# How to find the input impedance of a woodwind instrument? (playing frequency of a woodwind instrument)

I'm trying to reproduce the model described in this paper https://hal.archives-ouvertes.fr/file/index/docid/683477/filename/clarinette-logique-8.pdf.

The logical clarinet is a succession of 18 cylindrical pieces of tubing. Each segment has 3 parameters ; $$a$$ the radius of the cylinder, $$b$$ the radius of the tone hole and $$d$$ the length of the tube. From theses values we compute $$H$$ the transfer matrix of a cylindrical piece of tubing with $$L$$ the length and $$Z_c$$ the characteristic impedance. $$H= \begin{bmatrix}\cosh(\Gamma L) & Z_c \sinh(\Gamma L)\\\frac{1}{Z_c} \sinh(\Gamma L) & \cosh(\Gamma L)\end{bmatrix}$$

Each tone hole is modeled as a $$T-$$junction with $$Z_{st} = Z_s + Z_a/4 + Z_h$$, where $$Z_h$$ is the input impedance of a tone hole.

How can i obtain $$Z_{in}$$ the input impedance of the whole instrument? How can i get the playing frequency of the instrument?

Could you explain what means $$\Im[Z_{in}(\omega)] = 0$$?

Thank you.

[UPDATE]

Ok I think I got the model correct thanks to this other paper https://hal.archives-ouvertes.fr/file/index/docid/836022/filename/interactions_new.ps.

The transfer matrix for a piece of cylinder is $$T_{cyl}=\begin{bmatrix}\cosh(\Gamma L) & Z_c\sinh(\Gamma L) \\ Z_c^{-1}\sinh(\Gamma L) & \cosh(\Gamma L) \end{bmatrix}$$

The transfer matrix for a tone hole is $$T_{hole}$$=$$\begin{pmatrix}1&Z_a/2\\0&1\end{pmatrix}\begin{pmatrix}1&0\\Z_s^{-1}&1\end{pmatrix}\begin{pmatrix}1&Z_a/2\\0&1\end{pmatrix}$$

The pressure and velocity of the whole instrument is $$\begin{bmatrix}P_{in}\\U_{in}\end{bmatrix}$$=$$(\Pi_{i=1}^nT_i)\begin{bmatrix}Z_{rad}\\1\end{bmatrix}$$

Where $$Z_{rad}$$ is the radiation impedance and $$T_{i}$$ alternates between cylinders and tone holes matrices.

The impedance of the whole instrument is $$Z_{in}=P_{in}/U_{in}$$

Finally, the playing frequency of the instrument satisfy the equation $$\Im[Z_{in}(\omega)] = 0$$

The only problem is that when I print the values of $$\Im[Z_{in}(\omega)]$$ in function of the frequency, for a simple piece of tubing, i get this... and I would like to get this instead ! Any idea what is wrong? Thank you.

• Google for "transfer matrix methods" in acoustics. $\Im$ means the imaginary part of the complex impedance $Z_{in}(\omega)$ – alephzero Mar 19 at 0:01

This is the imaginary part of the impedance inside a cylinder of 15mm radius and 60cm length. 