Acoustic imaging in water traditionally relies on phased arrays of active electro-acoustic transducers to steer acoustic energy in specific directions. One potential alternative approach to steer acoustic beams is to use a single transducer attached to a dispersive antenna that radiates or receives acoustic energy from different directions as the frequency of operation changes. This is known as a leaky wave antenna (LWA). While LWAs have been proven effective in beam steering for electromagnetic and air-borne acoustic waves, the design of an analog device in water presents a unique challenge due to the low contrast in acoustic impedance between elastic solids and water, which necessitates the consideration of fluid-elastic coupling in the design of the elastic LWA. This work presents an approach to design an elastic metamaterial waveguide coupled to an external fluid domain as one means to create an acoustic LWA for underwater operation. Forward-to-backward radiation is achieved through the design of mass-in-cavity structures that produce simultaneous negative effective mass and modulus by considering fluid-elastic coupling. The design is presented through finite element analysis of individual unit cells and a water-loaded elastic LWA. A design example is presented that steers through backfire to endfire as a function of input frequency.

## I. INTRODUCTION

Leaky wave antennas (LWAs) make use of a dispersive waveguide structure to steer acoustic or electromagnetic energy as a function of frequency.^{1} LWAs were first introduced in the 1940s as a simple waveguide with open slits on one side that allowed for energy to radiate, or “leak,” into the surrounding medium in a direction that depends on frequency.^{2} LWAs steer energy using only one or a limited number of active elements, such as microphones or hydrophones for acoustics, coupled to an analog aperture. The limited hardware requirement makes LWAs an attractive design for cases where the high electronics cost or complexity of traditional actively phased arrays pose practical challenges.

Studies of electromagnetic LWAs have integrated metamaterial concepts such as negative effective properties and sub-wavelength unit cells to control dispersion. These negative effective properties include negative refractive index, enabling steering from backfire to endfire with a single active element coupled to an analog waveguide.^{3,4} Several review papers summarize the improvements made to the basic electromagnetic designs as well as modifications that extend the concept to two-dimensional and conformal geometries.^{5,6}

The concept of an acoustic LWA was introduced in 2014 by Naify *et al.*^{7} who utilized a metamaterial-based waveguide design described by Bongard *et al.*^{8} to steer air-borne acoustic waves from backfire-to-endfire using two independent structures to simultaneously generate negative (dynamic) mass density and negative (dynamic) bulk modulus. In the air-borne acoustic case, negative mass density was achieved using a series of compliant membranes, while negative bulk modulus was achieved by a series of openings spaced periodically along the waveguide, commonly referred to as shunts or stubs. The air-acoustic LWA has been evaluated in transmitting and receiving mode antenna performance^{9} and investigated for its imaging resolution compared to traditional active arrays.^{10} Further studies include the design of tapered,^{11} two-dimensional,^{12} non-reciprocal,^{13} and vortex wave configurations.^{14}

This work presents a design process to develop an acoustic LWA for use in water. In order to overcome the finite impedance hurdles associated with water-loaded designs, the design approach in this study includes an elastic, as opposed to purely fluid-acoustic waveguide as traditionally employed for LWAs operating in air. Effective mass elements are realized using an inertial mass-spring system while negative effective modulus is realized using a resonant cylindrical cavity, which acts as the primary acoustic source of the LWA.^{15} We introduce a method to account for the mass loading imposed by the external fluid on the resonant cylinder and a design candidate is analyzed.

A few other analog-type acoustic antenna approaches for water operation have included a slow waveguide antenna that radiated sound from a silicone cylinder without controllable directivity^{16,17} and the blazed array that uses a diffraction grating as a low power sonar aperture.^{18} Note that elastic waveguides have been exploited for directional underwater sound radiation, with designs that do^{19} and do not^{20} include steering capabilities. A key limitation of these previous designs, however, is their inability to steer backfire (toward the source end of a linear array) to endfire (away from the source end), a capability that is possible in our design due to the negative effective properties of a metamaterial leaky wave antenna.

This paper begins in Sec. II by describing how concepts of dynamic effective density and modulus from the air-acoustic LWA design can be adapted to geometry suitable for underwater use by accounting for elastic wave motion in the LWA. In Sec. III, a finite element analysis-based unit cell design process is presented including a method to account for the water loading of the structure. This is followed by a discussion of the dispersive band structure and calculation of effective properties. Using this design method, an example geometry is presented in Sec. IV. Analysis of the full antenna design is shown in Sec. V and beam patterns are calculated and compared to the unit cell dispersion relationship.

