Fig. 1. The origin of anomalous constitutive parameters in acoustics.

(A) A spring-coupled mass-in-mass oscillator. M₂ is assumed to slide without friction inside a cavity formed by M₁, and K is the spring constant. (B) The oscillator’s apparent mass $\overline{\mathit{M}}$ plotted as a function of angular frequency ω, where ${\mathrm{\omega }}_0=\sqrt{\mathit{K}/{\mathit{M}}_2}$ is the resonant frequency. $\overline{\mathit{M}}$ is divergent at ω₀ and can take negative values in a narrow frequency region that is shaded gray. (C) If there are two resonances, the average displacement 〈X〉 can cross zero at an antiresonance frequency $\tilde{\mathrm{\omega }}$, at which the effective mass/mass density displays a frequency dispersion similar to that shown in (B). Here, the red and green dashed curves show the displacement associated with the first and second resonances (denoted ω₁ and ω₂), respectively. The black solid curve represents the sum of the two displacements, and it crosses zero at $\tilde{\mathrm{\omega }}$.