Hofmann's Iron Law

A driver's reference efficiency can be written in terms of the driver:

\eta_0 = \frac{4\pi^2}{c^3}\cdot\frac{f_s^3 V_{as}}{Q_{es}}

or, equivalently, in terms of what the finished system achieves:

\eta_0 = k_\eta f_3^3 V_B

where f_3 is the system's −3 dB frequency in hertz and V_B the net box volume in cubic meters. The second form is the law. Efficiency is proportional to box volume and to the cube of the cutoff frequency, and the constant k_\eta has a ceiling no driver can exceed.

The consequence

Because of the cube, extension is brutally expensive. Taking a system an octave deeper at constant box size costs a factor of eight in efficiency: 9 dB of sensitivity. Equivalently, holding sensitivity and extension fixed, halving the box size is simply not available.

Richard Small worked out the ceilings. For an idealized closed box, \eta_{0(max)} = 2.0\times10^{-6} f_3^3 V_B; for an idealized vented box, 3.9\times10^{-6} f_3^3 V_B. The ratio of those two constants is 2.9 dB, which is the origin of the familiar claim that a vented box is about 3 dB more efficient than a sealed one of the same size and cutoff.

Real systems fall short of these ceilings, Small puts typical practical designs 2 to 3 dB below, because the ceilings assume no losses anywhere and a particular alignment.

Why it is called iron

Because there is no cleverness that avoids it. It is not a limitation of materials or of manufacturing; it falls out of the radiation physics of a small source in air. Every genuine improvement (a lighter stiffer cone, a stronger motor, a better suspension) moves a design towards the ceiling. None of them raises it.

The honest uses of the law are to sanity-check a specification before building, and to recognize when a disappointing result is not a mistake but physics. The dishonest use, common in marketing, is to claim to have beaten it.

Further reading

The derivation, the efficiency constants and the alignment charts behind them are in Richard H. Small, "Closed-Box Loudspeaker Systems Part I: Analysis", JAES vol. 20 no. 10 (December 1972), and "Vented-Box Loudspeaker Systems Part I", JAES (June 1973).