## II. DESIGN BACKGROUND

While the air-to-water transition may seem trivial at first glance, a few significant challenges appear when trying to realize a LWA in an underwater environment as opposed to air. The acoustic impedance of solids in water is finite, as opposed to in air, where the acoustic impedance can usually be assumed to be near-infinite and, therefore, modeled as acoustically rigid. Due to the low impedance contrast of solids in comparison to water, elastic waves in the waveguide material and, more importantly, waveguide motion associated with fluid-elastic coupling cannot be ignored as they are when designing LWAs for air-borne acoustics. It is for this reason that the physical behavior associated with unit cell designs for air-acoustic LWAs presented in previous works will not work underwater.^{7–9} In general, the demonstration of many acoustic metamaterial concepts for use in water have been very limited compared to those in air, despite the fact that acoustic waves in high impedance fluids like water have significant applicability for communications, imaging, and sensing.^{21}

Early underwater acoustic metamaterial studies generally tried to ignore elastic effects,^{22} whereas more recent efforts have exploited those elastic effects^{23,24} or used polymers whose shear modulus is orders of magnitude smaller than their bulk modulus. For that reason, the materials can reasonably be modeled as fluids and one can generally neglect elastic effects of solids in favor of scattering off of encapsulated air or other inclusions.^{25–27} A range of multiple scattering-based studies have also been done in water which, though they use elastic inclusions, utilize water as the fluid background medium and solids as subwavelength scattering objects.^{28–30} Conversely, the underwater elastic LWA presented here exploits fluid–solid interactions to create an elastic waveguide that interacts with the surrounding fluid to steer acoustic energy as a function of frequency.

The general idea of an acoustic LWA design capable of steering from backfire to endfire necessitates the use of metamaterial concepts, specifically negative effective mass density and negative effective modulus. In order for the antenna to smoothly transition from backfire to endfire scanning, it is necessary that the frequency bands of these negative effective properties converge at a single frequency. Similarly, in order for the antenna to scan at broadside, the upper-frequency cutoff for both the negative mass density and modulus must be exactly the same.^{5,7}

The effective dynamic mass density of a heterogeneous medium is found through volume averaging operations of the linearized momentum equation for an elastic solid; $\mu \u2192(x\u2192)=\rho (x\u2192)v\u2192(x\u2192)$, where $\mu \u2192(x\u2192)$ is the local momentum, $\rho (x\u2192)$ is the local mass density, and $v\u2192(x\u2192)$ is the local particle velocity field. The present study only considers one-dimensional (1D) wave motion along the axis of a cylindrical waveguide, so the local momentum field simplifies to $\mu (z)=\rho (z)v(z)$. For a heterogeneous 1D elastic waveguide with small heterogeneities compared to propagating wavelengths and without local asymmetry, the effective dynamic density, $\rho eff$, is determined by finding the ratio of the domain-averaged time-varying momentum, $\u27e8\mu \u27e9$, to the domain-averaged time-varying velocity field, $\u27e8v\u27e9$, as shown in^{31}

Note that we have employed $\u27e8\u22c5\u27e9$ to denote the domain averaging operation. When restricted to a single dimension, the domain average can be written explicitly as $\u27e8\chi \u27e9=L0\u22121\u222b0L0\chi (z)dz$ for an arbitrary field variable $\chi (z)$ and a unit cell of length $L0$. Note that a more general determination of the effective dynamic density can be provided as a function of the net dipole scattering by material heterogeneities, which can be determined using various analytical and numerical approaches.^{32,33}

The definition of the effective dynamic modulus in an elastic waveguide is found in a similar manner as one determines the effective dynamic bulk modulus in an acoustic waveguide, which is reviewed here for convenience. In an acoustic fluid, the relationship between the local acoustic pressure, $p(z)$, and volume strain, $\epsilon V$, can be written as $p(z)=\u2212K(z)\epsilon V(z)$, where $K(z)$ is the adiabatic bulk modulus. For a fluid-filled acoustic waveguide, the effective dynamic bulk modulus, $Keff$, can therefore be defined as the ratio of the domain-averaged time-varying acoustic pressure to domain-averaged time-varying volume strain,

