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Chris A

Why Horns? (a tutorial to accompany "High-Quality Horn Loudspeaker Systems")

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(This is a continuation of a deeper look into horn-loaded loudspeakers as a result of an in-depth review of Floyd Toole's book (Sound Reproduction, 3rd edition).


Many threads have been about horn (a.k.a., waveguide) loaded loudspeakers on this forum.  This isn't the first--nor will be the last. What's different about this thread is that it addresses those readers that are a little more than "just a little curious" about horns/waveguides and why a company like Klipsch produces loudspeakers that effectively 100% have some sort of horn loading of drivers.  (Even earbuds are "waveguide loaded": the ear canal is a waveguide). 


This thread is aimed at those people that would interested or motivated to buy this book and read it, but need just a little help getting up on the technical language.  In other words, this thread is a gloss or explanation/exegesis of this text:




This book is available from Parts Express (in the US) and elsewhere.  Here is the book's website for those living in other countries: https://hornspeakersystems.info/.

It's not an inexpensive book, but when you see it, you will understand the price--it's over 1000 pages--large pages and has many thousands of figures, etc.



"Why horns?"  That is the subject of this entire thread.  Based on another thread that reviews Floyd Toole's book, Sound Reproduction (third ed), it became clear that an in-depth discussion of horn-loaded loudspeakers, and horn/waveguide theory in particular, is needed for those that wish to learn more, and which are not deterred by the technical challenge of learning more about it.


Why here?  Klipsch has always been about horn-loaded loudspeakers (even when it wasn't cool--which was a lot of years).  As such, I consider the Klipsch name synonymous with horns.  If that changes, I would be extremely surprised.  I feel that an in-depth thread to counterbalance the "frequency response is king" direct radiating monkey coffin enthusiasts...is sorely needed.  I've been waiting for someone else to do it.  It hasn't happened, however, except for the author of the book that we'll be using (released in late 2019)...Bjorn Kolbrek. (The first part of the book--the history of horn loudspeakers, was mostly written by his co-author, Thomas Dunker).  This is as good a time as any to start in this forum.  I think PWK would actually be flattered to know his technical legacy survives. 


I hope that I'm up to the task.  I've had a lot of practice of late (tutorial threads) and good familiarity with the subject matter--albeit at an engineering ("educated consumer") level.  If anyone in Klipsch engineering wants to take on this task--I'll get out of their way.  Perhaps this thread might kick-start that involvement...or perhaps not.  Comments from Klipsch design engineering staff are always welcome. 


As this thread develops, I expect the format to be refined as we go along.  Questions are encouraged, but this is not a "social media thread":  if you want to talk about the social aspects of owning and listening to horn-loaded loudspeakers...start another thread!  You can link to those threads from this thread if you feel it's relevant to do so. 


If you are still bothered by the technical language even after reading the gloss explanation here, you still have a couple of choices:


  1. You can ask questions about those specific horn theory topics here or perhaps the mathematics of horn theory here, or
  2. If you are still bewildered by the whole subject, start another thread and talk about it, and you can link back to that thread here for reference.


My initial approach is to follow the book's table of contents, starting at chapter 9 (page 413) of the book.  If this proves to be too laborious or not useful from the perspective of "why horns?", I reserve the right to change it up to keep the thread moving forward.  Here is a table of contents of the book: https://hornspeakersystems.info/images/pdf/TOC.pdf.


The contents of chapter 9 are:

Title: Horn Loudspeakers - an Introduction
1 Efficiency
2 Impedance Transformation
3 The Horn Loudspeaker
4 Directivity
5 Basic Topologies:
  5.1 Front-Loaded Horns
  5.2 Rear Loaded Horns
  5.3 Full Range Systems
  5.4 Multi-Way Systems


Since there are 30 chapters (22 chapters to go, that are specifically related to this thread) and three appendices...comprising over 500 pages, this is going to take a while.  We're not in a hurry, however: the Klipschorn was designed over 75 years ago and it's still around. 😉




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9.1. Efficiency:

Section 9.1 introduces the concepts of:

