Numerical research on the focusing characteristics of steady state sound field in finite space by combining finite element and phase conjugation methods

Abstract

The research on sound source identification in an infinite free sound field has been relatively in-depth, but there is relatively little research on sound source identification and localization in limited spaces, especially those with sound absorption and insulation materials. The identification and reconstruction of the acoustical steady propagation field radiated from a single frequency finite space are studied numerically using discrete elements based on the phase conjugation method by FEM. Two different kinds of array forms of phase conjugation arrays are studied for sound source localization such as the planar array and the linear array. In addition, the influences of the existence of sound-absorbing material on the wall on the focusing properties are also discussed. The numerical results show that: The phase conjugation method can completely achieve the identification and location of the acoustical source finite space no matter using the planar array form or the linear array according to FEM. The linear array form can obtain the subwavelength focusing with fewer elements. The optimal distance between the array and the sound source is 0.5 $λ$ and 2 $λ$ to get the best reconstruction results. The smaller the absorption coefficient, the more to meet the multi-path phase compensation principle, the better reconstruction of the sound source.

Keywords

Acoustical steady field in finite space phase conjugation finite element method focusing properties sound source identification

Introduction

When sound waves radiate, they are divided into free sound fields, semi free sound fields, and finite space sound fields based on the properties of their propagation space.¹ In a free sound field, sound waves radiate from the sound source towards the surrounding space and are not reflected or scattered by boundaries or other objects. The sound pressure of the radiated sound waves is inversely proportional to the distance from the center of the sound source. And the sound intensity is inversely proportional to the square of the distance.

In reality, many noise sources’ radiation sites are located in bounded spaces, such as acoustic measurements in reverberation limited spaces, speeches in halls, radiation from machine noise in factory workshops or in the cabins of ships, planes, cars, and other means of transportation. Due to the presence of walls in limited space, sound waves must be reflected to form standing waves, which make the sound field in limited space quite complex. In addition, if the acoustic properties of the wall are not uniform everywhere, the shape of the room is irregular or there are other scatterers in the limited space, it makes the general limited space sound field more complex. There are two different phenomena in the propagation of sound waves in finite space compared to in a free field. Firstly, due to the continuous reflection of sound waves from the wall, when the sound source stops, there is still continuous sound waves in the finite space, which means that a standing wave field will be formed in the finite space. Secondly, due to the continuous reflection of the wall surface, in addition to the sound energy provided by the sound source radiation, additional sound energy will also be provided by reverberation² in a limited space.

There are currently two main methods for calculating the propagation of sound waves in the general finite space sound field mentioned above, which are wave theory³ and ray acoustics.⁴ With the development of computer technology and signal processing technology, the indoor sound field simulation technology is gradually mature. The geometric acoustic virtual source method, sound line tracking method, and the hybrid method of the two are only suitable for the simulation of the high frequency part of the indoor sound field. For the low-frequency part of the indoor sound field and small indoor space, the wave effect of the sound wave is more significant, such as the diffraction and interference phenomenon of the sound wave, the room mode, or the resonance effect, which can be used to simulate by the wave acoustic method. The commonly used indoor sound field simulation methods based on wave acoustics include finite element method (FEM), boundary element method (BEM), finite difference time domain method, and digital waveguide grid method (DWM).⁵

Acoustic properties for different systems: GUAN⁶ studied acoustic damping performances of double-layer in-duct perforated plates at low Mach and Helmholtz number to evaluate the effects of the porosities of the front and back plates, and the axial distance between these two plates. Zhao dan⁷ et al. studied the aerodynamic acoustic damping performance of the inner orifite of a double-layer pipe, and further noise absorption can be achieved by introducing an additional layer of inner orifite of the pipe. Ref. [8] measured the aerodynamic acoustic damping performance of 11 perforated plates in a cold flow pipeline with variable Mach number. These plates had the same porosity, but the number and geometry of the orifice were different. The results showed that the shape of the orifice plate had little effect on the power absorption. Mohammad Ghafouri⁹ developed an analytical strategy to take into account the theory of three-dimensional elasticity, proposing the vibrational acoustic behavior of sandwich plate structures under 3D-RACS. The state vector method is used to divide the structure into several layers and the global transfer matrix method is applied to all layers of the structure. Mohammad Hossein Asadi Jafari¹⁰ studied the acoustic performance of double-curvature sandwich shell structures with various truss cores based on diffusion field. Compared with single-layer sandwich shells without trussed cores, double-curved sandwich shells with trussed cores enhance the incident sound transmission loss in the entire frequency domain and transfer the curvature frequency to a lower frequency. M. R. Zarastvand¹¹ gives some explanations of the importance of the subject and the basic equations of the problem, according to all the work in the field of wave propagation through plate structures collected in various categories. Ref. [12], a comprehensive source under the title of review papers, discusses all existing articles on the acoustic properties of multilayer panel structures and analyzes the amount of sound transmitted to the structure.