For a 1D elastic waveguide, we first assume that the time-varying local stress-field is uniaxial such that $\sigma (z)=Y(z)\epsilon (z)$, where $\sigma (z)$ is the time-varing uniaxial stress, $Y(z)$ is Young’s modulus, and $\epsilon (z)$ is the time-varying strain along the axis of the waveguide and defined by $\epsilon (z)=\u2202u(z)/\u2202z$ with $u(z)$ being the local particle displacement. The effective dynamic Young’s modulus of the unit cell can therefore be found from the domain-averaged stress and strain fields with the following relationship:

As with the design of air-filled acoustic LWAs, front-to-back beam steering is only possible by structuring the waveguide such that its dispersive behavior includes bands with positive and negative phase velocity within the sonic cone of the external fluid.

Figure 1 illustrates the geometries by which negative parameters are implemented in a 1D waveguide geometry, with the example of representative air-acoustic elements used in the air LWA. Cartoons demonstrate how those effective parameters have been realized in the underwater LWA where we re-design each element to consider the complexity of a deformable elastic waveguide coupled to the external fluid.

Due to the complex motion of the elastic waveguide and the non-negligible interaction between the elastic waveguide and the acoustic medium in which it is immersed, the same analyses used in the air-acoustic LWA cannot be used to evaluate the presented underwater design. Rather, a finite element method (FEM) modeling approach was taken to explore the design space for a combination of parameters that exhibit simultaneous negative dynamic density and modulus. The elastic deformation of the waveguide itself leads to acoustic radiation from all sections of the LWA, with solid cylindrical segments radiating due to the Poisson effect associated with uniaxial stress. In addition, the negative dynamic modulus cylindrical cavity sections undergo dramatic changes in the radial velocity amplitude based on the modal behavior of the continuous elastic cylinder. The design of the underwater elastic LWA must consider both manufacturability (i.e., fewer parts and ease of assembly) and compactness of the parameter space. For an elastic waveguide consisting of shunting elements—to provide for dynamic modulus—and mass inclusions—for dynamic density—the design space has a dimensionality of 11. In Secs. II A and II B, the design geometry used for effective density and modulus in a water-acoustic LWA is presented and discussed in comparison to their air-acoustic counterparts.

### A. Dynamic density

Dynamic density in an air acoustic medium can achieve negative values when a series of thin membranes, clamped to the edge of a rigid waveguide, are arranged in series down the length of the waveguide. The flexural rigidity of the thin membrane provides the restoring force that causes the air between the membranes to oscillate, with axisymmetric modes under normal incidence, in order to provide a series compliance as seen in Fig. 1(a). Full development and discussion of this approach is provided by Bongard *et al.*^{8} In order to achieve a similar effect in our elastic waveguide design, a different approach is presented. The requirement of the series compliance is that the sub-wavelength inclusion must be actuated on by the pressure wave incident on the unit cell and continue to propagate along the axis of the waveguide to excite the next unit cell. The same normal-component acoustic velocity requirement in the air-acoustic case is also valid in the elastic case with $\sigma (x\u2192)$ and $v(x\u2192)$ characterizing the incident stress wave on the unit cell.

Since a thin membrane can be easily modeled as a mass-spring system in series (with the spring defined as the stiffness of the stretching or bending membrane), an equivalent mass-spring inclusion should be designed for the elastic waveguide. The conceptual elastic equivalent unit cell is shown in Fig. 1(b), which defines the mass, $mm$, of a solid inclusion and Young’s modulus of the spring elements. The equivalent mechanical stiffness of the spring elements, $ksp$, is a function of material properties and geometry of the spring element. This conceptual schematic is then converted into the continuous features of the elastic waveguide to create the dynamic density element show in Fig. 1(c) as a mass supported on spring elements, which is a thin annulus structure. While in a real system, the mass element is also compliant, the stiffness of the mass can be considered to be much greater than the spring elements, $ksp$. Thus, for design purposes, the mass is assumed to be a rigid body and the spring element stiffness is the only deformable element considered in the design of the effective dynamic density geometry. It is worth noting that in an open design space, these spring elements can take on a wide range of values as they do not have to be the same material as the host elastic waveguide, though for fabrication simplicity, they can be the same material.

