## The Simulation with Foam

The simulation model requires a small modification compared to the one used for the fibrous material. This is because foam, unlike fiber, occupies only part of the line volume. To recreate these conditions I “built” a second TL in parallel to the existing one to represent the empty volume. As for the other damping materials we can consider negligible the actual volume occupied by the foam: the density of the compact polyurethane (bulk density) is about 1200 kg/m3 while the same material in the form of foam has a typical density of 30 kg/m3 which leads, in case of 50% filling, to a percentage of volume occupied equal to 0.0125% of the total.

Figure 10. Test line impedance 50% filled with foam. a) Measurement. b) Simulation.

Figure 10 shows the behaviour of the impedance curve of a TL only partially filled with absorbent material and the excellent correspondence with the simulation. In Figure 11 the measured and simulated frequency response.

Figura 11. Risposta in frequenza linea test riempita al 50% con foam. a) Misura. b) Simulazione

In this model there are two flow resistances, one to represent the empty line losses and one to represent those of the portion of the line occupied by the foam. Contrary to what one might think, the speed of sound setting should be kept the same in both lines. The line filled with foam is conceptually very similar to Olney’s acoustic labyrinth [4] and those who have carefully read the first part will remember Olney’s doubts about the possibility of obtaining a model based on the one-dimensional equation of the acoustic wave (which assumes that the wave moving progressively along the duct is planar) to predict the behaviour of a tube lined with absorbent material. Olney had reasonably assumed that the speed of sound in the peripheral zone would be lower than that in the central zone, without damping material, and that the wavefront would become more and more convex as it moved along its path (Fig. 12).

Figure 12. (Olney).

Actually, this does not seem to happen, at least not as much as suggested by Olney. In fact, in the empty line impulse response, we have no peak at 2.75 ms to anticipate the slower acoustic wave traveling inside the foam. One could assume that the wave is deformed to the point where it starts to move in a partially incident way inside the tube, so as to cross both portions of the line, but this is pure speculation. The interesting thing to note is that, even in this case, once the principles of analogies are correctly applied, the model generates a simulation that predicts the measurements in a satisfactory way.

Andrea Rubino