## 1. Introduction

Periodic arrays of rigid scatterers embedded in a fluid are the analogues for the acoustic waves of the crystalline structures for the electrons or the photonic crystals for the electromagnetic waves. Such structures are known as Sonic Crystals (SCs) [

1] and the exploitation of the periodic distribution of scatterers in such structures have been intensively used to control the acoustic wave propagation. Perhaps the most known property of such systems is the presence of band gaps, ranges of frequencies in which the wave propagation is forbidden. The band gaps appear at high symmetry points in the Brillouin zone due to the presence of a degeneracy of the band structure produced by the Bragg interferences in the diffractive regime (

$\lambda \simeq a/2$,

λ being the wavelength of the incident wave and

a the lattice constant characterizing the periodicity of the structure). Many interesting physical phenomena arise from this particular dispersion relation such as wave localization [

2,

3], excitation of evanescent waves [

4,

5], and relevant applications concerns filtering [

6] and wave guiding [

7,

8,

9]. In particular, many approaches have been proposed to degenerate the band and thus enlarge the band gaps [

10,

11,

12]. Some possibilities consist of either reducing the total symmetry of the crystal in order to remove some band degeneracies, allowing the appearance of complete gaps [

13] or optimizing the shape and arrangement of the scatterers [

14,

15,

16,

17,

18]. The dispersion relation is governed particularly by both the periodicity and the shape of the scatterers providing different tools to tune the wave propagation.

Interesting properties can be obtained in the low frequency regime (

$\lambda \ll a$) in periodic structures if local resonators are used as scatterers. In acoustics, the pioneering works of Bradley [

19] and Sugimoto [

20] theoretically and experimentally examined the propagation of sound waves in a waveguide loaded periodically with local resonators (quarter wavelength and Helmholtz resonators). In these systems two different mechanisms are responsible of the generation of band gaps. The Bragg interferences produce the band gap due to the periodicity, while the resonances produce other band gaps when the frequency of sound waves coincides with the natural frequency of the resonators. The latter induces an hybridization between the resonance and the dispersion of the non resonant periodic structure. This feature has been used to introduce the concept of acoustic metamaterials with resonant band gaps at lower frequencies than the Bragg band gap [

21,

22,

23], as well as to improve the absorption capabilities of porous materials in the low frequency regime [

24].

In this article we exploit the idea of the coupling of the local resonant scatterers to generate multiple resonances that can be combined with the effect of periodicity in order to produce a broadband frequency region with high transmission loss. We experimentally and theoretically study the propagation properties of a three-dimensional SC made of square cross-section rod rigid scatterers incorporating a periodic arrangement of quarter wavelength resonators of circular cross-section. Particularly, we analyze different configurations in which the coupling between the resonators in the structure generates multiple resonances that are designed to be close to the Bragg band gap. This combined effect produces an overlap of the stop bands that can be used to strongly reduce the transmission in a broadband range of frequencies. In particular, we experimentally and theoretically show that the system can produce a broad frequency band gap exceeding two and a half octaves (from 590 Hz to 3220 Hz) with transmission lower than 3% in this whole frequency range. Finite element methods are used to study the dispersion relation of the locally resonant system. The visco-thermal losses are accounted for in the quarter wavelength resonators to study the wave propagation in the semi-infinite structures and to compare numerical results with the experimental ones performed in an echo-free chamber.

## 2. Experimental Set-Up

The resonant scatterers are infinitely long square-rod scatterers made of exotic wood (acoustically rigid for the ranges of frequencies analyzed in this work) with side length

l. Each scatterer incorporates a 1D periodic array of quarter wavelength resonators with periodicity

${a}_{z}$. These quarter wavelength resonators are made by drilling cylindrical holes of radius

R and length

L in one of the faces of the square-rod scatterer. The resonant square-rod scatterers are placed in a square array of periodicity

a.

Figure 1a shows the scheme of a resonant square-rod scatterer showing the parameters of the basic unit cell of the crystalline structure analyzed in this work.

Figure 1b shows the scheme of the finite array analyzed in this work as well as a picture of the SC in the anechoic chamber. As shown in the inset of the

Figure 1b the unit cell can be rotated by an angle

θ with respect to the center of the resonant square-rod scatterer, adding a degree of freedom to tune the dispersion relation of the system.