### B. Dynamic modulus

Dynamic modulus, enabled in an air-acoustic LWA using openings in the waveguide to create shunts to the external medium, is the ratio of dynamic average pressure to dynamic volumetric strain in the unit cell. Frequency-dependent effective stiffness can be achieved by introducing a radial degree of freedom for volume change of the LWA unit cell. In the air-acoustic case depicted in Fig. 1(d), the velocity across the shunt is nonzero while the velocity of the waveguide itself is assumed to be zero due to the impedance contrast between air and the solid.

In the case of the elastic waveguide, the assumption of zero velocity contribution from the waveguide is not valid, as shown in Fig. 1(e), where the waveguide itself will have radial motion due to the Poisson effect. It is possible to design the frequency response of this sub-wavelength element by removing some of the interior of the waveguide to create a cavity. The cavity will exhibit radial resonances where the local radial velocity is much higher than that of the waveguide sections. These cavity sections are analogous to the shunts in the air-acoustic case and, for certain combinations of wall thickness, length, and material properties, lead to negative effective dynamic modulus for axially propagating modes.

Although the shunt waveguide sections are specifically designed to have a non-zero radial velocity component at a specific frequency, the entire elastic waveguide is excited when waves propagate along the axis. Any motion of the non-shunt portion of the waveguide that has radial motion due to the Poisson effect will also radiate acoustic energy into the surrounding water. However, when properly designed, the magnitude of radiation from these sections will be small in comparison to that from the shunts. Therefore, it is important to ensure any numerical or FEM model takes the full elastic response of the waveguide into account.

## III. UNIT CELL ANALYSIS

It is very common to begin the design of metamaterial structures through the analysis of a unit cell with periodic boundary conditions. Figure 2 shows the generic unit cell geometry under consideration for the design of an underwater LWA where all constituent materials are assumed to be homogeneous, isotropic elastic solids. The periodic boundary conditions assume an infinite periodic structure and allow for efficient calculation of dispersive behavior using Bloch–Floquet analysis. Designing the LWA using a single unit cell with periodic boundary conditions is advantageous because it allows for rapid iteration of design parameters due to the small model size. For the geometry described above, we assume that the unit cell is axisymmetric and has a traction-free outer boundary. In an air-acoustic environment, this traction-free boundary is not an unreasonable assumption, but in a water environment, the radiation impedance from the water can alter the dispersion characteristics.

We employ the structural mechanics module of commercial FEM software COMSOL Multiphysics to perform the following analysis. Due to the large design space, an iterative approach was taken to rapidly converge to a set of design parameters that demonstrate dispersion in an elastic LWA that enables front-to-back steering of acoustic radiation. The design cycle is as follows:

Use a 2D axisymmetric model of a unit cell as seen in Fig. 3(a) to find the dispersion characteristics of the radial shunts by solving the eigenvalue problem with periodic boundary conditions (BCs).

Calculate transmissibility curves for a wide range of geometries and materials for the mass and spring elements using analytic mass-spring formulas to determine the approximate axial resonance frequencies.

Select appropriate mass and spring dimensions from step 2 given the dispersion characteristics of the radial shunt in step 1.

Create complete 2D axisymmetric model of the unit cell with prospective base material and geometry from steps 1 and 3 and evaluate the dispersion characteristics.

The 2D axisymmetric model of a unit cell is shown in Fig. 3(a). On either side of the unit cell, Floquet boundary conditions (BCs) are prescribed to reflect the periodicity of an infinite number of repeated unit cells. Floquet BCs are used to calculate dispersion curves of periodic waveguides as defined by

where $\beta $ is the Bloch–Floquet wavenumber, subscripts $dst$ and $src$ refer to the destination and source boundaries at either end of the unit cell, $Lwg$ is the length of the unit cell, $u$ is the displacement field, $z$ is the axial coordinate, and $ej\omega t$ is the assumed time convention. For a prescribed Bloch wavenumber, the model is converted to an eigenvalue problem and solved, resulting in a series of frequency-wavenumber solutions. From these models, we are able to directly evaluate dispersion curves for the system and identify branches with negative and positive phase speeds and positive group velocity. The positive group velocity modes within the acoustic radiation cone of the external acoustic medium represent branches that will radiate into the exterior fluid, with negative (positive) phase velocity branches radiating at negative (positive) angles with respect to the radial direction. Further, the unit cell analysis allows for the approximation of the effective density and stiffness using Eqs. (1) and (3).

