Sound Propagation Through Ducts

When sound propagates through a duct system it encounters various elements that provide sound attenuation. These are lumped into general categories, including ducts, elbows, plenums, branches, silencers, end effects, and so forth. Other elements such as tuned stubs and Helmholtz resonators can also produce losses; however, they rarely are encountered in practice. Each of these elements attenuates sound by a quantifiable amount, through mechanisms that are relatively well understood and lead to a predictable result.

Theory of Propagation in Ducts with Losses

Noise generated by fans and other devices is transmitted, often without appreciable loss, from the source down an unlined duct and into an occupied space. Since ducts confine the naturally expanding acoustical wave, little attenuation occurs due to geometric spreading. So efficient are pipes and ducts in delivering a sound signal in its original form, that they are still used on board ships as a conduit for communications. To obtain appreciable attenuation, we must apply materials such as a fiberglass liner to the duct's inner surfaces to create a loss mechanism by absorbing sound incident upon it.

In Chapt. 8, we examined the propagation of sound waves in ducts without resistance and the phenomenon of cutoff. Recall that cutoff does not imply that all sound energy is prevented from being transmitted along a duct. Rather, it means that only particular waveforms propagate at certain frequencies. Below the cutoff frequency only plane waves are allowed, and above that frequency, only multimodal waves propagate.

In analyzing sound propagation in ducts it is customary to simplify the problem into one having only two dimensions. A duct, shown in Fig. 14.1, is assumed to be infinitely wide (in the x dimension) and to have a height in the y dimension equal to h. The sound wave travels along the z direction (out of the page) and its sound pressure can be written as

(Ingard, 1994)

Figure 14.1 Coordinate System for Duct Analysis r

Figure 14.1 Coordinate System for Duct Analysis r

duct LINER,

where p = complex sound pressure (Pa) A = pressure amplitude (Pa)

q z and q y = complex propagation constant in the z and y directions (m-1)

The propagation constants have real and imaginary parts as we saw in Eq. 7.79, which can be written as

0 0