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}

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This post will be slightly different because it was written in German. German is the language of the original patents used and the main references of the literature used. Alone, to use the original terminology directly and pay homage to the inventors, I have decided to use the German language as a consistent continuation. I think that nowadays it is no longer an obstacle not to publish in English as powerful algorithms for automatic translation are available. For the captions, however, I have used English for the sake of simplicity.

As you might already guess by the name of this website, the spherical wave horn inspired my work a lot. If we assume expanding spherical wave fronts in round horns and want to stretch the round profile to an ellipse then we must inevitably think about ellipsoidal surface areas and of course the associated mathematics. My own learning phase was not easy either, until I found myself able to mathematically master the challenge. I will describe the results of my work in this post and try to give as many details as possible and describe as much math as necessary. The whole stuff is quite complicated and therefore we will simply start with the two-dimensional part and then gradually come to the ellipsoidal surfaces.