Acoustic Scattering Characteristics and Geometric Parameter Prediction for Underwater Multiple Targets Arranged in a Linear Pattern

Abstract

The construction of wind farm pilings, submarine pipelines, and underwater submarines involves multiple cylinders. However, there is currently a lack of economic research on predicting the mechanism and characteristics of mutual coupling of acoustic scattering from multiple cylindrical targets. This study investigates the mechanism and prediction method of acoustic scattering for the structural distribution characteristics of underwater multi-cylindrical targets. A model of a multi-cylindrical target’s two-dimensional acoustic field was established using the finite element method. Numerical calculations were then carried out to elucidate the scattering characteristics of the frequency–angle spectrum in far-field omnidirectional scattering. The simulation of echoes in the time domain explains how echoes propagate and interact with each other, and provides formulas for calculating interference and resonance frequencies. The frequency calculation formula extracts key features from the spectrum, providing a basis for predicting the characteristics of multi-cylindrical targets in terms of scale and spatial position. Measurement experiments were conducted on a double-cylindrical target in a water tank, and the theoretical calculations and experimental data were used to estimate the target’s radius and distance. The actual layout confirms the accuracy of the interference and resonance frequency prediction formulas. This study offers a valuable solution for refined feature extraction and spatial estimation of underwater targets.

1. Introduction

Scattered echoes from a target provide valuable information about its intrinsic properties and spatial relationships, including size, shape, material composition, distance, and other characteristics. A critical aspect of predicting and estimating the strength of underwater submarine targets is accurately understanding the contribution of different structures to the echo characteristics. In naval operations, for example, complex echoes generated by components such as underwater pipelines, periscopes, and communication antennas directly affect the effectiveness of underwater communication systems. Accurate prediction of scattering patterns is key to developing communication strategies that ensure reliable target acquisition in challenging underwater environments. In the field of marine engineering, the need for renewable energy has also raised concerns about the impact of underwater vehicles, pipelines, and wind farms on the marine environment, as these devices are usually arranged in a cylindrical shape. For offshore wind farms, although the existing literature focuses mainly on airborne noise [1,2,3], the underwater acoustic scattering characteristics of these large turbines are equally important. However, a comprehensive understanding of the underwater acoustic scattering characteristics of these turbines and pile drivers is essential. This understanding not only contributes to the structural integrity of these underwater devices but also plays a key role in ensuring their operational safety. By understanding the echoes of multi-cylinder underwater targets and the inter-coupled higher order scattering generation mechanisms, it is expected that refined predictions of complex underwater targets can be addressed.

Nevertheless, when confronted with multiple targets, the scattering problem becomes more intricate due to their mutual interference in the complex underwater environment. To calculate scattering from underwater simple geometric targets (e.g., spheres, cylinders, flat plates, etc.), early computational methods like Partial Wave Series (PWS) [4,5], Sommerfeld-Watson Transform (SWT) [6,7], and Resonance Scattering Theory (RST) [8,9] have been proposed and extensively employed for addressing acoustic scattering problems. Conversely, when dealing with acoustic scattering from complex targets, the widely utilized methods are the finite element method (FEM) [10] and the boundary element method (BEM) [11]. In the exploration of acoustic scattering from cylinders, researchers have employed numerical methods to analyze the vibration response and acoustic radiation from cylindrical shells [12]. Zou [13] conducted a numerical evaluation of the acoustic radiation from an elastic spherical shell in an infinite fluid using a three-dimensional sono-elastic method, subsequently validating the results with experimental data. The results demonstrate a satisfactory agreement between numerical and experimental findings. Qi [14] analyzed the acoustic radiation curves of stiffened cylinders with single and double shells under two conditions: longitudinal unit force excitation and vertical unit force excitation. Through the analysis of peak values and their corresponding characteristic modes, key structural models contributing to acoustic radiation are identified. Zou [15] proposed a formulation for calculating underwater acoustic radiation from an infinite cylindrical shell containing an internal flexural floor based on the reciprocity theorem. This transformation essentially converts the acoustic radiation problem into an acoustic scattering problem. Kha [16] employed Flügge’s shell theory to model the equations of motion for thin shells. Adopting the image source method to implement fluid–structure coupling and consider the infinite reflection of acoustic waves off the waveguide boundary, an analytical vibroacoustic model for an infinite cylindrical shell under line-distributed harmonic excitation is presented. Zou [17] introduced a mixed analytical–numerical acoustic–vibration interaction method to model and calculate vibration and acoustic radiation from a locally damped cylindrical shell immersed in water. The study’s results offer technical insights for vibration and noise reduction in cylindrical underwater vehicles.

