Rayleigh made the earliest study of the acoustical performance of sound‐absorbing materials in 1883 [125]. That was followed by a theoretical study of the propagation of sound in rectangular lined ducts by Sivian in 1937 who used the impedance of the lining material as the acoustical boundary condition for his model [126]. Morse [127] extended Sivian’s model in 1939. Sabine produced a simple empirical model for the attenuation of lined ducts in 1940 [128]. Many other researchers have continued these studies since then [130–148]. Most of these studies involve propagation in rectangular ducts. Scott continued these early studies in 1946 and made measurements of sound absorption in circular ducts [129]. Zwicker and Kosten published an important book on sound‐absorbing materials in 1949 [130]. Cremer went on to develop Morse’s model in 1953 [131].
The theoretical models are based on the assumptions that the sound absorption in ducts depends on lining thickness, material flow resistivity (flow resistance per unit length), lining thickness, wavelength of the sound, porosity (fractional volume of fluid in the porous material) and duct length. Lined ducts are normally used where mean flow is present and sound attenuation is found to depend not only on flow velocity magnitude but also on direction. (Such lined ducts are usually placed on the intakes and exhausts of fans and blowers, compressors, power plant turbines, etc.) Meyer et al. [132] produced one of the first studies on the influence of flow on sound attenuation in absorbing ducts in 1958.
Most early theoretical studies assumed that sound cannot propagate in the material and that the lining material can be completely described by its wall impedance [127, 131]. The material is then known as “locally reacting.” In Scott’s analytical model, sound can propagate in the lining in the direction parallel to the duct axis and the material is known as “non‐locally reacting” or “homogeneous.” Kurze and Ver [133] have developed a theory for the attenuation of ducts lined with non‐isotropic sound‐absorbing materials, which in the extreme cases gives the results of models for the locally reacting lining models of Morse [127] and Cremer [131] and the homogenous (non‐locally reacting) lining model of Scott [129]. A number of theoretical and experimental studies on lined ducts and ducts with multiple parallel absorbing elements (usually termed baffles) have been published in recent years [134–148].
In 1978, Ver published a review of the attenuation of sound in lined and unlined ductwork of rectangular cross‐section [134]. The review was mostly concerned with rectangular ducts lined on four sides used in air‐conditioning systems. In 1982, Ver published another paper, but this time it was concerned with parallel‐baffle mufflers such as those used to quiet turbines in power plants or large fans or blowers [136]. Several other authors since then have conducted theoretical studies to predict the sound attenuation of lined ducts [137–140]. The paper by Astley and Cummings [140] in 1987 used a finite element scheme. They provided a series of design charts for air‐conditioning ducts, giving the attenuation per unit length of the ducts lined on four sides for a range of geometries and flow resistivities. They also included the effect of mean flow [140].
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