Time-domain illposedness of effective frequency-domain boundary conditions for quiescent viscothermal acoustics

Accurate simulations of sound propagation in narrow geometries need to account for viscous and thermal losses. In this respect, effective boundary conditions that model viscothermal losses in frequency-domain acoustics have recently gained in popularity. Here, we investigate the time-domain analogue of one such boundary condition. We find that the thermal part of the boundary condition is passive in time domain as expected, while the viscous part is not. More precisely, we demonstrate that the viscous part is responsible for exponentially growing normal modes with unbounded temporal growth rates, which indicates ill-posedness of the considered model. A finite-difference-time-domain scheme is developed for simulations of lossy sound propagation in a duct. If viscous losses are neglected the obtained transmission characteristics are found to be in excellent agreement with frequency-domain simulations. In the general case, the simulations experience an instability much in line with the theoretical findings.