For the study of the scattering problem between multiple cylinders, Young [18] analyzed the effect on the multiple scattering phenomenon between two columns under different arrangements and dimensional variations by combining direct matrix inversion and iterative method to measure the multiple scattering variations of the double columns using TCG (Two Cylinder Gain). Sherer [19] presents an analytical method for calculating the scattered acoustic field generated by an arbitrary parallel arrangement of cylinders exposed to two axisymmetric sound sources (a line source and a spatially distributed Gaussian source). The method employs the Hankel transform to ascertain the incident field and the separation of variables method to derive the scattered field from each cylinder in the cylindrical ensemble. Wu [20] explored the multiple scattering of plane waves by plasma cylinders distributed in parallel using the boundary value method. The study resulted in an approximate expression for the far-field scattering arising from interactions between plasma cylinders. Zhang [21] integrated the graphical acoustic computing method with the shooting bouncing ray method (GRACO-SBR) to determine the target intensity of complex underwater structures with concave surfaces and address higher-order scattering on concave structures. This method proves more efficient than the traditional finite element method in solving the scattering intensity of complex targets and overcomes the limitation of the traditional graphical acoustic computing method in handling second-order scattering. Santini [22] examined the scattering of planar electromagnetic waves on an infinite and perfectly conducting elliptical cylinder with incomplete reflection. The study introduced a novel analytical solution validated through numerical experiments. The proposed theoretical and numerical methods are versatile and can be applied across a broad spectrum of applications, spanning from microwave to optical technologies. Wang [23] suggested an iterative physical acoustic (IPA)-based method for calculating the multiple scattering acoustic field on a rigid surface at high frequencies and predicting the target strength of a concave target. This method imposes no limitations on the curvature of the concave surface, accommodating higher-order scattering on the concave surface. The scattering problem for linearly arranged multi-cylindrical targets can be extended to sonic crystals structures. Moheit [24] uses the Astley–Leis infinite element method to solve acoustic exterior problems, highlighting frequency-independent normal modes in elliptical domains. It explores their applicability, focusing on sonic crystals and meta-atoms, providing insights into their sound-insulating effects, and proposing a novel approach for design using normal modes in exterior acoustics.

Linearly distributed multi-cylinders are common structures in underwater multi-target scenarios. They are frequently encountered during underwater operations and target identification. Therefore, it is crucial to establish an accurate model for predicting omnidirectional target characteristics. This will help in analyzing the features of underwater multi-cylinder targets and subsequently identifying them. To comprehensively understand the overall characteristics of underwater multi-cylinders, it is essential to acquire omnidirectional, wideband, and finely spaced frequency target echo characteristics. This takes into account the influences of signal incidence direction, frequency, number of targets, radius, and target distance in echo characteristic measurements. The study uses the finite element method to create a multi-cylinder target model and calculate its frequency response in different directions when stimulated by a wideband signal from a far-field monostatic transducer. The omnidirectional frequency response characteristics of the target reveal the mechanism of the primary components of the target echo and the resonance characteristics resulting from the coupling between echoes. The text presents theoretical numerical relationships that are linked to the structural characteristics of the target. These relationships offer a theoretical foundation for predicting the structural and spatial distribution characteristics of the target. The accuracy of the theoretical relationship is subsequently validated through underwater experiments. The research results are important for studying how complex, non-standard shaped multi-targets scatter underwater and for predicting their structural characteristics.