### A. Compensation for fluid loading

For the geometry described above, we assume that the unit cell is axisymmetric and has a traction-free outer boundary. The axisymmetric geometry ensures that transverse and torsional modes will not be scattered by material heterogeneity, and thus this design only considers symmetric longitudinal modes. In an air-acoustic environment, this traction-free boundary is a very good approximation. However, when modeling an underwater elastic LWA, the impedance from the water may significantly alter the dispersive wave motion in the elastic waveguide. In the case where radiation from the shunt section dominates over radiation from the elastic waveguide, the influence of fluid loading can be accounted for by considering the acoustic radiation impedance on the shunts. Radiation impedance is represented as a frequency-dependent resistance and reactance, where the resistance is related to the radiation of acoustic energy to the acoustic domain and the reactance is determined by energy transferred to lossless inertial and compliance mechanisms in the surrounding fluid.^{34} Radiation resistance can therefore only affect the magnitude of motion associated with forced resonances, but not the frequency dependence of the system response. On the other hand, the radiation reactance can significantly alter the resonant response of a system and is therefore important for the design of a fluid-loaded elastic LWA. When the radiating section is small compared to acoustic wavelengths and the external domain is infinite, the reactance is well-approximated as an additional mass that is entrained by the motion of the radiating surface.^{34} It is, therefore, possible to approximate the influence of fluid loading in the unit cell analysis by adding a radiation mass calculated from the appropriate radiation impedance model.

To determine the mass required to account for the fluid loading, we employ a Fourier series approximation for the calculation of the radiation impedance for pulsating finite cylinders developed by Butler and Butler.^{35} This calculation is based on an infinite structure comprised of alternating rigid and uniformly pulsating cylinders. Near the cavity resonance, this is an accurate approximation for the behavior of the LWA because the displacement of the shunt is significantly higher than that of the elastic waveguide section.

The radiation impedance on the shunt is a function of the sound speed, density, and wavenumber in the external fluid. It is preferable to assume a constant radiation mass for the unit cell design. The wavenumber used to calculate the radiation impedance is at the average frequency of the negative stiffness band of the unit cell analysis, which represents an axisymmetric, breathing mode of the shunt where the LWA most resembles a structure comprised of alternating rigid and uniformly pulsating cylinders. The radiation impedance is also a function of the outer radius of the shunt, the length of the shunt, and the replication distance, which is equal to the length of a unit cell. The Fourier series solution assumes that the pressure is small at the replication distance, so this solution neglects pressure coupling (i.e., mutual radiation impedance) between adjacent shunts.

The frequency-dependent complex-valued radiation impedance, $Z(\omega )=R(\omega )+jX(\omega )$, contains a resistive and a reactive component as shown in Fig. 4 for the design example presented in Sec. IV. While the former is related to the acoustic power transferred to the fluid, indicating that the shunt will radiate efficiently, this term does not influence the apparent mass load on the shunt. The latter component is related to the restoring and inertial forces of the fluid. However, because the waveguide is modeled in a domain with open boundaries, there is no physical method for the fluid to provide compliance-like impedance. The reactive portion of the radiation impedance is therefore purely due to the accession to inertia of the shunt.^{34} In the case that the replication period is small, it is necessary to retain a very large number of terms in the Fourier series to correctly approximate the radiation impedance.^{35} For the design geometry presented in Sec. IV, the reactive portion of the radiation impedance asymptotes such that we can then use this asymptotic value to calculate the radiation mass, $mrad$. The general form of impedance due to an additive radiation mass is $Zm=j\omega mrad$, so the radiation mass can be calculated through the relationship defined in

### B. Calculation of effective parameters

To find branches of the dispersion relations that display simultaneous negative dynamic density and modulus, and thus propagating modes with positive group velocity and negative phase velocity, we extract the effective dynamic parameters from the FEA unit cell model. Beginning with Eqs. (1) and (3) from Sec. II, the relationships for effective dynamic density and Young’s modulus are modified for a one-dimensional elastic structure. Only the axial dimension is considered because the radiation angle depends on the phase speed in this direction.

The effective density, given by Eq. (1), uses the domain-averaged momentum and velocity. An equivalent method for calculating the effective density uses the domain averaged rate of change of momentum and velocity. Since $F=d\mu /dt$ and $a=dv/dt$, where $F$ and $a$ are the net time-varying force and acceleration acting on the unit cell, the effective density can be written as $\rho =\u27e8F\u27e9/\u27e8a\u27e9$. In a one-dimensional elastic solid in a uniaxial stress state, the effective Young’s modulus can be found from Eq. (3) by considering the stress and strain along the axis of propagation.

