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}