The remainder of the paper is organized as follows. Section 2 outlines the sound field modeling and numerical calculations, followed by an in-depth analysis of scattering mechanisms in Section 3. Section 4 extends the investigation to predict target geometry and distance. Experimental validation is presented in Section 5, and the study is discussed in Section 6 by summarizing key findings and their implications.

2. Sound Field Modelling and Numerical Calculations

2.1. Geometric Spatial Distribution of Targets

The monostatic configuration involves modelling multiple cylindrical targets arranged in space using linear elasticity. The scattering far field is then calculated using the finite element method. If the length of the cylindrical targets is significantly greater than the radius, they are approximated as cylinders of infinite length. The assumptions for the modelling and investigation are as follows: the fluid (water) is assumed to be homogeneous and clear. The current model does not include reflection from other solid surfaces, such as the seabed, coast, or close underwater constructions. To account for these targets, a two-dimensional finite-element acoustic field model can be used. In this study, a physical field model is constructed using COMSOL Multiphysics 6.0 and the pressure acoustics module, along with the solid mechanics module for frequency domain calculations. The software calculates the sound pressure equations based on preset equations. Equations (1)–(3) show the calculation process. Equation (1) solves the Helmholtz equation in the frequency domain:

$$\mathsf{\nabla }\cdot \left(-\frac{1}{{\rho }_0}\mathsf{\nabla }{p}_{\mathrm{t}}\right)-\frac{{\omega }^2}{{\rho }_0{c}^2}{p}_{\mathrm{t}}=0$$

(1)

where ▽ denotes the gradient operator, ρ₀ is the density of the medium in the acoustic field, ω is the angular frequency, and c is the propagation speed of the acoustic signal. Where p_t = p_b + p_s, p_t is the total acoustic pressure in the acoustic field, p_s is the scattered acoustic pressure, and p_b is the background acoustic pressure. At the boundary of the acoustic-structural coupling, the boundary conditions are as follows:

$$-\mathit{n}\cdot \left(-\frac{1}{{\rho }_0}\mathsf{\nabla }{p}_t\right)=-\mathit{n}\cdot {\mathit{u}}_{tt}$$

(2)

$${\mathit{F}}_A={\rho }_t\mathit{n}$$

(3)

where u_tt is the structural acceleration, n is the surface normal direction, and F_A is the load (force per unit area) acting on the structure.

The basic process of model finite element numerical computation simulation includes creating model components, adding physical fields, numerical computation, and data post-processing. The model components are created in the geometric model creation module, as shown in Figure 1, where an illustration is provided with three cylindrical targets arranged linearly. The distance between these targets is denoted as d, R represents the radius of the cross-section of the cylinders, and θ denotes the angle of incidence of the sound wave. The cylinders are sequentially labeled as targets #2, #1, and #3, while S represents the sound source in Figure 1. The radius of the three cylindrical targets is R = 0.25 m, the spacing is d = 0.25 m, the radius of the water space is 1.3 m, the thickness of the perfect matching layer is 0.1 m, and the acoustic-structural boundaries are set up on the contact surfaces between the targets and the water.

A mesh convergence study was conducted to determine the appropriate grid size for numerical calculations. The maximum cell grid size is determined by the wavelength/N condition, and the number of grids increases as N increases. Figure 2 demonstrates that the sound pressure calculated by this model converges when N is 5. To ensure accuracy in the calculation, the maximum cell grid size is uniformly set to wavelength/6 in subsequent numerical calculations. The target and water regions are divided by a free triangular grid, and the perfect matching layer is divided by a swept grid with eight swept layers. The incidence of plane waves is used. The target cylinders are made of stainless steel with a longitudinal wave velocity of 5940 m/s, a transverse wave velocity of 3100 m/s, and a density of 7800 kg/m³.

