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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...

enter image description here

and I would like to get this instead !

enter image description here

Any idea what is wrong? Thank you.

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  • $\begingroup$ Google for "transfer matrix methods" in acoustics. $\Im$ means the imaginary part of the complex impedance $Z_{in}(\omega)$ $\endgroup$ – alephzero Mar 19 at 0:01
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The problem was that I was using the speed of light instead of the speed of sound and the viscosity was not expressed properly. This is working now !!!

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

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