Simulation analysis of underwater structure based on local cylindrical acoustic holography

2.1. Deduction process

The measurement of the cylindrical near-field acoustic hologram is shown in Fig. 1: the cylinder with the radius of $r_H$ is the holographic measuring surface, the length of the measuring surface along the axial direction is 2$L$, and the measuring surface is expanded as shown in the right side of the figure; The cylinder with radius $r_S$ is the reconstruction surface.

Fig. 1Schematic diagram of measurement and reconstruction of cylindrical acoustic holography

As shown in Fig. 1, the axial measurement interval on the holographic surface is $∆z=2L/(2m+1)$; The circumferential angle of the measurement is $∆\varphi =2\pi /(2N+1)$, where $M$ and $N$ are positive integers, the $(2M+1)\times (2N+1)$ point discrete sound pressure is obtained on the holographic surface. And can be expressed as $-N\le n\le N,-M\le m\le M$. Then, $p_n(r_H,k_z)$ discrete space Fourier transform that can approximate the holographic surface sound pressure of $p(r_H,n∆\varphi ,m∆z)$:

where $m$ and $j$ are integers from one $M$ to $M$; $n$ and $j$ are integers from one $N$ to $N$; $∆k_z=\pi /L$; $w_{2M+1}$ and $w_{2N+1}$ are respectively $e^{-2\pi i/(2M+1)}$ and $e^{-2\pi i/(2N+1)}$. From the discretized $p_n(r_H,i∆k_z)$ and Eq. (4):

the sound pressure cylinder spectrum on the source surface can be approximated as:

Therefore, the sound pressure at the discrete points on the reconstructed surface can be obtained by the discrete space Fourier inverse transform:

From the discretized implementation Eqs. (3), (5) and (6), the cylindrical near-field acoustic holography can be transformed by the computer to obtain the sound pressure field on the reconstructed or predicted surface.

2.2. Cylindrical spectral filtering

Wavenumber domain windowing filtering can be used to suppress the amplification process of high spatial frequency component errors. The window function used is similar to the planar near-field acoustic holography, just replace $k_x$ and $k_y$ in $k_r$ with the axial wave number $k_x$ and the circumferential wave number $n/r$:

In the formula, $k_r=\sqrt{k_z^2+(n/r{)}^2}$; $a$ is the steepness coefficient of the window function; $k_r$ is the cutoff wave number.

This is a circularly symmetric exponential filter. The choice of $k_c$ is very important and directly affects the accuracy of the reconstruction results. A method for selecting the holographic surface and the radius of the reconstructed surface and the signal to noise ratio of the holographic measuring system is given below.

Let the sound pressure signal-to-noise ratio of each point of the holographic surface be SNR, $p_{nt}(r_s,k_z)$ is the cylindrical spectrum of the theoretical sound pressure $p_t(r_H,k_z)$ of the reconstructed surface with radius $r_s$, known by Eq. (8):

The theoretical sound pressure of the holographic surface with radius $r_H$ is the relationship between $p_{nt}(r_H,k_z)$ and $p_{nt}(r_s,k_z)$:

Let $p_{nt}(r_H,k_z)$ be the spiral spectrum of the noise signal component $p_t(r_H,\varphi ,\mathrm{z})$ contained in the holographic surface sound pressure, then the cylindrical spectrum $\stackrel{\cup }{p}(r_H,k_z)$ of the actual sound pressure on the holographic surface can be expressed as the sum of the theoretical sound pressure cylinder spectrum and the noise cylinder spectrum:

According to Eqs. (8), the reconstructed surface acoustic pressure spiral spectrum is:

Available from Eqs. (10) and (11):

At this point, the reconstructed surface noise spiral spectrum $p_{nt}(r_S,k_z)$ can be expressed as:

To ensure that the reconstruction results are meaningful, it must be ensured that the theoretical sound pressure cylinder spectrum on the reconstructed surface is not overwhelmed by noise, that is, it should be satisfied:

According to Eq. (13), it needs to be satisfied:

According to the previous analysis, in the process of sound wave propagation from reconstruction to holographic surface, the amplitude of the propagation wave is attenuated by the square root of the distance, and the amplitude of the evanescent wave component is attenuated exponentially with the propagation distance. Therefore, the following formula is established:

Eq. (16) shows that as long as the wavenumber component of $\sqrt{k_z^2+k_o^2}>k_q$ in the cyber surface acoustic pressure cylinder spectrum is filtered out, it can be ensured that any cylindrical wave component obtained by reconstruction is not overwhelmed by noise. Thus, according to the Eq. (16), it can be determined that the highest wave value used in the reconstruction process is $k_q$, and this value is used as the cutoff wave number of the wavenumber domain filtering in the reconstruction process.

3.1. Simulation conditions and steps

A group of wideband signals is simulated, and the sound field of each frequency point is reconstructed by local aperture acoustic holography algorithm. The sound field reconstruction accuracy of local aperture acoustic holography algorithm is analyzed, and the influence on the calculation of relative local radiated sound power level is evaluated. Sound source: point source, coordinates (0.75, 075); simulation size: $S_1=$ 28×16 = 13.5 m×7.5 m; array spacing: detal = 0.5 m; the distance of the holographic surface from the sound source: $Z_h=$ 0.5 m; reconstruction distance from the sound source: $Z_s=$ 0.4 m; simulation frequency: $f=$ 20 Hz-1 kHz.

3.2. Calculation steps

1. Simulate a set of Gaussian noise, through low-pass filtering, the upper limit of the low-pass filter cutoff frequency is 1 kHz.

2. Setting parameters and bringing into the broadband near-field acoustic holography algorithm module.

3.Calculate the reconstructed sound field and relative local radiated sound power level for each frequency point, and the total relative local radiated sound power level of the reconstructed sound field and the theoretical sound field within the bandwidth.