## The Speed of Sound in the Transmission Line Pt.1

As we have seen in the introductory article, Bailey, in [6], argues that long-fibre natural wool, if correctly positioned along the transmission line, has the property of slowing down the sound wave by a factor between 0.7 and 0.8 compared to its speed in free air. Bradbury [7], attracted by the possibility of reducing the size of transmission line and horn-loaded loudspeakers, went deeper into the subject and found slowdowns in the speed of sound of up to 50%. Although these results date back to 1976, they are still the subject of debate today; however, the most respected opinions tend to refute Bradbury’s thesis. Martin King, for example, pointed out that Bradbury may have used a tapered transmission line for his tests without taking into account the impact that this geometry would have had on the frequency of the first resonance modes. The Bradbury treatment is basically theoretical and the type of TL used for the tests is not clearly described. In the final part of his article, however, the data provided by a model “similar to the enclosure of Bailey” is analyzed. The section of the speaker in question can be seen in Figure 1.

Although it is not a “tapered” transmission line, it is certainly true that the coupling chamber of about 50 litres, located behind the woofer, would have had equally marked effects on the frequency position of the resonances measured at the opening and, in those circumstances, the application of the classic equation f=c/(4xL), where c is the speed of sound and L the length of the duct, to calculate the speed of sound at those frequencies, would have given misleading results.

I have never tested the properties of natural wool inside the TL, but I have reconstructed Bailey’s speaker with the simulator. A comparison of the frequency response graph of the simulated system with the one in the article confirms a reduction in the speed of sound by a factor of 0.8 as indicated by Bailey himself.

Augspurger in [3] also argues that wool is not able to halve the speed of sound and that, even if it were, the results would not be those hypothesised by Bradbury. With his software he therefore simulates a TL tuned at 100Hz and, keeping the amount of absorbent material constant, varies the speed of sound in a range from 0.4 to 0.8 times that in air. The result is visible in Figure 2.

Figure 2. Response of a TL tuned at 100 Hz with variable sound velocity.
Relative speed: 0.80, 0.63, 0.50 and 0.40 (Auspurger).

Augspurger points out that although there is a slight change in system response, this becomes significant only one octave below f at -3 dB, which remains ‘stubbornly’ fixed at 100 Hz. My software, like Augspurger’s, allows you to set the speed of sound and the amount of damping material independently of each other. Launching a multiple simulation (in Spice you can implement it very easily with the .step param function) with sound speed of 344, 300 and 250 m/s you get the graph, similar to Augspurger’s, visible in Figure 3.

But if we look at the results of the same simulation from the perspective of the loudspeaker impedance curve we realize that the variations are really important (Fig. 4) and cannot in any way be ignored. This is all the more true the more the absorbent affects the behaviour of the TL, as in the case of polyurethane foam, which we will see in the second part.

Andrea Rubino