns, it is still very useful. When the mouth is small enough to not have significant directivity, the predictions can be quite close to reality. The free software Hornresp has been successfully used by many to design low frequency horns. Since the throat impedance can be predicted with reasonably good accuracy, the power response of a speaker can be calculated. This applies to any frequency range.

The horn equation is also useful for instrument modelling, as the wave propagation in wind instruments is essentially one-dimensional, and the important property is the location of resonance frequencies.

In summary, Webster’s horn equation was a great help to the early electroacoustic engineers in the “roaring twenties”, when loudspeaker design progressed from a pure art to something that may be called science. But even today, it is not completely outdated. It still serves horn designers, and serves them well, if they ask the right questions. It cannot do everything, and it never claimed that either.

Does Webster deserve to have his name attached to it? I definitely think so. Webster was not the first to derive this equation, but he introduced it to the community of electrical engineers when it was needed, in a language they could understand. And to this day, nearly every paper dealing with the theory of horns, will cite his paper. It has become his equation.

New interesting compression driver

Celestion recently came out with a new compression driver, the Axi2050. For people like me who are interested in large midrange horns covering most of the vocal range (and I'm not talking about the 300Hz-3kHz telephone range, but more like 100Hz to 5kHz), this driver is very interesting. Especially since the high end response is very good for such a large unit. It's more of a mid-high than a midrange driver. Take a look at the frequency response curves, taken from the Celestion data sheet:

Axi2050

The PWT response is only shown down to 100Hz, but it is basically flat down to that frequency. In addition, the HF response looks good up to about 10kHz. That's two decades! And the HF sounds pretty good too; I got a change to hear one such driver on a straight circular Hypex horn at the Celestion factory, and it actually had a nice, smooth high end. Moreover, the driver is rated for 150W down to 300Hz, and the diaphragm is large, so for domestic use, using it down to 100Hz or even lower would not be a problem. 

I would love to try a pair of these drivers on my midrange horns, but unfortunately they are listed as OEM only. Also, I would have to design a new throat and middle segment for the horn, as the driver has 2" exit, and the horn is currently less than 2" in height for about half its length. 

I first became aware of this driver at the 2015 AES convention in NYC, where Jack Oclee-Brown and Mark Dodd presented two papers on the design of this driver: 

Wideband Compression Driver Design, Part 1: A Theoretical Approach to Designing Compression Drivers with Non-Rigid Diaphragms (preprint 9386) and

Wideband Compression Driver Design. Part 2, Application to a High Power Compression Driver with a Novel Diaphragm Geometry (preprint 9391).

At the 2016 AES Convention in Paris, I got the chance to look at the driver itself. It's fairly big, but not very heavy for its size (about 7kg). 

And just for the record: when writing this, I'm still at the university. I'm not trying to sell the driver or to advertise for Celestion, I just want a pair!

There appears to have been some problems with the contact form. Or, apparently, the form works, but I don't get the e-mails! So if you have tried to contact me over the last couple of months and not received a response, please try again. 

European Triode Festival 2015

For the first time since 2001, I went to the European Triode Festival, ETF. This year it was in Tisvildeleje in Denmark, and it was a great success. Five of us from Norway drove for 8 hours with three cars full of horns and tube amps. And one of us took the plane from Bergen.

Since the festival has been well covered by Thomas Mayer starting here (with mentions of our system here andhere), I will not try to cover everything. So this blog post will mainly cover the system we set up in the room we had. 

Below is a picture of the system on Thursday afternoon. Thomas Dunker is writing up the details of the system on the whiteboard. 

