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Attenuation in acoustic waveguides with lossy impedance boundary conditionsare associated with non-Hermitian and non-normal operators. This subject hasbeen extensively studied in fundamental and engineering research, and it hasbeen traditionally assumed that the attenuation behavior of total sound powercan be totally captured by considering the decay of each transverse modeindividually. One of the classical tools in this context is the Cremer optimumconcept that aims to maximize the attenuation of the least attenuated mode.However, a typical sound field may be a superposition of a large number oftransverse modes which are nonorthogonal, and the individual mode attenuationmay have little to do with the total sound power attenuation. By using singularvalue decomposition, we link the least attenuated total sound power to themaximum singular value of the non-normal propagator. The behaviors of the leastattenuated total sound power depend only on the lossy boundary conditions andfrequency, but are independent of sources. The sound may be almost non-decayingalong the waveguide transition region for any lossy impedance boundaryconditions although all modes attenuate exponentially. This spatial transientappears particularly strongly if the impedance is close to an exceptional pointof the propagator, at which a pair of adjacent modes achieve maximumattenuation predicted by Cremer optimum concept. These results are confirmedusing non-modal numerical calculations and a two-by-two toy model.

Spatial transient behavior in waveguides with lossy impedance boundary conditions