  • an electrical circuit,
  • Lorenz force in a coil of wire carrying an electrical current in a permanent magnetic field (voice coil inside a circular groove in a stationary magnet),
  • radiation mass and radiation resistance (i.e., what we are trying to achieve to generate acoustic output of the driver/horn),
  • velocity of a vibrating acoustic diaphragm driven by a changing Lorenz force (linear motor) is inversely related to frequency (higher frequency=lower amplitude),
  • mass control --when the mass of the vibrating diaphragm and air controls the amplitude of motion,
  • impedance (denoted by the letter "Z")--the frequency-based counterpart to direct current electrical or mechanical resistance.
  • efficiency of an electromagnetic driver
  • horn loading


The purpose of this section is to introduce the concept of electromagnetic-acoustic efficiency and initially define the concept of horn loading.  The mathematics are simple: multiplication, division, and ratios.  The concepts above are all intuitive: an electrical circuit, electromagnetic force of a voice coil in a permanent magnetic field to move a diaphragm, accounting for the impedance losses (like DC resistance) of a electromagnetic--acoustic driver as the output of the driver acoustically as compared with the input forces and energy. Nothing here is relying on higher mathematics.


So why bring up efficiency?  The concept of horn loading requires it and is defined by it. Why is this important?  The balance of the book will answer this question.  A clue: Klipsch's law states that the higher the efficiency of a loudspeaker, the lower its distortion (modulation distortion, but also compression and phase distortion, and to much lesser degree, harmonic distortion). There's a lot more to the answer to the question, however.



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wave lens...

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Following.... may buy the book though I bought a lot of records over the holidays.  1,000 pages is more of an actual coffee table than a coffee table book.


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9.2 Impedance Transformation:


Before we get into the text (which I think isn't the clearest way to start this important subject), I think a Bell Labs film from 1959 will help much more about thinking about impedance transformation:



I also recommend going back to the first of this film to understand the concept of impedance and wave behavior in systems.



This book section is introduced with an analogous example: air moving through a duct which encounters a discontinuity: the duct suddenly gets larger from left to right.  We all can intuit the idea that air moving through a small diameter duct requires more "push" and that the flow of air of same mass of air flowing through a larger duct--larger ducts are more efficient at allowing the air to flow without incurring the same losses (and fan requirements are lower to push the air). 


What might be less intuitive is that there are very large flow losses associated with the sudden change in "flow impedance' of the duct carrying the air.  Gradual expansions and contractions are more efficient than sudden, abrupt transitions.


Terminology includes:

  • "S" (cross section area of the duct),
  • the purely resistive part of air resistance "R", and
  • the complex part "X" also known as the "reactance"--the part of the flow resistance corresponding to the effects of the mass of air and springiness of the air itself. 

This flow resistance is a function of distance along the duct, just upstream of the discontinuity and just downstream.  The impedance transformation in figure 9.2.2 (in the book) is a function of the distance "l" times a constant "k" (from where the plot gets its "kl" horizontal axis, below) from a point just upstream of the duct's discontinuity.




The flow impedance resistance and reactance upstream to downstream of the discontinuity oscillates with distance along the path. The discontinuity is at point "100" in the figure, below.  Notice that air is a continuum, and the effects begin upstream of the actual change in cross sectional area, then gradually damp down over distance, 1/6th of a "wavelength", denoted by the horizontal axis "kl".  The constant "k" is defined by the formula: 2πf/c, where "f" is frequency. This can also be written in terms of wavelengths:


k = 2πl/λ


where λ is the wavelength of the impedance (the applying to the real and the reactance parts curves) oscillations from the figure below.  The mass and springiness effects of reactance (bottom trace) slightly lead the oscillations of real resistance (top trace), below:




In this example, there is no "impedance transformation", only a sudden change in the impedance from a point just upstream of the duct discontinuity (point "1" in the top figure, above):


Z0 = ρoc/S0 


where ρo is the density of air, and "c" is the speed of sound in air, and S0 is the (smaller) area of the duct, to the point downstream of the discontinuity ("2" in the top figure, above), where the duct size is bigger:


Z1 = ρoc/S1


The video above shows the effects of an intermediate section (1/4 wavelength in length) that approximates a steady-state "matching transformer", and the effect of using a fully tapered intermediate section that acts as an impedance transformer over a much wider span of frequencies.  You can think of this intermediate tapered section just like an acoustic horn does its job, except the horn is also using the volume of air inside, as well as the walls of the horn (for higher frequencies) to match impedances.




