Spiral Functions for Horns – The Sici Spiral

The last post in this series deals with the Sici spiral (Link1). It is similar to the Nielson spiral. I already mentioned that this spiral is my personal favourite of the three spiral functions presented on my page. On the one hand this is because the curve reminds me of JMLC horns and on the other hand because of the simple relationship of the tangent with respect to the basic rotation angle. In addition, we will see that the tangent vector, when the parameters are selected appropriately, results in a nearly constant length over large areas of the horn curve and only expands towards the horn mouth.

The cartesian parametrization look simple on the first view

\tag{1a}x =-a \cdot Ci({\phi})

\tag{1b}y = a \cdot \left( \frac{\pi}{2}-Si({\phi})\right)

but Ci(\phi) and Si(\phi) are the cosine and sine integrals (Link2). These integrals need to be solved but again as for the Cornu spiral these integrals can be developed as a series expansion. With proper offsets defined the Sici spiral becomes usable as horn profile function.

\tag{2}x_0 =-a \cdot Ci({\phi_s})

\tag{3a}y_0 = y(\phi_s) =  a \cdot \left( \frac{\pi}{2}-Si({\phi_s})\right)

\tag{3b}y_0 = y(\phi_0) =  a \cdot \frac{\pi}{2}

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Just another HiFi blog?

Rather not. This blog is intended to give my work on horns a suitable frame and to present information in a coherent form.

I remember very well my first experiences with horn loudspeakers and was fascinated by their sound from the first moment on. These experiences were based on the visit of various HiFi shows or special demonstrations. Inevitably one meets also the usual representatives of round horns, i.e. Tractrix, JMLC or similar/equal to Klangfilms spherical wave horn (Kugelwellenhorn). An awesome and almost comprehensive overview about the Kugelwellenhorn can be found here: Ref. 1.

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