The driver

The DX3 is one of the best drivers to have come out of Lowther Loudspeaker Systems to date, and is amazingly easy to handle, compared to earlier drivers. It has a very pleasant and smooth sonic character, and an unsurpassed level of HF detail. Also, the basket and magnet assembly takes up much less space than previous units. This allows for a minimal back-load cavity (the inevitable volume of air that exists between the driver and the horn throat), thereby gaining further horn efficiency.

Design

Designing a horn involves a number of decisions that each lead to new decisions with regards to compromise. The basic design process includes these steps (in no specific order):

1. Deciding on a throat area (At) that matches the driver.
2. Selecting a lower cut-off frequency (Fc).
3. Selecting a horn curvature (contour).
4. Deciding on a box size and shape, thereby introducing necessary compromise considerations to the design.
5. Folding the horn into the cabinet

As a rule of thumb, throat areas (St) with a throat to driver area (Sd) ratio of 0,30 to 0,70 are selected. Selecting a low ratio extends the horn’s bandwidth (up and down), at the cost of efficiency. Selecting a high ratio increases efficiency, but narrows the bandwidth. A complex relationship between the driver's physical properties (in particular Qes/Qts, Fs, Sd, and Vas) and the throat area determines the system's efficiency Vs bandwidth performance. The theory of these relationships will not be covered here, but they have been explored in great detail by ***, see the reference list for in-depth reading. Throat area (St) was set at 75cm2, which should ensure a good compromise between efficiency and freq. response. The selected throat area corresponds to a St/Sd of 0,36. Note that this ratio requires that a very efficient, low-Q 8'' driver be used, and may not be appropriate for other 8'' drivers.

Since size was restrained from the outset in this design, the lower cut-off frequency (Fc) was selected through an iterative process where different frequencies where literally "tried on for size". I initially aimed for a cut-off at 38Hz. This was finally adjusted to 40Hz, to make sure that the rules of thumb would be observed with good margin. Just to give you an idea what this means, the lowest string (E) of a correctly (440Hz) tuned bass guitar reverberates at a steady 41,2Hz.

There are basically three types of horn curvatures that are applicable for bass reproduction: hyperbolic, exponential, and tractrix curves. These all have distinct features with regards to acoustic properties.

• The hyperbolic is the most effective, and results in the longest horns.
• The Exponential contour is the most widely used curvature and is very easy to calculate.
• Tractrix curve horns are shorter than exponential horns, and considered to be a valid alternative to exponential contour horns. It is widely used for mid-range horns, both in professional and domestic appliances. Loudspeakers from Klipsch and Dr Edgar Bruce being excellent examples of the latter. However, very little research material and documentation is available for bass-horn employment.

Lowther Loudspeaker Systems inc. (formerly Lowther Voigt inc.) is one of the biggest proponents of tractrix designs and indeed the whole P.G.A.H. Voigt legacy. However, in talks I had with their chief engineer, Mr. Roy Hopps, he claims that for bass horns, exponential contours tend to be more efficient than the tractrix. I suspect that this performance problem has to do with mouth/impedance matching difficulties due to foreshortening, rather than the tractrix contour itself. Interestingly enough, Dr Bruce Edgar’s bass horn designs are also based on exponential and hyperbolic contours, although he swears by tractrix mid-range horns. On a side note, it is interesting to observe the fact that the more an exponential horn is foreshortened, the more it resembles a tractrix horn.

 Based on these facts, I decided to employ an exponential curvature in this project.

 The exponential curve is easily calculated as:
A(x) = At * e^(M*X), Where:

X = Distance from throat,
A(x) = Area at X,
At = Area at throat,
M = Fc * 4 * p/c, and
c = Speed of sound