The famous acoustician Morse introduced the theory of normal modes into finite space acoustics and obtained the solution of forced vibration in finite space based on the wave equation. Over the past few decades, this theory has had a profound impact on the study of steady-state sound fields in finite space.^13–15 For several bounded spaces with regular shapes, such as rectangles, spheres, and columns, strict theoretical solutions can be obtained through wave acoustics. However, for spaces with complex shapes or spaces with certain requirements for boundary conditions, this method is not applicable. And ray acoustics is based on the assumption of energy averaging and is only suitable for high-frequency calculations, which is not suitable for calculations in large low-frequency spaces. In order to solve the distribution of sound field in finite space under complex room shapes and boundary conditions, many scholars have begun to introduce the finite element method into the study of finite space acoustics.^16–18 At present, the application of finite element in finite space acoustics is mainly applied to steady-state fields, such as ship radiated noise,¹⁸ vibration and noise analysis of automobiles,¹⁹ and acoustic modal analysis in steady-state fields. Ref. [20] establishes a finite element model of the sound field for high-strength focused ultrasound based on the sound wave equation and its boundary conditions.

The noise source identification technology is of great significance in structural noise control and acoustic fault diagnosis. Choosing a reasonable noise source identification technology is crucial,²¹ especially for the sound source identification technology of a steady-state sound field in a limited space. Lochner, J.P.A.²² et al studied the relationship between speech articulation and signal-to-noise ratio and obtained that the signal-to-noise ratio can reflect the relationship between speech articulation and reflected sound. He pointed out that the comprehensive use of direct sound and early reflected sound of speakers can improve speech articulation. Craggs²³ established an acoustic finite element model to analyze the sound field in a finite space with irregular shape. Petyt²⁴ discussed the internal sound field formed by the vibration of a cylindrical shell under force excitation. Richards analyzed the internal sound field of a car and conducted a car model test.²⁵ Nefske uses direct method and modal superposition method to solve the problem of acoustic-solid coupling in vehicle interior and carries out an example analysis. Ma Dayou once pointed out that in the classical theory of finite space sound fields, there is only normal mode reverberation sound and no direct sound, which is not in line with reality.¹⁸ His research shows that in normal mode theory, sound sources have a dual role: sound sources radiate sound waves in free space and remain unchanged as long as they do not encounter the walls or boundaries of the room. After the free radiation reaches any boundary, it is reflected on the wall as a series of wavelets with irregular distribution in time, space, and direction. And the source of the wavelets can be traced back to the sound source. These wavelets continue to propagate and reflect, ultimately forming standing waves that are allowed in a finite space which is normal modes. So the sound source is both a direct source of sound waves and a source of standing waves in limited space.

The phase conjugation (time reversal) method²⁶ is a new discipline developed in recent years, which can achieve reverse propagation and adaptive focusing of sound waves and be used for sound source localization.²⁷ This is mainly due to the good robustness and temporal stability of the time reversal method. This method for identifying and locating sound sources in ocean waveguide media has advantages that cannot be compared to other positioning methods.

Peter Blomgren and George et al.²⁸ proposed that because of the non-uniformity of the ocean medium and the presence of ocean interfaces, multi-path channels can achieve high-resolution sound source localization results. Jackson and Dowling²⁹ have demonstrated that time reversal in the time domain is equivalent to Phase Conjugation (PC) in the frequency domain that is also a forward propagating sound field, and a backward propagating sound field. According to the above theory, there is also a multi-path phenomenon of sound waves reaching the measurement array in a finite space sound field due to wall reflection, which is very similar to the propagation of sound waves in ocean channels. Therefore, the phase conjugation method can be fully applied to finite space sound field for sound source identification. Cai Yefeng et al.³⁰ studied the feasibility and mechanism of the application of time reversal method in complex sound field microphone array pickup. He provided general rules and performance. This paper uses finite element method based on the principle of phase conjugation to identify and locate a single frequency sound source in a steady-state sound field in a finite space. Two different array forms are used for sound field measurement and conjugation and forming a focus at the sound source, namely, linear array and planar array. In addition, this paper also explores the influence of the presence of sound-absorbing materials on the phase conjugation focusing characteristics of the wall.