2.2. Numerical Calculation and Characterization of Target Scattering Intensity

An acoustic field model is constructed, and numerical calculations for omnidirectional multi-target scattering in the far-field are performed using the finite element method. The incident signal is an LFM (linear frequency modulation) signal ranging from 100 Hz to 10 kHz with a frequency interval of 20 Hz. The incident direction θ ranges from −90° to 90° at 2° intervals. It is specified that the angle of incidence is 0° along the x-axis. Figure 3 and Figure 4 show the frequency–angle spectra of two and three cylinders, respectively, for different distances d (where d = 0R is the fit arrangement between the targets). The plots use the angle of incidence of the acoustic wave on the horizontal axis and the frequency range of the incident signal on the vertical axis. Brightness values in the plot signify the target intensity.

Figure 3 and Figure 4 show the directivity distribution of scattered sound pressure for a line array of multi-cylindrical targets. The spectra display distinct light and dark stripes that vary in distance and intensity depending on the angle of incidence, signal frequency, and distance between the targets. The stripes are formed by the main and side lobes scattering in different directions. At lower frequencies, the main lobe dissipates less energy than the side lobe. The lobe width narrows as the number of frequency resonance peaks increases at higher frequencies. The main lobe concentrates sharp energy while the side lobe increases. Comparing the scattering characteristics in Figure 3 and Figure 4 at different distances reveals an increase in the number of light and dark stripes as well as a decrease in distance with an increase in the number of targets. Therefore, analyzing the spectral characteristics of the scattered sound pressure in relation to the number of targets, the distance between targets, and the angle of incidence of the sound source can serve as a basis for multi-target detection and identification.

The signal’s frequency response was extracted sequentially for two targets and three targets with distance d = 1R at incident angles θ = 0°, 30°, 45°, and 90°, as shown in Figure 5. Figure 5 shows that as the number of targets increases, so does their scattering intensity. At θ = 0°, the scattering intensity of the three targets differs significantly from that of the two targets, as all three targets receive the emitted signals. This difference is more noticeable than at other oblique incidence angles. At 90°, the targets’ intensities are similar for the three cylinders and the double cylinder due to occlusion. Target #3’s specular reflection is the main contributor to the overall intensity at this point. The slight difference in intensity is due to differences in coupling between the targets, resulting in different echo compositions for the two scenarios.

3. Scattering Mechanism

3.1. Time Domain Echo Simulation

Acquiring the time–domain echo plot of the target is essential in order to identify the factors that influence the echo scattering intensity of multi-targets. The goal is to identify the key echo components that impact the distribution characteristics of the target’s intensity. The process from signal transmission to reception can be treated as a linear time-invariant system, assuming linear acoustics. The system’s response to the incident signal is represented by the echo signal. To determine the time–domain echo signal, we convolve the transmitted signal and the system impulse response. Using the convolution theorem, solve the inverse transform of the product of two functions through the fast Fourier transform to determine the convolution operation in the spatial domain.

Assuming that x(n) is a LFM signal of length N, it is transformed into a frequency–domain representation by Fourier variation:

$$X\left(k\right)={\sum }_{n=0}^{N-1}x\left(n\right)W_N^{kn},k=0,1,\dots ,N-1$$

(4)

where $W_n=e^{(-2\pi i)/N}$. The target frequency response calculated by COMSOL finite element results in H(k), the omni-directional system transfer function, which is multiplied with the frequency response of the LFM signal to obtain the frequency response of the target echo signal:

$$Y\left(k\right)=X\left(k\right)·H\left(k\right)$$

(5)

Performing a Fourier inverse transformation reconverts the frequency domain signal back to the time domain:

$$y\left(n\right)={\sum }_{k=0}^{N-1}Y\left(k\right)W_n^{-kn}$$

(6)

Equations (4)–(6) can be used to obtain the time domain echo simulation signal of the target. The simulation results of the time domain echoes for two and three cylinders with different distances are shown in Figure 6 and Figure 7, respectively. The vertical axis of the figure represents the double propagation distance of the target time domain echo, while the horizontal axis represents the angle of signal incidence. To elucidate the frequency response and resonance mechanism of the multi-cylinder, the composition of the scattered waves in different directions of incidence is detailed. This information, combined with the target attitude change and position data from the simulation process, forms the basis for understanding the shift in bright spot positions for different echoes as a function of angle of incidence, as shown in the time domain echo plots.