As a leading-order approximation, the values for force, acceleration, stress, and strain were evaluated on the boundaries of the unit cell ($z=\xb1Lwg/2$) and the effective density was then corrected for by dividing by the volume of the unit cell, $V$.^{31} This homogenization process, which treats everything inside the boundaries as a black box, is only accurate when the unit cells are very small compared to propagating wavelengths. While this approximation is not perfect for wideband dynamic homogenization, it is acceptable for use in design studies that use the FEM-based unit cell analysis to identify frequency bands of positive and negative index in elastic LWAs.

Using these modified relationships, the effective parameters of the unit cell are calculated by

where $Fz$, $az$, $\sigma z$, and $\epsilon z$ are the magnitudes of force, acceleration, stress, and strain acting on the boundaries of the unit cell in the axial direction. The phase difference between the force and acceleration is defined by $\theta F\u2212\theta a$ while the phase difference between the stress and strain is defined by $\theta \sigma \u2212\theta \epsilon $.

Phase speed can then be calculated in two different manners: (i) directly from the dispersion diagram via the relationship: $cph=\omega /\beta z$ and (ii) using the effective parameters via

## IV. DESIGN EXAMPLE

Thus far, the design process has been material, dimension, and frequency agnostic. This section presents an example case that demonstrates forward and backward steering within a narrow frequency band in the metamaterial limit. For this example, the geometry will be described, the effective parameters and steering angle will be calculated, and the performance will be discussed.

The geometry selected for analysis in this study is shown in an exploded view in Fig. 2. The design consists of an elastic waveguide (dark gray), a shunt element created by a void in the waveguide (light gray), a spring element as an annular ring (blue), and a mass element (red) which in this case is a cylindrical disk.

### A. Steel waveguide

In this design space, the waveguide and mass are steel while the springs and shunt are lead. Dimensions of the unit cell geometry, as illustrated in Fig. 3(a), are listed in Table I. To account for the radiation impedance of the surrounding fluid, an added mass of 0.134 kg was calculated using the method described in Sec. III A and added as a distributed load along the shunt surface. It is assumed that all materials used have a linear elastic response with properties listed in Table II. Note that the shear moduli for all materials are considered in all models presented. For linear elastic materials, the shear modulus is related to Young’s modulus and Poisson’s ratio through the relationship $G=E/2(1+\nu )$. The FEM model for the linear elastic constituents of the LWA, therefore, considers the presence of shear deformation in all elastic components. It is important to note, however, that it is the Poisson effect of the waveguide section of the LWA that excites the radial motion of the shunt, which is central to the generation of negative effective dynamic modulus in this LWA architecture. The complex dispersive wave motion of hollow circular elastic cylinders, which has been analyzed in detail by Gazis,^{36} demonstrates the dependence of this resonant behavior on both Young’s and shear moduli.