The experimental prototype consists of a $16\times 6$ array, located on a square lattice with constant $a=7.5$ cm with a vertical periodicity of the quarter wavelength of ${a}_{z}=5$ cm. The scatterers have a side length $l=5$ cm. The quarter wavelength resonator has a diameter of $2R=3.5$ cm and length $L=4$ cm. Finally, the scatterers are 2 m long and incorporate 29 quarter wavelength resonators in their central parts.

All the acoustic measurements are performed using a microphone B&K 1/4” type 4135 (B&K, Naerum, Denmark). The acoustic source was the loudspeaker Genelec 8351A (Genelec, Olivitie, Finland). The movement of the microphone in the anechoic chamber is controlled by a 1D robotized arm (Zaber LSQ, Vancouver, Canada) designed to move the microphone over a 1D trajectory with a step of 1 cm. The acquisition of the acoustic signal is preformed using a Stanford SR 785 analyzer (Stanford Research Systems, Inc., Sunnyvale, USA). The movement of the robotized arm and the acquisition are synchronized by a computer. Once the robotized system has positioned the microphone, the acoustic source generates a swept sine signal and the microphone detects it. The analyzer provides the frequency domain signals (module and phase for each frequency).

In the approach considered here, the loudspeaker used to generate the acoustic field in the anechoic chamber is placed in

${x}_{s}=0$ and a single microphone was used to measure the transfer functions between the signal provided to the loudspeaker and the sound pressure at four locations shown in

Figure 1b. The loudspeaker and the microphone are aligned in the middle height of the structure and only propagation in the

$x-y$ plan,

i.e., along the

$\mathrm{\Gamma}XM\mathrm{\Gamma}$ is considered. Those transfer functions are denoted

${P}_{1}$ to

${P}_{4}$. The wave front can be considered planar because the loudspeaker is placed far enough from the SC although it produces a spherical wave front. However, the amplitude decay of the wave is considered as

$1/\sqrt{{x}_{i}}$ where

${x}_{i}$ is the distance between the loudspeaker and the

i-th location of the microphone. For the present purposes

${P}_{1}$ to

${P}_{4}$ may be considered to represent the complex sound pressure at the four measurement locations

${x}_{1}$ to

${x}_{4}$,

i.e.,

Here,

k represents the wavenumber in the ambient fluid and

${e}^{+\u0131\omega t}$ sign convention has been adopted (

$\omega =2\pi f$ is the angular frequency with

f the frequency). The four complex pressures,

${P}_{1}$ to

${P}_{4}$, comprise various superpositions of positive- and negative-going waves in the up- and downstream segments of the anechoic chamber. In the range of frequencies of interest in this work,

$fa/c\le 1$, only the fundamental grating plane mode is reflected back (specular reflection), so the reflected waves can be approximated as plane waves. The higher grating order modes are evanescent and their amplitudes decay rapidly away from the SC so that they disappear in the vicinity of the SC. The complex amplitudes of those waves are represented by the coefficients

A to

D. Equation (

1) yield four equations for the coefficients A to D in terms of the four measured sound pressures,

i.e.,

The latter coefficients provide the input data for subsequent transfer matrix calculations. Here, the transfer matrix is used to relate the sound pressures and normal acoustic particle velocities on the two faces of the SC respectively located at

${x}_{0}$ and

${x}_{d}$ as in

Figure 1b,

i.e.,

In Equation (

3),

P is the exterior sound pressure and

V is the exterior normal acoustic particle velocity. The pressures and particle velocities on the two opposite surfaces of the SC may easily be expressed in terms of the positive- and negative-going wave component amplitudes,

i.e.,

where

ρ is the ambient fluid density and

c is the ambient sound speed. Thus, when the plane wave components are known from measurements of the complex pressures at the four locations, the pressures and the normal particle velocities at the two opposite surfaces of the SC can be determined.

It is then of interest to determine the elements of the transfer matrix since, as will be shown below, the elements of that matrix may be directly related to the properties of the SC. Then, instead of making a second set of measurements it is possible to assume the reciprocal nature of the SC. Thus, given reciprocity and symmetry, it follows that

The transfer matrix elements can be expressed directly in terms of the pressures and velocities on the two opposite surfaces of the SC. Therefore, if the transmission coefficient is defined as

$T=C/A$, and if we consider anechoic termination,

i.e.,

$D=0$, it can be expressed in function of the elements of the transfer matrix

In what follows, we will focus on the transmission coefficient, all the configurations not being necessarily symmetric. In particular, the configuration depicted in

Figure 4 is reciprocal but not symmetric.