In Figure 6, the green solid line and the green dashed line represent the positions of the specular reflection wave of the two cylindrical targets themselves. In an elastic medium surrounding the target, the slow Franz wave propagates in the tangent direction near the surface. The black solid and black dashed lines in Figure 6 correspond to the target’s own Franz wave. The red solid line in Figure 6 represents the position of the second order scattering wave resulting from the coupling between the two targets.

As the number of targets increases, the light and dark stripes in the frequency angle spectra become more complex due to the increased number of echo components. Figure 7 shows a simulation of the time domain echoes for three line-array aligned targets. In addition to the specular reflection and Franz waves emanating from each target, Figure 7 clearly shows the second-order scattered waves that occur between any two targets.

3.2. Echo Analysis and Numerical Simulations

As observed in Figure 6 and Figure 7, under the conditions of a monostatic transducer, the scattered wave components resulting from the acoustic excitation of multiple elastic cylindrical targets are mainly composed of the specular reflection wave of each target (Figure 8a), the Franz wave propagating along the tangent direction of the edge of each target (see Figure 8b), and the second-order scattering due to the coupling between the targets (see Figure 8c). These represent different echoes that interact with each other during the propagation process, leading to a resonance phenomenon in the case of isotropic bit superposition.

Considering the three targets in Figure 8 as an example, using the center of target #1 as the reference point,

Table 1. Propagation paths for various waves.

Number	Path	Type of Wave	Propagation Path (m)
r1	S-②-S	Specular reflection	$2r_1$
r2	S-①-S	Specular reflection	$2r_2=2\left(r_1+\left(2R+d\right)sin\theta \right)$
r3	S-③-S	Specular reflection	$2r_3=2\left(r_1-\left(2R+d\right)sin\theta \right)$
r4	S-A-B-S	Franz wave	$2r_2+2R+\pi R=2\left(r_1+\left(2R+d\right)sin\theta \right)+2R+\pi R$
r5	S-C-D-S	Franz wave	$2r_1+2R+\pi R$
r6	S-E-F-S	Franz wave	$2r_3+2R+\pi R=2\left(r_1-\left(2R+d\right)sin\theta \right)+2R+\pi R$
r7	S-④-⑤-⑥-⑦-S	Second order scattering	$\pi R+2r_1+d+\left(2R+d\right)\mathit{sin}\theta +2R$
r8	S-⑦-⑧-⑨-⑩-S	Second order scattering	$\pi R+2r_1+d-\left(2R+d\right)\mathit{sin}\theta +2R$
r9	S-④-⑤-⑥-⑧-⑨-⑩-S	Second order scattering	$2\pi R+2r_1+2d+2R$

displays the propagation path of each echo component relative to the target center:

4. Target Geometry and Distance Prediction

Using three cylinders as an example and considering the relative paths of the different echo signals in Table 1, if the path difference between the echoes is an integer multiple of the wavelength of the signals, the echoes will couple together to produce interference and the corresponding position in the spectra will represent the peak. We determine the mathematical relationship between the peak interference frequency resulting from the mutual coupling between different echo components and the angle of incidence of the signal, as well as the target radius and distance. The formula for calculating the frequency of the peaks produced by the interaction of two adjacent target specular reflections (r₂-r₁, r₃-r₁) is shown below.

$$f_1=\frac{n_{1}c}{2\left(2R+d\right)\mathit{sin}\theta }, n_1=1,2,\dots ,⌊\frac{2\left(2R+d\right)\mathit{sin}\theta \cdot f_{\mathrm{m}\mathrm{a}\mathrm{x}}}{c}⌋$$

(7)

where the ⌊ ⌋ symbol denotes downward rounding, f_max denotes the highest frequency of the received wideband signal, and n₁ denotes a multiplicative relationship (same for n₂, n₃ below).

