Fig. 2

Nodal rings for sound waves in 3D phononic crystals. a Photograph of the 3D sample, with an enlarged front-view shown in the inset. b Schematics of a magnified side view of the sample, which is constructed by layered stacking structures. The thickness of the plate h_plate = 1.80mm in which staggered holes with diameters d₁ = 2.16mm and d₂ = 3.84 mm are penetrated and among which square pillars of height h_rod = 3.36mm and side length d₀ = 7.20mm are periodically arranged. The distance between the staggered holes is a = 12mm, and the total height of a single layer h = h_plate + h_rod. c The simulated (purple lines) and measured (colors) first and second bands along the high symmetry lines in the first BZ. Linear crosses occur at a frequency of approximately 10.6 kHz. d Simulated nodal rings distributed in reciprocal space. The colors denote the frequency variations, which range from 10.51 to 10.68 kHz. e Experimentally measured nodal rings at a frequency of 10.6 kHz shown at the planes k_zh/π = 0 and k_zh/π = 1, which agree with the simulated data (marked by red dashed curves)