ETFsystem1

A block diagram of the system is shown below (click on the image for a larger version). Thanks to Lars Tørresen for the drawing.

anleggETFinvert

Sources:

  • Modified Dual 701 turntable with a SME II arm and a Denon DL103 pickup (Thomas Dunker)
  • Micromega Minimum DVD player as CD transport (TD)

 

Preamps:

  • Allen Wright style RTP RIAA (Bjørn Kolbrek)
  • Modified Behringer DCX2496 with external linear power supply, input selector, passive analog stages and output volume control (BK)

 

Power amps:

  • Kenwood KA-9100 for bass duty (TD)
  • 300B Push-pull amps for midrange duty (BK)
  • 809 SE amps for tweeter duty (BK)
  • SE 801A amps (midrange, not used due to technical problems, TD)
  • SE 46 amps (tweeters, not used, TD)

 

Speakers:

  • JBL 4560 cabinets (TD) with JBL woofers by Torbjørn Lien. The woofers were a hybrid design, described by Torbjørn as "the same diaphragm as 2220-D130, magnet as D130 but with a copper voice coil, overhung, 1/2" long, i.e. the exact same motor as D140."
  • "Tentamenshorn" midrange horns, as described here. These were Thomas' horns, with his modified drivers. More about that later.
  • Downscaled Iwata horns (1") with TAD 2001 drivers (TL)
  • Dipole tweeters with DIY 19mm 9µ titanium foil diaphragms (TL)
  • Modified Coral H100 tweeters (TL)

 

The tweeters were crossed in passively, and mostly provided some "air". 

ETFsystem2

Several people asked for drawings for the midrange horns. We hope to be able provide that, but the drawings we've got need some work before we can make them publicly available. Please be patient. 

Thomas had brought two pairs of modified Altec 288B: one pair with Neodymium magnets, giving a gap B of 2.15T, and one field coil pair that were finished just in time to ETF. We played on the first pair until mid-Friday, then we switched to the field coil version. There was perhaps not a great difference, but at least both drivers sounded great! And the FC drivers look cool too. After some hours they were not so cool, though...

288FC

Friday evening we had live music.

ETFmusic1

The system of our room mates: Thomas Mayer (amps) and Wolf von Langa (speakers).

ETF Mayer Langa

We were not the only ones to bring measurement gear.

ETFmeasurementeq

On saturday, Thomas Dunker and I also held a lecture on horns and horn loading. This was a quite basic lecture on the horn design philosophy of the 1920s and the benefits of horn loading, using our big midrange horn as an example. 

ETFhorntalk

Saturday evening we listened to Charles King's tapes for what must have been several hours. His Stellavox tape machine and low-generation tapes sounded magnificent and gave us a sound quality experience we will not soon forget. 

ETFtape1

 

Sunday, and time to leave.

ETFleaving2

ETFleaving

Higher Order Modes

Higher order modes describes the deviation from simple wave propagation in horns and sound fields in general. If the wave front in the horn is not plane, cylindrical or spherical, it can be described as a sum of mode functions. In one of my earlier blog posts, and also in the reports available under Horn Theory, I have discussed this in some more detail.

Since it appears that many believe that the existence of higher order modes in horns (often just called HOM) is a quite recent discovery, I decided to put together a compilation of references on the topic. This list will probably be updated contiously as I find more references, but here are the first ones. Some of them have URLs, but you may not have permission to download the papers from the journals. If you have access to a university library, you may be able to download them there. I will not download them for you...

V. A. HoerschNon-Radial Harmonic Vibrations within a Conical Horn, Phys. Rev. 25, 218–224 (1925)

Perhaps the first analysis of modes in conical horns. The boundary condition at the horn mouth is not realistic (zero pressure condition), but the analysis is valid. It illustrates that the existence of HOMs were known at a very early stage in electroacoustics.

K. Sato: On the Sound Field Due to a Conical Horn with a Source at the Vertex, Japanese Journal of Physics 5, no 3, 103-109, 1929

The sound field internal and external to a conical horn is given as an expansion in spherical harmonics (modes in spherical coordinates). Directivity plots for nearfield and farfield are given.

R. J. Alfredson: The propagation of sound in a circular duct of continuously varying cross-sectional area, Journal of Sound and Vibration, Volume 23, Issue 4, 22 August 1972, Pages 433-442

The horn is divided into short cylindrical segments, where a modal description is used to compute the sound field. Modes are matched at the discontinuities, the number of modes determined iteratively.

