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In your expression, the term $m\ddot x$ is the "inertial term". It describes the force needed to accelerate ($\ddot x$) the mass $m$. Hence, "inertial". It can be neglected if you know that that term is small compared to other terms in the expression (for example, the velocity term with $\dot x$ in it). That depends on the context.


If $x$ is displacement and $m$ represents mass and the mass is traveling at near constant velocity, then the acceleration is small, so the first term may be neglected.


If the duct structure has passive modes that are within the bandwidth of the 'noise' you are imposing, and the active controller they can sap and release energy resulting phase shifts. Also the tube like structure whether closed or open will impose a standing wave resonance with all the possible harmonics that may be near or not near to the frequency you are ...

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