The position of the peak interference frequency resulting from the interaction between the specular reflection of target #2 and the specular reflection of target #3:

$$f_2=\frac{n_{2}c}{4\left(2R+d\right)\mathit{sin}\theta }, n_2=1,2,\dots ,⌊\frac{4\left(2R+d\right)\mathit{sin}\theta \cdot f_{\mathrm{m}\mathrm{a}\mathrm{x}}}{c}⌋$$

(8)

Location of the valley of the resonance frequency of the mutual coupling of the cylindrical target’s own specular reflection with its own Franz wave of (r₄-r₂, r₅-r₁, r₆-r₃):

$$f_3=\frac{n_{3}c}{\pi R+2R}, n_3=1,2,\dots ,⌊\frac{\left(\pi R+2R\right)\cdot f_{\mathrm{m}\mathrm{a}\mathrm{x}}}{c}⌋$$

(9)

The interference frequencies resulting from the second-order scattering between the targets, coupled to the specular reflections (r₇-r₂, r₈-r₁) from the targets are as follows:

$$f_4=\frac{n_{4}c}{\pi R+2R+d-\left(2R+d\right)\mathit{sin}\theta }, n_4=1,2,\dots ,⌊\frac{\left[\pi R+2R+d-\left(2R+d\right)\mathit{sin}\theta \right]\cdot f_{\mathrm{m}\mathrm{a}\mathrm{x}}}{c}⌋$$

(10)

The interference frequencies of the second-order scattering of targets #2 and #3 coupled with the specular reflection (r₉-r₂) of the targets are as follows:

$$f_5=\frac{n_{5}c}{2\pi R+2R+2d-2\left(2R+d\right)\mathit{sin}\theta }, n_5=1,2,\dots ,⌊\frac{\left[2\pi R+2R+2d-2\left(2R+d\right)\mathit{sin}\theta \right]\cdot f_{\mathrm{m}\mathrm{a}\mathrm{x}}}{c}⌋$$

(11)

Considering Equations (7)–(11) above, it is obvious that the calculation formula for the coupling frequency is related to the distance d of the multi-cylindrical target as well as the target radius R. Consequently, the prediction of the target radius and distance can be made based on the peak of the interference frequency and the valley of the resonance frequency between the different echoes. Using the frequency calculation formula for the mutual coupling between different echoes, the numerical prediction results of the frequency positions where the different echoes of the three targets interact are presented based on Figure 4, as shown in Figure 9.

As can be seen in Figure 9, there are two types of stripes in the spectra of the three cylinders. One is the bright and dark stripes (e.g., the red solid line and the black dashed line) with different widths curving towards the 0° incidence direction, where the narrower bright stripe (black dashed line) is due to the strong amplitude characteristics exhibited by the coupled interference of target #2 and target #3 with each other. The wider bright streak (red solid line) is mainly due to the interaction of specular reflections from two adjacent targets (targets #1 and #2, targets #1 and #3), as well as the interaction of specular reflections from distant targets (targets #2 and #3), which together form a secondary amplitude superposition of energy and thus exhibit the strongest amplitude features. The other streak (white dashed line), curved from the 0° incidence direction to the 90° and −90° directions, originates from second-order scattering from the target, and the intersection with the specular reflection coupling produces the highest target intensity values.

In addition, the position of the green dashed line in Figure 9 indicates the variation in target scattering intensity resulting from the resonance of the target’s own specular reflection echo with the Franz wave. When the Franz wave and the specular reflection wave overlap, there is a valley in the target scattering intensity near a fixed frequency.

