Acoustic propagation in ducts with varying cross sections

The method of multiple scales is used to derive the equations that describe the spatial and temporal variation of the amplitudes and phases of a wave packet propagating in slowly varying hard-walled or lined ducts. The analysis is carried out for rectangular as well as circular ducts. These equations are statements of the conservation of energy. For large admittance or high-frequency modes, an approximate expression is obtained for the attenuation. This expression shows that all possible acoustic modes are attenuating. The results also show that decreasing the cross sectional area can lead to elimination of some of the acoustic modes.