The Multiple Flare
In 1938, Harry F. Olson of RCA Manufacturing Co., Camden, N.J., published a paper titled "A HORN CONSISTING OF MANIFOLD EXPONENTIAL SECTIONS".╣ One of the problems with a horn is that to get high efficiency at low frequencies, a small throat imedance , or resistance, is needed. Conversely, the opposite holds true at high frequencies. In short, ya can't have both simultaneously, even if the high frequency limit is only 400 Hz. One way is to lower the upper frequency limit but then the acquisition of a moderately priced squawker going as low as 200 Hz or lower will cost an arm and a leg. Paul Klipsch realized this, ergo the Klipschorn« has a dual flare.▓
The idea behind the dual flare, simply put, is that it "fools" the driver into thinking it is operating at a lower impedance than it actually is at the lower frequencies. So, the throat can be made smaller than required for a given low cutoff as the driver is operating into a horn designed for a higher low frequency. The initial flare is usually short, the mouth of which couples to the throat of the lower frequency horn. If my understanding of this is correct, it goes something like this.
As the frequency of operation changes, as it usually does with music, the load imposed upon the diaphragm (cone) changes. Consider a dual flare with a transition of 100 Hz. The initial throat can be made small enough to properly load the driver down to that frequency, below which the driver "sees" a higher load. However, the initial flare, being short, will couple to a mouth large enough to load the driver properly below that frequency. As the frequency changes between the absolute low frequency cutoff of the larger flare, say 40 Hz. to the upper limit of the smaller flare, 100 Hz., so does the load imposed upon the diaphragm. It kinda works as if the throat area changes with frequency between 40 Hz. and 100 Hz. Mr. Klipsch quite appropriately referred to this phenomenon as a "rubber throat."▓
Let's consider, again, our 40 Hz. horn. Let's also say that the throat is determined to be 88 sq. in. The cross sectional area will double every 18.6 inches and the total horn length, unfolded and intended for use outdoors, hence no corner with which to play acoustical tricks. In this example, the horn will have a total length of 12.4 ft.
Now, we introduce an initial flare with, say, a 100 Hz. lower cutoff. The doubling length of the initial flare is 7.4 inches and terminates in a mouth of 88 sq. in., which becomes the throat of the second horn. So, we've just reduced the total horn length by 11.2 inches.
Adding a third flare will reduce the horn more. It can be seen that the horn shortens as the number of flares is increased. Therefore, for you mathematical purists, if the number of flares is increased without bound, the length of the horn will reduce to zero. How practical such an endeavour would be is left to one's imagination. :)
1 Society of Motion Picture and Television Engineers, (S.M.P.T.E.) January 18, 1938.
2 P.W.Klipsch, "A Low Frequency Horn of Small Dimensions" Journal of the Acoustical Society of America, Vol. 13, No. 2, pp.137-144, August 7, 1941.
2 P.W.Klipsch, "Improved Low Frequency Horn" Journal of the Acoustical Society of America, Vol. 14, No. 3, pp.179-182, September 10, 1942.
Horn Theory - Chapter Index