Dimension (mm) . | |
---|---|

r_{wg} | 49.6 |

L_{wg} | 72.7 |

r_{sh} | 41.6 |

t_{sh} | 7.9 |

r_{sp} | 37.1 |

t_{sp} | 3.1 |

h_{sp} | 8.7 |

h_{m} | 12.2 |

r_{m} | 40.3 |

Dimension (mm) . | |
---|---|

r_{wg} | 49.6 |

L_{wg} | 72.7 |

r_{sh} | 41.6 |

t_{sh} | 7.9 |

r_{sp} | 37.1 |

t_{sp} | 3.1 |

h_{sp} | 8.7 |

h_{m} | 12.2 |

r_{m} | 40.3 |

. | Pb . | Steel . |
---|---|---|

ρ (kg/m^{3}) | 11 340 | 7850 |

Y (GPa) | 16.3 | 200 |

ν (—) | 0.44 | 0.3 |

η (—) | 1 × 10^{−2} | 1 × 10^{−4} |

m_{rad} (kg) | 0.134 |

. | Pb . | Steel . |
---|---|---|

ρ (kg/m^{3}) | 11 340 | 7850 |

Y (GPa) | 16.3 | 200 |

ν (—) | 0.44 | 0.3 |

η (—) | 1 × 10^{−2} | 1 × 10^{−4} |

m_{rad} (kg) | 0.134 |

The dispersion diagram found using the unit cell analysis for both the unloaded and added mass-loaded cases is provided in Figs. 5(a) and 5(b), respectively, and the modal response is shown in blue. The red lines indicate the acoustic radiation cone for the external fluid (water). All modes propagating in the LWA that are outside the cone are evanescent, i.e., they do not radiate into the external fluid. The yellow line marks the frequencies such that any propagating modes have half-wavelengths shorter than the unit cell length. One would, therefore, anticipate that the LWA will radiate grating lobes for frequencies above the horizontal yellow line. Under the assumption that the antenna is being excited at one end ($z=0$) and no end reflections occur, waves will only travel in the positive $z$-direction. In this scenario, propagation is only possible for modes with a positive group velocity, which are highlighted in dark blue.

While the no-mass design indicates there should be a bandgap from approximately 8.3 to 9.1 kHz, by incorporating the fluid loading approximation into the design, the magnitude of that bandgap decreases to approximately 150 Hz. The added mass primarily affects the third mode of the structure since that is the axisymmetric “breathing” mode of the shunt.

The calculated effective density and modulus for the analysis with the added mass are shown in Figs. 6(a) and 6(b), respectively. The frequency range in which the structure is double-negative, and thus will have negative index behavior, is highlighted in orange. The frequency range in which the structure is double-positive, and thus will have positive index behavior, is highlighted in blue. The bandgap region, also evident when effective parameters are calculated, will be expected to have no radiation when the unit cells are assembled into an array.

Figure 7 shows the calculated phase speed in the unit cell using two methods: (i) $cph=\omega /\beta z$ directly from the dispersion diagram and (ii) the effective parameters as described in Eq. (8). The dispersion and effective parameter methods agree in their prediction of positive and negative phase speeds, as well as the overall trends (increase and decrease within each mode). There is some discrepancy, however, in the magnitude in frequencies outside of the shaded regions. This discrepancy originates from the effective parameters calculation, which evaluated the parameters at the boundaries of the unit cell as described in Sec. III B and thus is only valid when wavelengths of the propagating modes are much larger than the unit cell length.

## V. FINITE ANTENNA DESIGN

Thus far, the design process has been limited to single unit cell analysis to facilitate rapid design space exploration while relying on FEA models. However, the periodic boundary conditions and lack of explicit fluid-elastic coupling cannot consider some highly important aspects of a realistic fluid-loaded elastic LWA. We therefore consider a finite-length antenna to investigate important effects that are neglected in the unit cell analysis, such as the attenuation of elastic wave energy associated with acoustic radiation into the fluid, mutual radiation impedance between elastic elements, and reflections from the end of the antenna. In addition, the full FEA model of the finite antenna is used to calculate the radiated acoustic energy and beam patterns. The beam patterns from the finite array include the complex sidelobe structure, which can vary significantly depending on the number of unit cells.^{10}

The finite antenna was implemented in COMSOL Multiphysics using the structural mechanics and pressure acoustics modules. The process was as follows:

Create a structure comprised of $N$ unit cells from Sec. III. In this example, $N=40$.

Apply a source condition to excite the elastic waveguide along the magenta line shown in Fig. 3, representing the base of the antenna.

Surround the structure with water.

Apply a perfectly matched layer (PML) as the far-field radiation boundary to simulate a free field environment.

A significant detriment to finite antenna performance is the reflection of propagating modes from the end of the antenna, that is, opposite to the source. Reflections can result in standing waves in the LWA rather than traveling waves, which effectively eliminate the frequency-dependent angular radiation capability of the LWA. The standing wave response can be thought of as the excitation signal interfering with the reflected signal, which causes irregular sidelobes in the radiated beam pattern. In the air-acoustic LWA case, these reflections can be mitigated relatively easily by using an absorbing foam for an anechoic termination. For an elastic LWA in water, however, it is necessary either to attenuate the signal using a designed impedance-matched, absorbing medium at the termination, which is a significant technical challenge, or to design the length of the antenna such that the energy is dissipated before it reaches the end via a combination of the radiation leakage and the material loss of the elastic waveguide. The LWA example considered here uses the latter approach.

