Nonlinear Propagation of a Wave Packet in a Two-Dimensional Lined Duct
T method of multiple scales is used to analyze the nonlinear effects of the gas motion and the acoustic lining material on the propagation and attenuation of a wave packet in a two-dimensional lined duct of a uniform cross section. The partial differential equations describing the space and time variations of amplitude and phase are obtained and used to show that both the monochromatic waves and the pure amplitude-modulated wave packet are stable in the frequency range of interest. The spatial attenuation vs frequency curve for the pure amplitude-modulated wave packet is the flattest one compared with those of linear and nonlinear monochromatic waves. For the pure amplitude-modulated wave packet, a threshold exists near the resonant frequency below which the nonlinear effect is favorable for spatial attenuation, although adverse for temporal attenuation, and above which the nonlinear effects are reversed for spatial attention and temporal attenuation, respectively.