During my research about the spherical wave horn I also played around with the tractrix profile. While investigating the wave font expansion of the spherical wave horn I came up with the idea to investigate the same expansion for the tractrix horn and how far the surface area of the propagating wave front follows the exponential expansion coefficient m:
\tag{1}m = \dfrac{4\pi \cdot f_c}{c_s}
\tag{2}S_{exp}(x) = S_0 \cdot e^{m \cdot x}
The formulas for the traxtrix horn are known since a long time:
\tag{3}r_0 = \dfrac{c_s}{2 \cdot \pi \cdot f_c}
\tag{4}x=r_0\ln\left(\dfrac{\sqrt{r^2-y^2}+r_0}{y}\right)-\sqrt{r_0^2-y^2}
r_0 is the mouth radius of the tractrix horn. It can be also denoted as the construction vector of the tractrix horn. If we put y=y_0 in eq. (4), where y_0 is the throat radius, then we get the length of the tractrix horn. At the throat we are able to calculate the initial surface area of the spherical wave front very similar to the spherical wave horn:
\tag{5}S_{trx, 0} = 2 \cdot \pi \cdot r_0 \cdot h_0
\tag{6}h_0 = r_0-\sqrt{r_0 ^ 2-y_0 ^ 2}
With the same procedure the surface area of the wave front can be calculated at every point of the horn axis as we know the radius and the section wise corresponding y value.
\tag{7}S_{trx}(x) = 2 \cdot \pi \cdot r_0 \cdot h(x)
If we compare the surface area expansion it becomes obvious that the tractrix horn does not expand as fast as described in eq. (2). The surface area expansion is too small and the biggest difference occurs at the mouth. The following diagram is showing the comparison between both surface area expansions:
What can we do now if we want the tractrix horn to expand following to eq. (2) while preserving the tractrix character of the curve? To my surprise the solution is quite easy and I will call it “Constraint Boundary Linear Combination of Tractrix Horns”.
- At the beginning of the procedure a tractrix horn with its general properties is selected for a chosen cut-off frequency which gives r_0. Then the length of the horn l_{trx}, S_0, h_0 can be calculated as described above. These values, except r_0, will be set as reference and not changed anymore.
- An iterative procedure is initiated along the horn axis and at a distinct position x, which will be held fix, the two surface areas eq. (2) and eq. (5) will be made equal. To achieve this a larger tractrix horn is needed at every x position except for the throat. This can only be determined by iterating (increasing) the radius r_0 at the current horn axis position. With increased r_0 the new resulting tractrix horn is aligned to the initial horn with both horn mouths in the same plane. Then at the same x position the temporary horn with a longer radius will have a larger surface area. The iteration is repeated until S_{etr} = S_{exp}. Now, we have found the desired y-value.
What we get now is a tractrix like profile that opens faster than the well-known common tractrix horn. If we assume spherical wave fronts then the new horn fulfils eq. (2). Of course, the new proposed horn has no constant radius anymore. Especially in the last section of the new true expansion tractrix horn the radius is increasing very fast up to the distinct end value. The corresponding cut-off frequency is considerably lower at mouth compared to the common tractrix horn:
How the new horn profile compares to the common tractrix profile can be gathered from the last picture. The black lines are the common tractrix horn profile with 400 Hz cut-off frequency and the red lines the new proposed true expansion tractrix horn.
I have a spread sheet for this new horn profile but it is based on a quite old program code version and for example does not contain the final version of the stereographic projection. So, I will not publish it at this time.