We investigate the wave characteristics and homogenization theory of soft-matter layered structures in the low-frequency P-wave limit. Using the potential method, we derive a closed-form dispersion relation and identify three distinct wave modes: quasistatic, resonance, and slip. These modes are governed by equivalent interface conditions—namely, a continuous interface for the quasistatic mode, a springlike interface for the resonance mode, and a sliplike interface for the slip mode. Furthermore, we propose a simplified model that captures P-wave characteristics in the high-frequency S-wave regime. Our findings provide a unified framework for wave–structure interactions across solids, liquids, and soft-matter composites, enabling predictive design of metamaterials with programmable wave responses. This study offers insight into the fundamental understanding of layered media and provides direct design principles for applications in acoustic cloaking, vibration damping, and biomedical imaging.

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• Acoustic wave phenomena
• Cloaking
• Functional materials
• Surface instabilities
• Wave-structure interactions
• Acoustic metamaterials
• Composite materials
• Elastomers