Hofmann's Iron Law

Dayton Audio Max-X 10" Subwoofer in a 1.5 cuft Ported Enclosure

 

Josef Anton Hofmann was a physicist, audio expert and entrepreneur.  

In addition to earning  Ph.D. in physics from Harvard and working on the Manhattan Project he was a co-founder or partner in no less then three groundbreaking audio companies; KLH (the H stood for Hofmann), Advent, and Acoustic Research.  These companies, which today are all shells of their former selves, were the giants in hi-fidelity home audio back in their heyday.  Acoustic Research, for example, was the first company to produce an acoustic suspension (i.e. sealed) loudspeaker.  Interestingly enough a simple sealed speaker enclosure was not invented until the 1950's by Edgar Villchur, the founder and president of Acoustic Research, while the vented enclosure was patented by a scientist Bell Labs in the 1930's.  

Kicker Comp R 12 in a 2.5 cuft Ported Enclosure

 

We would not have the field of acoustics, much less the market for hi-end audio equipment, as we know it today with out Dr. Hofmann.  His work would would become the backbone that Neville Theil and Richard Small would use to create what is now known as the Theil-Small parameters.  

When explaining the mathematics behind enclosure design people often ask me why this stuff seems like rocket science.   Simple answer:  It is.  I cannot find any direct evidence that Dr. Hofmann was a literal rocket scientist.  However, he was clearly an intellectual giant and could have easily pursued this field.  Instead he focused on things that were equally as complex like semiconductors, solid state devices, and of course, the pursuit of audiophile quality sound. 

 

Hofmann's Iron law is elegant in it's simplicity.  

The law focus on the interplay between enclosure size, efficiency, and low end extension.  

Efficiency is a way of quantifying loudness relative to power level.  A more efficient speaker can play louder with the same amount of amplifier power as a less efficient speaker.  With today's modern class D designs amplifier power has become cheap.  But imagine trying to fill a room with sound using a tube amp from the 1950's or transistor radio from the 1970's.

Low end extension is just what it sounds like, the ability to play bass frequencies.  With the massive subwoofers available today this is trivial.  But again, think the post WW2 era.  Long before Cerwin-Vega started producing strokers for movie theaters.  Long before Kicker started putting enclosures behind the seats of regular-cab pick-up trucks.  Back then not even the best hi-fidelity sound systems could play 20 HZ with authority.

Imagine that you are Dr. Hofmann and your boss, the inventor of the sealed loudspeaker, needs you to build a hi-fidelity speaker that could reproduce quality music, covering wide band of frequencies, in a way that had never before been done.  

That is where enclosure size and Hofmann's iron law comes into play.  Dr. Hofmann determined that if you wanted and enclosure that was efficient and had good low end extension the key to getting low and loud was a large enclosure.  Especially at at time when adding more power was not easy or even feasible.  Hofmann's Iron Law has been summarized like this:  with a given amount of power you can be low, you can be loud, or you can have a small enclosure.  Pick two.  This means that a small enclosure can either be efficient or it can have low end extension, but not both.  An efficient enclosure can either be small or it can have low end extension but not both. 

I have been unable to find any specific mathematical formulation of Hofmann's Iron Law written by Dr. Hofmann.  A mathematical formulation does exist, but it makes use of T/S parameters.  For those that are interested you can find some equations on the DIY Audio Forums.  If any reader has information on these formulas or happens to have access to any of Hofmann's original research please let me know.  I would love to make a post breaking these down into common-sense terms.  In the mean time the best explanation of the math that I have found has been this facebook post.

Implications for today

Simple.  It takes space to make bass.  Especially ultra low frequencies.  Consider this review of DIY home theater subwoofers:

In that video Nick from Toid's DIY Audio showed off a 15" subwoofer in a 5 cubic foot enclosure.  I have the 10" version of that subwoofer.  The manufacturer recommends 1.1 cubic feet.  My plans call for a 1.5 cubic foot enclosure.  Why so big?  Because Joseph Anton Hofmann is smarter than I am, and he is probably smarter than you as well.   So if you insist on installing a 12" subwoofer in a pitiful box like this one go ahead.

Cheap, Undersized, Flimsy Subwoofer Enclosure.

I am going to build a box that is about 50% bigger and my box will be at lest 3DB louder at every frequency, plus my port will be tuned lower so I can hit frequencies that your box cannot and I can do all this without the subwoofer unloading. Your only chance of keeping up with me is to double your amplifier power. When you do that you will need to go ahead and buy an upgraded aftermarket alternator.  The only downside to a bigger box (besides the space taken up by a bigger box) is cone control.  Under high power the driver will exceed X-max and distort.  Fortunately, the larger box is more efficient, so Hofmann's Iron Law tells us that we don't need as much power.

If you find yourself in a situation where you are forced to use a small box, like an under-seat setup in an crew cab pickup truck then you are stuck.  You can either choose to sacrifice low frequencies for efficiency or efficiency for low frequencies.  The iron law has you.  You can't do anything about it, other than throwing power at the problem.

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