Have a look at the section on the input impedance of a transmission line in https://en.wikipedia.org/wiki/Transmission_line.

the propagation constant gamma is alpha + j beta, where alpha is the attenuatuion in nepers per metre and beta is the wavenumber, in radians per metre. The article is about electromagnetism but the same equations apply in that section for acoustics.

Big gamma is the reflection from the far side, which can be positive as well as negative, depending on the ratio of the characteristic impedances on the two sides.

The Z calculated is actually the Z just inside the front of the material.

Please make sure that you are using the right modelling approach - the "normal" impedance modelling fits well for homogeneous thickness (e.g. variations of thickness small to the relevant wavelength) for plane waves and perpendicular direction. Basically, what you would measure in an impedance tube for plane waves.

So, your problem is probably more about modelling the sound field within your pourous medium (taking ito accout its geometry) - however, that is not such an easy problem to solve...

If you only know the surface impedance of your porous slab for a given thickness, you cannot predict exactly that of a porous slab made of the same material and of different thickness.

A good guess is that the resistance will increase for increasing thickness and that the reactance curve will be shifted towards the low frequencies.

As mentioned by @Malcom, the surface impedance of a porous slab of thickness L, at normal incidence and rigidly backed is : Z = -i Zeq cot(keq L), with Zeq and keq the characteristic impedance and the wave number of the medium (the porous medium is considered as an equivalent fluid), respectively.

To predict Z for different L values, you need to know keq and Zeq. These 2 complex and frequency dependent variables can be retrieved by measuring 2 acoustic properties of your porous slab, e.g., its reflection and its transmission coefficients.

Alternatively, if your porous material is conventional, you can certainly find its transport parameters (porosity, resistivity...) in some datasheet or paper and use an equivalent fluid model (Miki, Johnson...) to predict its keq and Zeq.

Jean Boulvert

Thanks for your reply. can the equivalent fluid model be used for predicting the equivalent values for keq and zeq?

do you have the full name for the paper by (Miki, Johnson...)

Yes.

If you want to learn more about the acoustics of porous media, you can refer to the book of Allard and Atalla, "Propagation of sound in porous media" or to the Matelys lab website https://apmr.matelys.com/

The propagation models are listed and explained in both references.