The frequency responses corresponding to θ at 0°, 30°, and 45° in Figure 9b are calculated sequentially, as shown in Figure 10, where the frequency response peaks marked by the red inverted triangles (▼) are generated by the coupling of the specular reflected waves from two neighboring targets (#1–#2, #1–#3). Due to the mutual coupling of the echoes, the interference occurs when the difference in the propagation paths between the different echoes is an integer multiple of the wavelength (or an integer multiple of the phase difference of 2π), resulting in the formation of a peak in the frequency response. This is shown as bright interference fringes in the frequency–angle spectrum in Figure 9. At the same time, there are smaller frequency response peaks marked with black squares (■) on the plot, which are produced by the coupling of specular reflection waves from two targets (#2–#3) and appear as slightly darker fringes in Figure 9. Among these frequency response peaks, there are unusually strong peaks (indicated by ○ with a blue circle in Figure 10), which are due to the superposition of energy caused by the interaction of the specular reflection and the second order scattering echoes between the targets. The frequency response valleys are marked with a green asterisk (*) in Figure 10.

By integrating the pattern in Figure 10 with the above Equations (7)–(11) for calculating the echo coupling frequency, the distance and radius of the target can be estimated numerically. This can be achieved by identifying the peaks resulting from the interaction between the specular reflection wave and the second-order scattering wave, and the valleys in the frequency response resulting from the interaction between the specular reflection wave and the Franz wave in the frequency spectra of the linearly arranged multi-cylindrical target.

5. Experimental Setup and Results

5.1. Experimental Systems and Data Acquisition

Based on the frequency–angle spectrogram of the target and the echo coupling frequency calculation formula, the characteristics of underwater multi-cylinder targets with unknown radius and distance can be numerically predicted. To verify the accuracy of the echo coupling frequency prediction formula, a double-cylinder echo experiment was conducted in a water tank. A 100–200 KHz LFM signal with a pulse width of 0.02 ms and a sampling frequency of 1000 KHz was used, and the experiment was carried out in two configurations: two cylinders close together and two cylinders 0.15 m apart. The two cylinders were suspended vertically in a water tank with a length of 80 cm and a diameter of 5 cm, and it was specified that the two cylinders facing the hydrophone simultaneously should be at 90° and the single cylinder facing the hydrophone should be at 0°. The exact layout of the experimental test is shown in Figure 11. In this double cylinder echo test experiment, the signal source transmits a LFM signal, which is amplified by a power amplifier, and finally the signal is transmitted to the target by a transducer, and the echo signal of the target is received by a hydrophone, and the data is collected by a data acquisition card, and finally the echo signal is processed by a signal processing center. By rotating these two cylinders, the characteristics of the target echoes in the range of −90°–90° at the receiving and transmitting positions were obtained, and the time domain echoes (Figure 12a and Figure 13a) and frequency-angle spectra (Figure 12b and Figure 13b) were used to represent the two-cylinder targets under the two configurations.

At the same time, finite element numerical calculations are carried out for the echo signals of the double cylindrical target under the same arrangement and size. Extracting the omnidirectional frequency response plots of the numerical calculation and experimental results at any same frequency (160 KHz and 180 KHz), as shown in Figure 14, it can be seen that the number and angular positions of the resonance peaks of the theoretical model and the practical model at the same frequency are in agreement with each other, and the slight difference lies in the amplitude, which does not affect the use of the echo prediction formulas. This phenomenon supports the feasibility of the prediction formula.