Note that impedance transformers (such as a horn) works just as well in reverse as it does in the forward direction.  Horns have been used in battlefield service to serve as direction finding devices for counterbattery artillery fire:








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9.3 The Horn Loudspeaker (pg. 418):


After the preceding discussion of the concept of impedance transformation, this section begins the discussion of the balance of the book: horn loudspeakers.  It introduces the concepts of:

  1. standing waves inside the horn as related to its "mouth bounce"--reflected waves from the horn mouth termination (...the big end...😉)
  2. the "two chambers" of a horn loudspeaker: the volume of air inside the horn and the back chamber, and how they both act on the horn's frequency bandwidth
  3. reactance annulling of the back chamber to the driver and horn characteristics (to be expanded in chapter 19)
  4. moving mass of the driver MAD and its effects on the high frequency roll-off point of the driver/horn (i.e., the "mass corner frequency")
  5. the concept of a second order low-pass filter (not explained here)
  6. "Q" or quality factor (also not explained here)
  7. Adjustment of the equivalent parallel capacitance around the driver to adjust the high frequency roll-off point
  8. larger the throat areas (up to the size of the driver's exit-diaphragm ) increases the horn's efficiency, but reduces the horn's bandwidth
  9. the concept of the efficiency-bandwidth product of a horn system (i.e., including the driver, back chamber, and front chamber)
  10. the problem with using Thiele-Small (T/S) parameters for horn loudspeakers (i.e., the differences in the assumed air masses coupled to the diaphragm)
  11. calculating horn system efficiency (including front, back chambers and driver)
  12. directivity index (DI) of a horn
  13. the importance of strong motor (B*l)2 product of the driver's electro-magnetic circuit for horn operation
  14. the driver's low frequency resonance (Fs) does not determine the horn system's resonant frequency-the horn system Fs can be lower than the free air Fs
  15. the resonant frequency of the horn system, F0, that shows up at a higher frequency than Fs
  16. small-mouth horns will have higher SPL response ripple than larger mouth horns (important when looking at Klipsch horn-loaded bass bins)--with explanation/plots
  17. horn/driver compression ratio
  18. the ultimate value of horn distortion along the horn's length tends to reach a stable value
  19. distortion reduction via increase in throat area (related to the so-called "rubber throat that PWK talked about)


Why did I list the concepts of this section, which is largely text-only with few equations? 


I think it's now evident that a long list of concepts have been introduced, above, some of which are not explained in the text.  This is reason to slow down and examine each concept until a foundational understanding has been achieved for each, before moving on to later chapter 9 sections.  For those that wish further explanation of these ideas, it is better for them to ask questions here/now than for me to try to detail each concept--most of which I hope will already be familiar to the reader, but I know many of these concepts will be new to readers.


Also note: the fact that so many concepts have been introduced in such a short space of text is typical with the beginning of any technical subject textbook.  The reason why I slow down at this point is to help those needing help to "jump aboard" and not be left behind because many of the initial concepts were not understood.  Some of these concepts will be further explained as time passes, but many will not.  It's important to slow down and gain some feel or understanding before moving on. 


And don't stop here--stay with us.  The rewards are yet to come.  There is much to gain from the investment you made in this extensive textbook. 



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9.4 Directivity (pg. 422):


The idea of directivity is a separate concept than the one about impedance transformation, but in reality, they are inverses of each other.  Horns must reduce the output coverage volumes in order to produce impedance transformation, and impedance transformation itself will result in a reduction of the polar volume of output covered by an  acoustic driver. 


There is an intuitive feel for directivity--like placing your hands around your mouth in order to shout louder at someone far enough away that you need to restrict the output volume in order to increase the SPL of the person shouting.  (What's perhaps not as well known is that this also increases the efficiency of the throat/voice box due to impedance transformation.)