One side effect of the fact that the LWA is designed to be long enough to allow propagating modes to attenuate before reaching the end of the antenna is that radiation is dominated by the radial motion near the source. The resultant axial dependence of the radial velocity of the LWA will alter the radiation pattern and the directivity of the LWA.^{11} One can predict the portion of the array that primarily contributes to acoustic radiation through the attenuation coefficient for propagating modes in the LWA. The amplitude of elastic waves in the waveguide attenuates as $e\u2212\alpha z$, where $\alpha $ is the attenuation coefficient with units of Np/m. The directivity of radiated sound is therefore a function of the attenuation coefficient. For the design considered here, the attenuation coefficient was determined by extracting the radial velocity of the midpoint of the shunts from the finite antenna model as a function of distance from the source (Fig. 8). The corresponding distance where the amplitude of elastic waves decreased 99% from the value at the source indicates that a waveguide of 40 unit cells (2.9 m) is sufficiently long to prevent end reflections (Fig. 8). Note that the spike in attenuation coefficient at 8.45 kHz aligns with the bandgap in the dispersion diagram in Fig. 5(b) demonstrating an agreement between the mass-loaded unit cell model and the finite antenna model.

In a real world scenario, the source condition would likely be a piezoelectric transducer bonded to the base of the antenna and isolated from the external fluid. For this study, the excitation has been modeled as a velocity source that is perfectly bonded to the LWA but isolated from the external fluid. This boundary is used to excite one end of the elastic waveguide as a uniform source acting purely in the axial direction. Note that it is the source condition that allows transverse and torsional modes to be neglected in the analysis of this LWA. Further, both transverse and torsional modes are poor acoustic radiators, where the former radiates as a dipolar source and the latter does not couple to the surrounding fluid whose shear modulus is zero. For these reasons, only axial modes are considered in the analysis of the LWA.

To simulate an open water environment, the finite antenna FEM uses a perfectly matched layer at the boundary of the water domain to prevent a reflection of the radiated beam pattern. This allows for an accurate calculation of radiated beam patterns through COMSOL’s exterior field pressure calculation, which solves the Helmholtz–Kirchhoff integral over the outer boundary of the antenna. Calculated beam patterns are shown for the design example in Fig. 9. The angle predicted by the mass-loaded unit cell analysis is overlaid to demonstrate the agreement between both analyses. As predicted based on the calculated dispersion of a single unit cell, the antenna has radiation behavior that sweeps from backfire (negative) to endfire (positive) as frequency increases. Radiation patterns of select frequencies are shown in order to demonstrate the angular range of the chosen design.

Figure 9 clearly shows the spatial-to-spectral mapping achieved using the fully elastic LWA coupled to the external fluid. Figure 9(a) shows the full-field frequency and angle relationship, but features of the beam are difficult to discern due to the large dynamic range of the radiated acoustic field. Figure 9(b) narrows that range down to $\u22126$ dB so that distinct beam features are seen. The $\u22126$ dB plot shows that this design has continuous backfire-to-endfire scanning over a narrow frequency range. There are sidelobes at both the backfire and endfire frequencies and those sidelobes are easily seen at frequencies of 8.26 and 8.6 kHz. The antenna also shows near-broadside radiation at 8.38 kHz with a 6 dB beamwidth of $2\xb0$.

## VI. SUMMARY AND CONCLUSIONS

This work presented a design approach for a one-dimensional axisymmetric elastic leaky wave antenna capable of steering backfire to endfire in a water environment. The methodology first utilizes finite element analysis of the unit cells that make up the LWA in order to design the dispersive behavior of elastic wave motion in the LWA that accounts for fluid loading. This is followed by a simulation of the finite LWA to predict beam patterns for acoustic radiation. The design we have presented indicates a clear path to the design of underwater LWA, but the example case was constrained to only consider readily available materials, which limits the range of constituent moduli and densities available. By utilizing a wider design space, including architected materials such as lattices to finely control the sound speed of the waveguide material, we anticipate that it is possible to create a customized elastic underwater LWA to steer from backfire to endfire over the desired frequency range without a bandgap. To the authors’ knowledge, this is the first acoustic leaky wave antenna design for use in water that considers the complex elastic wave motion in the waveguide and acousto-elastic coupling between the LWA and the exterior fluid. The design presented here demonstrates that elastic LWAs can be designed for underwater use, which opens up the potential for this technology to be used in fields such as sonar or ultrasonic imaging.

## ACKNOWLEDGMENTS

C.W.B. and M.R.H. acknowledge partial support from the Office of Naval Research through YIP Grant No. N00014-18-1-2335.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*Proceedings of MTS/IEE OCEANS*(IEEE, 2001).