A. Cummings: Sound transmission in curved duct bends, J. Sound Vibr. 35, no 4, 451-477 (1974)

A modal description of the sound field in a rectangular bend. Mode functions are cosine and Bessel functions.

A. H. Benade and E. V. Jansson: On plane and spherical waves in horns with nonuniform flare. I- Theory of radiation, resonance frequencies, and mode conversion, Acoustica 31, 79-98 (1974)

The existence of HOMs and their effect on the resonance frequencies of musical instruments are discussed in general terms. Some simplified calculations. Discussion of mode conversion.

E. R. Geddes: Acoustic Waveguide Theory Revisited, J. Audio Eng. Soc. 41, no 6, 452-461 (1993)

A more detailed analysis of the Oblate Spheroidal waveguide than the original 1989 paper. The existence of HOMs in the waveguide is taken into account and analyzed. 

V. Pagneux, N. Amir and J. Kergomard: A study of wave propagation in varying cross‐section waveguides by modal decomposition. Part I. Theory and validationJ. Acoust. Soc. Am. 100, 2034 (1996)

V. Pagneux, N. Amir and J. Kergomard: A study of wave propagation in varying cross‐section waveguides by modal decomposition. Part II. ResultsJ. Acoust. Soc. Am. 101, 2504 (1997)

V. Pagneux, N. Amir and J. Kergomard: Wave propagation in acoustic horns through modal decomposition, Transactions of the Wessex Institute, 1997

The papers by Pagneux et al descibe computational methods for the calculation of horn performance based on mode matching of the sound field. The focus is on musical instruments, but the method is of course also applicable to other horns. Axisymmetric and 2D horns are described.

S. Félix, V. Pagneux: Sound propagation in rigid bends: A multimodal approachJ. Acoust. Soc. Am. 110, 1329 (2001)

Multimodal description of sound propagation through bends in tubes. 

J. A. Kemp: Theoretical and experimental study of wave propagation in brass musical instruments, PhD dissertation, University of Edinburgh, 2002

A multimodal method for computing the throat impedance and mouth volume velocity of horns. Partly based on the work of Pagneux, Amir and Kergomard.

E. Geddes: Audio Transducers, 2002

Describes mode based methods for several geometries, including the oblate spheroidal coordinate system. The measurement chapter partly describes how to measure the modal amplitudes at the mouth of a horn or other transducers.

Behler, Gottfried K.; Makarski, Michael: On the Velocity Distribution at the Interface of Horn Driver and Horn, AES Convention:116 (May 2004) Paper Number:6097

Measurement of the sound field at the interface surface between a compression driver and horn, and the decomposition of this sound field into modes described in cylindrical coordinates.

Makarski, Michael: Do Higher Order Modes at the Horn Driver's Mouth Contribute to the Sound Field of a Horn Loudspeaker?, AES Convention:117 (October 2004) Paper Number:6188

Description of how a non-planar velocity wave front at the horn throat influences the sound field at the horn mouth, and the radiated sound field. 

Makarski, Michael: Tools for the Professional Development of Horn Loudspeakers, PhD thesis, RWTH Aachen, 2006

The PhD thesis contains the information from the above two articles, and more. Modal decomposition of sound fields, and modal exitation of horns are described un detail. If you want to measure HOMs, this is a good place to start.

Leo L. Beranek and Tim J. Mellow: Acoustics: Sound Fields and Transducers, Academic Press, 2012

This is an update of the classical book Acoustics by Beranek from 1954, and it's the most useful book on acoustics that I have. If you plan to buy only one book on loudspeakers and sound radiation, this is the one. (For room acoustics, noise control etc, there other books that have more information, but for a loudspeaker designer, this one is a must). It covers many analytical and semianalytical methods to calculate sound fields, many of them are based on eigenmode or eigenfunction expansions. 

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