# Scattering and Directionality Effects of Noise Generation from Flapping Thrusters Used for Propulsion of Small Ocean Vehicles

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## Abstract

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## 1. Introduction

## 2. Noise Generation from Flapping Thruster

## 3. Green’s Function in the Ocean Acoustic Waveguide

#### 3.1. Normal-Mode Series

#### 3.2. Multiple Image Series

_{0}= −300 m, in an isovelocity c = 1500 m/s ocean acoustic waveguide of depth h = 3000 m. In this case, the number of propagating modes from Equation (25) is ${M}_{p}=20$ and the acoustic field calculated by the normal mode series is obtained keeping 40 terms in the series Equation (20), and is plotted in the left subplots. For comparison, in the right subplots, the same field obtained from the mirror series using the above first six terms is presented. It is observed in these plots that, except the vicinity of the source (at submergence depth z

_{0}near the origin), the two series expansions of the monopole and dipole source fields provide the same result, up to a horizontal distance of $\left|x\right|=3000\mathrm{m}$, after which the normal mode series provide accurate results. On the contrary, it is the image series that provide accurate results in the whole intermediate region $\left|x\right|<3000\mathrm{m}$, including the excellent representation of the point singularity in the near-source field region $\left|x\right|<300\mathrm{m}$.

## 4. Three-Dimensional Acoustic Wave Scattering Problem

#### 4.1. Formulation of the Scattering Problem

#### 4.2. Acoustic Boundary Element Method

## 5. Numerical Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Calculation of the Flapping-Foil Hydrodynamic Loads by 3D BEM

## References

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**Figure 1.**(

**a**) Flapping thruster used for the propulsion of an AUV or small ocean vehicle. (

**b**) Consecutive positions of foil oscillatory motion due to combined heaving h(t) and pitching θ(t) motion.

**Figure 2.**Time snapshots of acoustic field generated by dipole sources in the case of flapping thruster operating in water (c = 1500 m/s), in the case of foil with NACA0012 sections of Figure 1 flapping at Str = 0.23, h

_{0}/c = 0.75, ${\theta}_{0}$ = 23 deg, using the calculated hydrodynamic loads from a pressure integration 3D BEM method. (

**a**) real part, (

**b**) imaginary part.

**Figure 4.**Calculated acoustic field corresponding to the monopole source, Equation (20). Source frequency 5 Hz, at submergence depth z

_{0}= −300 m, in an isovelocity c = 1500 m/s ocean acoustic waveguide of depth h = 3000 m. Left column result obtained by the normal mode series, keeping 40 terms: (

**a**) real part, (

**b**) imaginary part. Right column: field obtained from the mirror series using six terms: (

**c**) real part, (

**d**) imaginary part.

**Figure 5.**Calculated acoustic field corresponding to the horizontal dipole source, Equation (23). Source frequency 5 Hz, at submergence depth z

_{0}= −300 m, in an isovelocity c = 1500 m/s ocean acoustic waveguide of depth h = 3000 m. Left column result obtained by the normal mode series, keeping 40 terms: (

**a**) real part, (

**b**) imaginary part. Right column: field obtained from the mirror series using six terms: (

**c**) real part, (

**d**) imaginary part.

**Figure 6.**Calculated acoustic field corresponding to the vertical dipole source, Equation (24). Source frequency 5 Hz, at submergence depth z

_{0}= −300 m, in an isovelocity c = 1500 m/s ocean acoustic waveguide of depth h = 3000 m. Left column result obtained by the normal mode series, keeping 40 terms: (

**a**) real part, (

**b**) imaginary part. Right column: field obtained from the mirror series using six terms: (

**c**) real part, (

**d**) imaginary part.

**Figure 7.**Transformation of the boundary element ${E}_{j}$ from the Cartesian coordinate system $\left(x,y,z\right)$ to the local curvilinear system $\left(\xi ,\eta \right)$.

**Figure 8.**Vectors ${\mathbf{e}}_{1}$,${\mathbf{e}}_{2}$ tangential to the elementary surface $d{E}_{j}$ and the normal vector $\mathsf{\nu}={\mathbf{e}}_{1}\times {\mathbf{e}}_{2}$.

**Figure 9.**Calculated acoustic field in the semi-infinite ocean waveguide considering scattering of a monopole source field by an acoustically hard ellipsoid. The real part and the modulus of the field are presented along the vertical plane passing from the source position and cutting through the major axis of the ellipsoid and on the transverse plane passing from the source position in front of the ellipsoid. (

**a**) Real part and (

**b**) modulus of the acoustic field on the vertical plane. (

**c**) Real part and (

**d**) modulus of the acoustic field on the transverse plane.

**Figure 10.**Calculated acoustic field in the ocean waveguide considering scattering of a monopole source field by an acoustically hard ellipsoid, with the effect of rigid sea bottom at depth h = 200 m. (

**a**) Real part and (

**b**) modulus of the acoustic field on the vertical plane. (

**c**) Real part and (

**d**) modulus of the acoustic field on the transverse plane.

**Figure 11.**Same as Figure 10, but for a horizontal dipole source. The axis of the dipole is parallel to the major axis of the ellipsoid (x-axis). (

**a**) Real part and (

**b**) modulus of the acoustic field on the vertical plane. (

**c**) Real part and (

**d**) modulus of the acoustic field on the transverse plane.

**Figure 13.**Calculated acoustic field in the semi-infinite ocean waveguide considering scattering of a monopole source field by an acoustically soft ellipsoid. (

**a**) Real part and (

**b**) modulus of the acoustic field on the vertical plane. (

**c**) Real part and (

**d**) modulus of the acoustic field on the transverse plane.

**Figure 14.**Calculated acoustic field in the ocean waveguide considering scattering of a monopole source field by an acoustically soft ellipsoid, with the effect of rigid sea bottom at depth h = 200 m. (

**a**) Real part and (

**b**) modulus of the acoustic field on the vertical plane. (

**c**) Real part and (

**d**) modulus of the acoustic field on the transverse plane.

**Figure 15.**Same as Figure 14, but for a horizontal dipole source. The axis of the dipole is parallel to the major axis of the ellipsoid (x-axis). (

**a**) Real part and (

**b**) modulus of the acoustic field on the vertical plane. (

**c**) Real part and (

**d**) modulus of the acoustic field on the transverse plane.

**Figure 16.**Same as in Figure 14, but for a vertical dipole source. The axis of the dipole is parallel to the z-axis. (

**a**) Real part and (

**b**) modulus of the acoustic field on the vertical plane. (

**c**) Real part and (

**d**) modulus of the acoustic field on the transverse plane.

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**MDPI and ACS Style**

Belibassakis, K.; Prospathopoulos, J.; Malefaki, I.
Scattering and Directionality Effects of Noise Generation from Flapping Thrusters Used for Propulsion of Small Ocean Vehicles. *J. Mar. Sci. Eng.* **2022**, *10*, 1129.
https://doi.org/10.3390/jmse10081129

**AMA Style**

Belibassakis K, Prospathopoulos J, Malefaki I.
Scattering and Directionality Effects of Noise Generation from Flapping Thrusters Used for Propulsion of Small Ocean Vehicles. *Journal of Marine Science and Engineering*. 2022; 10(8):1129.
https://doi.org/10.3390/jmse10081129

**Chicago/Turabian Style**

Belibassakis, Kostas, John Prospathopoulos, and Iro Malefaki.
2022. "Scattering and Directionality Effects of Noise Generation from Flapping Thrusters Used for Propulsion of Small Ocean Vehicles" *Journal of Marine Science and Engineering* 10, no. 8: 1129.
https://doi.org/10.3390/jmse10081129