Theory

Acoustical FEM in the room

In the volume V, surface S of the closed space, the sound source was assumed to simple harmonic motion, and the active acoustic wave equation in the room is³¹

\nabla^{2} p + k^{2} p + i p_{0} ω q (r) = 0

(1)

The acoustic absorption condition is

i k p + ξ \frac{\partial p}{\partial n} = 0

(2)

where k is the wave numbers,

k = ω / c

ω

is the circular frequency, c is the sound speed,

ρ_{0}

is the air density,

ξ

is the acoustical impedance ratio,

ξ = z / (c ρ_{0})

, (Z) is the surface impedance, and q(r) is the density function of the sound source.

The sound field space V was disserted into a number of units. For any unit, the sound pressure p at any point within the unit could be represented by the sound pressure nodes ${P}_{e}$

\bar{p} = [N] {p}_{e} = [\begin{array}{l} N_{1} & N_{2} & \dots & N_{m} \end{array}] {\begin{cases} p_{1} \\ p_{2} \\ ⋮ \\ p_{m} \end{cases}}

(3)

where

[N]

is the element shape function, and corner mark 1,2,… m indicates the node number. By substituting equation (3) into equation (1), the discrete wave equation is obtained

\begin{array}{l} \sum_{e} \int_{l e} ({[Δ N]}^{T} [Δ N] d V_{e} {p}_{e} - \sum_{e} \int_{l e} k^{2} ({[N]}^{T} [N]) d V_{e} {p}_{e} \\ - \sum_{e} \int_{l e} (i ρ_{0} ω {[N]}^{T} q (r)) d V_{e} + \sum_{e} \int_{s} \frac{i k}{ξ} ({[N]}^{T} [N]) d S_{e} {p}_{e} = 0 \end{array}

(4)

where

V_{e}

and

S_{e}

are the volume and area of the units. If

\int_{l e} ({[\nabla N]}^{T} [\nabla N] d V_{e} = {[k]}_{e}

(5)

\frac{1}{c^{2}} \int_{l e} ({[N]}^{T} [N] d V_{e} = {[m]}_{e}

(6)

\frac{1}{c ξ} \int_{S e} ({[N]}^{T} [N] d S_{e} = {[d]}_{e}

(7)

\int_{l e} i ω ρ_{0} {[N]}^{T} q (r) d V_{e} = {[f]}_{e}

(8)

Then equation (4) can be expressed as

\sum_{e} ({[k]}_{e} + i ω {[d]}_{e} - ω^{2} {[m]}_{e}) {p_{e}} = \sum_{e} {f_{e}}

(9)

{[k]}_{e}

{[d]}_{e}

{[m]}_{e}

, and

{f_{e}}

are referred to as element stiffness matrix, element damping matrix, element mass matrix, and element equivalent nodal force, respectively. Each unit is integrated in the sense of formula (9), as follows

([K] + i ω [D] - ω^{2} [M]) {p} = {F}

(10)

where

[K]

[D]

[M]

, and

{F}

are, respectively, called the global stiffness matrix, global damping matrix, global mass matrix, and global load matrix, and

{p}

is the finite space node sound pressure matrix. Considering harmonic excitation, the above equation is equivalent to

[M] {\ddot{p}} + [D] {\dot{p}} + [K] {p} = {F}

(11)

Since the absorption of air is not taken into account, only the units on the boundary contribute to $[D]$ . Equations (10) and (11) are the finite element basic equations of steady-state sound field in finite space.

Phase conjugation method

The reason why the time reversal method can realize the back propagation and adaptive focusing of sound wave is that the linearized wave equation (1) only contains the second derivative of sound pressure with respect to time, so $p (r, t)$ and $p (r, - t)$ are both solutions of the wave equation, where $p (r, t)$ represents the sound field radiated outward and propagating forward by the sound source. $p (r, - t)$ represents the sound field of backpropagation (i.e., the sound waves that travel to and converge on the source region). Time reversal in the time domain is equivalent to the frequency domain phase conjugation, namely, the $p (r, t)$ and $p (r, - t)$ is equivalent to the $p (r, ω)$ and $p^{*} (r, ω)$ , where $p^{*} (r, ω)$ is for the $p (r, ω)$ complex conjugate. Similarly, $p (r, ω)$ is the sound field of forward propagation, and $p^{*} (r, ω)$ is the sound field of reverse propagation. For the finite space sound field, the sound pressure radiated by the sound source can be measured at the measurement array, conjugated, and re-emitted back to the finite space sound field. At this time, the sound source term in the wave equation (1) needs to be removed, that is, the Helmholtz equation

\nabla^{2} p + k^{2} p = 0

(12)

In the sound pressure measurement surface array described by Helmholtz equation, after the sound pressure is measured, the array is moved a very small distance, and then the measured sound pressure gradient is brought into the above equation, and the phase conjugation formula applicable to the monopole source and dipole source can be obtained

