Natural sonic crystal absorber constituted of seagrass (Posidonia Oceanica) fibrous spheres

We present a 3-dimensional fully natural sonic crystal composed of spherical aggregates of fibers (called Aegagropilae) resulting from the decomposition of Posidonia Oceanica. The fiber network is first acoustically characterized, providing insights on this natural fiber entanglement due to turbulent flow. The Aegagropilae are then arranged on a principal cubic lattice. The band diagram and topology of this structure are analyzed, notably via Argand representation of its scattering elements. This fully natural sonic crystal exhibits excellent sound absorbing properties and thus represents a sustainable alternative that could outperform conventional acoustic materials.


Scattering properties
(a) depicts the refection and transmission coefficients in amplitude and the absorption coefficient as experimentally measured, numerically simulated via Finite Element Method -FEM-(see Section Numerical Modeling of the main text), and calculated via the Multiple Scattering Theory -MST-(see Methods). All curves are in good agreement, with an exact superposition of the FEM and MST simulations. For comparison purpose, the reflection and transmission coefficients in amplitude calculated with the MST in the case of perfectly rigid cylinders are also provided. In this latter case a clear band gap is visible around f B = c 0 /2d ≈ 3950Hz, with a very low transmission and large reflection due to the Bragg interference. In the lower pass band, the transmission is maximum, i.e., vanishing reflection, at six different frequencies corresponding to the Fabry-Perot interferences occurring within these six layers of periodic cylinders. Of particular interest is the fact that only four peaks in transmission are visible within the second pass-band because of a strong overlapping of two Fabry-Perot interferences. In the soft and lossy case, the transmission coefficient only presents a smooth drop from a frequency band that is slightly lower than f B , implying the almost absence of Bragg band gap. Only the reflection coefficient presents a peak within this frequency range, thus exhibiting the Bragg interference feature. This is explained by the weak impendance contrast bewteen the air medium and that of the scatterers. In addition, only the reflection coefficient presents drops at the location of the Fabry-Perot interferences. This feature is also testified by the dispersion relations presented in Figure 2 vanishes within the band gap, with a large imaginary part in the rigid case, it is almost linear in the soft and lossy case. Only an inflection point is noticed in its imaginary part at the location of the Bragg interference. Please note that the numerically reconstructed wavenumber from the finite depth sonic crystal match that evaluated for the infinite periodic arrangement via MST.

Argand diagram, symmetry inversion and acoustic absorption
The most interesting feature is the absorption encountered in the 2-dimensional man modified sonic crystal, which is higher than 0.8 for frequencies higher than 1000 Hz. At the location of the Bragg interference, it presents a local minimum associated with a maximum of reflection. Of particular interest is the fact that the absorption is quasi-perfect at a frequency within the second pass band. To get a physical insight of this feature, Fig. 2(d) depicts the Argand diagram of R and T , i.e., the value of a quantity in the complex plane in function of the frequency. All elements are located in the unitary circle, because of the energy conservation, and each element describes counter-clockwise ellipical loops, because of the time Fourier convention (e −iωt ). The results are relatively similar to those of the 3-dimensional fully natural sonic crystal discussed in the main text. The movie FPcoalescence.mp4 depicts reflection, transmission and absorption coefficients and Argand digramm of R and T in the lossy and lossless cases as calculated with MST for scatterer radii varying from 14.5 mm to 15.6 mm. This movie points out the coalescence of Fabry-Perot interferences around the symmetry inversion frequency (≈ 5500 Hz). The lossless case has been calculated by considering only the real parts of the density and bulk modulus used in the lossy case. The coalescence of two Fabry-Perot modes is clearly seen around 5500 Hz, especially on the reflection coefficient in the lossless case. The absorption coefficient at low frequency can be further enhanced by increasing the filing fraction ff . The calculated absorption coefficient of the closely packed 2-dimensional man modified sonic crystal, i.e., when r = d/2 thus leading to ff = π/4, is depicted on Figure 2(a). The absorption coefficient of this closely packed configuration is clearly higher than that of the corresponding bulk material for all frequencies but those around the Bragg interferencies, while less material is used and less volume is occupied.

Sample preparation
Half of the 15 remining samples was cut in a cylindrical shape with 30 mm in diameter and ≈ 21.5 mm in height. These cylindrical samples were then cut in half along their axis, resulting in semi-cylindrical samples.

Multiple Scattering Theory
The Multiple Scattering Theory (MST) is applied to calculate the reflection R MST and transmission T MST coefficients and the acoustic parameters of both 2-dimensional finite and infinite depth structures. The notation adopted in 2 is used. As far as it concerns the finite-depth structures, the upstream and downstream pressure fields read as: where k 1q = k i 1 + 2qπ/d and k 3q = k 2 0 − k 2 1q , with Re k 3q ≥ 0 are the Bloch waves and R q and T q are the reflection and transmission coefficients of the q-th Bloch wave. The incident plane wave is k i = k i 1 x 1 + k i 2 x 2 = k 0 x 2 at normal incidence. The reflection and transmission coefficients are calculated in the Cartesian coordinate system considering a super-cell comprising 6 identical cylinders via the Multiple Scattering Theory as proposed in 3 . The Schlömilch series are evaluated with the formulae provided by Twersky 4 and the scattering coefficients of a single cylinder takes the form: where α j = k j R, β j = α j /ρ j , J n (x) and H (1) n (x) are respectively the first kind Bessel and Hankel functions of order n, anḋ χ(x) = dχ(x)/dx. Below the Wood anomaly, R MST = R 0 and T MST = T 0 .
As far as it concerns the infinite structure, the dispersion relation is determined by solving the following eigenvalue problem where a ± u (resp. a ± u ) are the complex amplitudes of the waves at the upper (resp. lower) interface of d-thick row of d periodic arrangement of cylinders along the positive and negative x 3 directions, T is the matrix of components T Q q the transmission 3/4 coefficient of the q-th Bloch wave when the layer is excited by the Q-th Bloch wave, R is the matrix of components R Q q the reflection coefficient of the q-th Bloch wave when the layer is excited by the Q-th Bloch wave, 0 is the zero matrix, Id is the identity matrix, and k B 3 is the normal component of the Bloch wavenumber to be determined. Along the ΓX direction, k B 3 = k B the Bloch wavenumber.