Acoustic modes in a rectangular flow duct with a wall of finite acoustic impedance value

The acoustic field in a rectangular flow duct of infinite length was numerically investigated. One of the walls of the duct was assumed to have finite acoustic impedance while the other three were rigid walls. Uniform flow profile was also assumed. The propagation constants, namely, complex wave numbers, were calculated for various impedance values and the Mach numbers of the flow from the boundary condition on the finite impedance wall derived by Myers. First, we calculated the propagation constant without uniform flow. If all four walls are rigid and the frequency of the sound is under the cut-off frequency, only zeroth mode propagates along the duct and higher modes decay rapidly. In contrast, if one wall has a finite impedance value, two modes can have comparable amount of attenuation. The acoustic field in the duct would be a superposition of those less decaying modes depending on the inlet condition. As for the flow effect, the attenuation tends to be decreasing in fair wind, and vice versa. This tendency, however, is not general because the propagation constant moves on the complex plane in intricate manner.