5.2. Estimation of Target Radius

To verify the accuracy of the above echo prediction formulas, the target scattered intensity is subjected to the valley value of the scattered intensity due to the mutual coupling of the specular reflection wave and the Franz wave. Taking the 0–90° range as an example, the location of the valley frequency of the target scattered intensity is extracted, as shown by the location of the purple spotlights in Figure 15a,b. According to the formula given in Equation (9) for calculating the resonance frequency of the target’s specular reflection and Franz wave, the radius of the cylindrical target was estimated based on the frequency of the location of each valley point, and the estimation results were statistically analyzed for the two configurations. The probability density distribution of all the estimation results is shown in Figure 15c, and taking the center of the probability density distribution as the evaluation prediction value, the radius result is about R = 0.0251 m. This estimation result is very close to the actual target radius.

5.3. Estimation of Target Distance

When the specular reflection echoes of the two targets couple and interfere with each other, the frequency response peak is formed, which represents the brightest stripes in the frequency angle spectra. The locations of the brightest interference stripes in the spectra of the two configurations were extracted separately, and the peak bright spots in the frequency range with clear bright interference stripes were extracted, as shown in Figure 16a,c. We found the frequencies and incident angles corresponding to different peak points. Based on the above estimation results of the radius, the target distance is estimated by substituting into Equation (7). The results are shown in Figure 16b,d. The red dashed lines are the average results of the peak spotlight estimation results for all stripes, and the two target distances are calculated to be d = 0 m and d = 0.149 m, which are very consistent with the actual experimental configuration. The numerical prediction results of this experiment confirm the accuracy of the target echo interference resonance frequency calculation formula.

6. Discussion

The investigation analyzed the scattering echo characteristics of underwater multi-cylindrical targets using a two-dimensional simulation model constructed with the finite element method. The obtained omnidirectional scattering frequency response characteristics allowed for a comparative analysis between double-cylindrical and triple-cylindrical targets. In order to investigate the composition of the scattering echo components of the multi-cylindrical target. By comparing the spectra of double-cylindrical target and triple-cylindrical target, the scattering intensity of the target increases with the increase in the number of targets. In addition to the difference in scattering intensity, the echo contribution of coupling resonance is different, which makes the frequency response show different characteristics. The time domain echo simulation is carried out on its frequency response characteristic plots; according to the time domain echo plots, it can be seen that the main echo components of the target include the target’s specular reflection wave, the Franz wave, and the second-order scattering wave between the target, and the propagation path by different echoes is given according to the formula of the actual simulation. When the propagation path difference between different echoes is an integer multiple of the signal wavelength, the echoes couple with each other to produce interference and resonance phenomena. On the basis of this characteristic, the article gives the formula for calculating the peak frequency of the coupled interference between different echoes, which is an effective prediction of the distance and radius information of multi-cylindrical targets.

The article describes an experiment that uses an omnidirectional double cylinder echo test in a water tank. The experiment extracts the frequency position of the peak and valley of the interference stripes in the spectra under two configurations: double cylinders close and double cylinders separate. The article then estimates the radius of the double cylinders and the distance between them using the theoretical coupling frequency formula. The results closely aligned with the actual experimental setup, affirming the validity of the proposed formula.

7. Conclusions

This study significantly advances the understanding of scattering echo characteristics in underwater multi-cylindrical targets, specifically examining a configuration with three linearly arranged cylinders. Through meticulous analysis, the study elucidates the reverberations of mirror reflection, Franz waves, and second-order scattering. Additionally, it details the propagation paths for nine distinct echoes among the targets, providing a foundation for comprehending the resonance mechanisms behind five specific resonance peaks. The integration of numerical simulations, analytical formulations, and experimental validation establishes a robust predictive framework for target attributes and highlights the practical efficacy of the proposed methodology in accurately estimating radius and range.

The article validates the proposed predictive formulas using echo signals from double cylinders in a water tank. The predicted radius of the targets has a relative error within 0.4%. When double cylinders are separated, the relative error in the predicted distance is 0.67%, while the estimation result for the distance when double cylinders are closely aligned is zero. The predictive model accurately and reliably captures the details of multi-cylindrical targets in different configurations. This provides valuable insights into the performance of the proposed methodology under varying conditions. The study successfully enhances the detection and identification precision of small underwater targets within clusters.