Mode hybridisation strategies for subwavelength sound attenuation in miniaturised Helmholtz resonators.

Controlling wave propagation is a topic of fundamental importance across many areas of physics. Photonic crystals have proven highly effective for this purpose. However, because they rely on Bragg interference, they typically require relatively large structural dimensions. Metamaterials, on the other hand, attenuate sound by breaking the mass-density law, and can exhibit deep subwavelength attenuation. Miniature acoustic resonators capable of broadband, tuneable sound manipulation are essential for next-generation compact noise-control systems. Here, we investigate metamaterials for subwavelength operation based on resonant unit cells composed by Helmholtz resonators. We interrogate the introduction of structural compliance as means to decrease the operating frequency of the device. Starting from a conventional HR, this work shows that geometric scaling alone is constrained by enhanced thermoviscous losses, limiting achievable miniaturisation. By incorporating compliant elements in the unit cell, it is possible to couple acoustic and mechanical degrees of freedom, producing hybridised modes that enable frequency reduction and direction-dependent while remaining within the deep subwavelength regime. Complex frequency analysis reveals that compliance and viscoelastic losses jointly govern the motion of resonant poles and zeroes, offering a unified framework to interpret the balance between miniaturisation, absorption, and bandwidth. The proposed metamaterials achieve subwavelength at a factor of λ/38, and their design is made possible by innovative multi-material light-based 3D-printing techniques. The results establish compliance-driven modal hybridisation as a general principle for engineering compact and tuneable acoustic systems, bridging traditional HRs and emerging soft metamaterials. This framework offers a physically grounded route toward scalable, broadband, and directionally responsive acoustic systems.