This section talks about the utility of horns over using multiple drivers in arrays, and controlling their relative phase and amplitude (a phased array--just like a phased array radar, and the hanging line arrays commonly seen in rock concerts and arenas, etc.).  The downside of phased arrays is complexity and inefficiency (using output from some drivers to cancel the output from other drivers--the diffraction problems).  This creates a too-narrow coverage angle, and therefore, areas within the venue that have not-so-good sound quality.  Many people have experienced this--being seated outside of the "sweet spot" of a line array in an audience and hearing not-so-good sound quality.  Also, the loudness (SPL) in the audience drops off much more rapidly with distance if you are seated more than the length of the line array, so those further back in the audience are hearing usually significantly lower SPL levels than those in the front of the seating section.  This isn't a good thing--because those in the front of the seating section usually get blasted with far too much SPL during typical higher SPL popular music concerts (e.g., "rock concerts").



Another directivity topic is the effect of frequencies within the output patterns of horns.  At the highest frequencies, the polar patterns will begin to "beam" like a flashlight if the internal size of the horn throat is greater than one wavelength of sound.  For 2" throat drivers, this is ~6.8 kHz; for 1" throat drivers it's 13.6 kHz where the output pattern of sound is controlled only by the phase plug geometries. (More on this later in the book--but this has large implications for high frequency horns of any type.)  Anything above these high frequency break points will experience a separating of the main portion of sound field from the walls of the horn throat and "ray tracing" acoustics energy coming out of the horn above these frequencies will increasingly be controlled only by the phase plugs and the residual diffraction of the throat area of the horn.  So now we begin to understand one of the comments made in the earlier section: larger throated horns will be limited in bandwidth, but their efficiency within their "passband" will be higher.  We want the directivity of horns to be constant with frequency, like the following (notional) polar sonogram:




(Much more on this subject in chapter 15.)





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  9.5 Basic Topologies

     9.5.1 Front Loaded Horns

     9.5.2 Rear Loaded Horns

     9.5.3 Multi-Way Systems

  9.6 Summary

Chapter 10 Symbols


The preceding sections, just above, and chapter are short and straightforward to read.  We'll move on to the next chapter of interest:


Chapter 11. Basic Theory


A listing of chapter headings:


11 Basic Theory
  11.1 Peak and RMS values
  11.2 Complex Numbers
  11.3 Dynamical Analogies
    11.3.1 Impedance and Admittance
    11.3.2 Acoustic Impedance
    11.3.3 Sources
    11.3.4 Resistive Elements
    11.3.5 Inductive Elements
    11.3.6 Capacitive Elements
    11.3.7 Acoustic Transformer
    11.3.8 Coupling Between Domains -- Transducers
  11.4 Linear Time-Invariant Systems
    11.4.1 Minimum Phase and All-Pass
    11.4.2 Transient Response and Delay
  11.5 Filters
    11.5.1 Image-parameter Method
    11.5.2 Modern Filter Synthesis
  11.6 Transmission Lines
  11.7 Resonant Circuits
    11.7.1 Resonance
    11.7.2 Q
  11.8 Impedance Matching
    11.8.1 Maximum Power Transfer
    11.8.2 Efficiency
  11.9 Two-Port Networks
    11.9.1 Transmission Matrix
    11.9.2 Impedance Matrix
    11.9.3 T, 𝜋 and Lattice Networks
  11.10 Principles of Sound Propagation
    11.10.1 The Wavenumber
    11.10.2 The Wave Equation -- Simple Derivation
    11.10.3 Velocity Potential
    11.10.4 The Wave Equation -- More Detailed Derivation
    11.10.5 Sound Intensity
    11.10.6 Sound Power
  11.11 One-Dimensional Solutions to the Wave Equation
    11.11.1 Plane Waves
    11.11.2 Spherical Waves
    11.11.3 Cylindrical Waves
  11.12 Simple Sources
  11.13 Modal Description of the Sound Field
    11.13.1 Modes in Rectangular Coordinates
    11.13.2 Cylindrical Coordinates
    11.13.3 Spherical Coordinates
  11.14 Air Nonlinearity
  11.15 Summary



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Thanks for this Chris. I'd love to buy a copy of the book but the price of USD150 is a little steep for me

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