I_{P C P} = \int_{s} [P * (r') \frac{\partial G (r, r^{'})}{\partial n} - G (r, r') \frac{\partial P * (r^{'})}{\partial n}] d s^{'}

(13)

In the free field, the measured sound pressure of the far field and the normal derivative of Green’s function can be seen as

\begin{array}{l} \frac{\partial P}{\partial n} = i k P + O (\frac{1}{r^{2}}) \approx i k P \\ \frac{\partial G}{\partial n} = i k G + O (\frac{1}{r^{2}}) \approx i k G \end{array}

(14)

where

O (1 / r^{2})

is the extreme value of

\frac{1}{r^{2}}

, and when r is the position at infinity,

ν = P / ρ c

, the normal derivative of the measured sound pressure can be obtained as

\partial P / \partial n = i k P

, where c is the sound velocity in the free field, ρ represents the density, w represents the angular frequency, and v represents the vibration velocity of the particle.

The PCM method for calculating the phase conjugate sound field based on the monopole sound source can be obtained by the simultaneous above formula

I_{P C M} = i k \int_{s} [P * (r^{'}) G (r, r')] d s'

(15)

The PCD method based on the phase conjugate sound field of the dipole source is

I_{P C D} = \frac{2 i}{k} \int_{s} [\frac{\partial^{*} P (r^{'})}{\partial n} \frac{\partial G (r, r^{'})}{\partial n}] d s^{'}

(16)

The numerical calculation steps of sound source identification in finite space steady-state sound field using phase conjugation method are as follows:

(1) For the positive sound field problem, the sound pressure value or the sound pressure gradient at the phase conjugate measurement array is calculated by the finite element method;

(2) Conjugate values of the sound pressure or sound pressure gradient of the measurement array are substituted into the acoustic governing equation as known boundary conditions of sound pressure, and finally focus is realized at the sound source.

Numerical simulations

The feasibility of identification and location the sound source by using phase conjugation method was realized in the size of 5.0 m * 4.0 m * 3.0 m room as the example. The coordinate origin was taken as the geometric center of the room. Tetrahedral grid was used for room division, containing 68,673 domain units, 4004 boundary units, and 204 side units. The maximum cell size was 0.09 m, the minimum cell size was 0.09 m, the maximum cell growth rate was 1.5, the mesh curvature factor was 0.6, and the narrow area resolution was 0.5. According to the fluid unit limit of 6 units in one wavelength, sound source located at the point (−1.9, 0, 0) and sound pressure amplitude was unit. All walls in the room were taken as rigid walls. The sketch of the room is shown in Figure 1:

Figure 1.

The diagram of the acoustical field in the room and the measurement array.

Calculated by finite element method, the maximum eigenvalue iteration times is 300, the eigenvalue linearization point value is 100, the order 392 modes are calculated, and the maximum natural frequency is 370 Hz. The six surfaces of the space are set as the boundaries of the hard sound field. According to the content of Ref. [31], the same model was established. The first three modes are calculated by the method in this paper and compared with the first three modes in Ref. [31], as shown in the following table. Compared with Ref. [31], it can be seen that the finite element simulation method can provide effective data in this model. The comparison parameters and pictures are shown in Tables 1 and Table 2.

Table 1.

The natural frequency of the model, Hz.

Numbers of mode	The current model	The Ref.[31] model
1	1.4478*10⁻⁴
2	19.545	19.547
3	27.891	27.887

Table 2.

Calculation of the comparison of finite element method.

Numbers of mode	The current model	The Ref. [31] model
1
2
3

The obtained natural frequencies of some finite space sound cavities are shown in Table 3:

Table 3.

The natural frequency of the room acoustic cavity.

Numbers of mode	Natural frequency (Hz)
2	34.32
14	99.116
41	148.73
79	200.14
235	300.08

According to the calculation, the natural frequencies under different modes are obtained. The speed of sound in air is 340 m/s. The calculated frequency is f = 300 Hz corresponding to the 235th modes, then the corresponding acoustic wavelength $λ$ = 1.13 m. Modal shape is shown in Figure 2:

Figure 2.

The mode shape corresponding f = 300 Hz.

Under the soft sound field boundary condition, the natural frequency sound pressure distribution map when f = 300 Hz is shown in Figure 3:

Figure 3.

The outdoor sound field sound pressure distribution map f = 300 Hz.

Another sound source is added at a symmetric position with the existing sound source, and is introduced as information to simulate background noise. When the noise source and the original sound source are both 1 Pa, the resulting sound pressure distribution map is shown in Figure 4.

Figure 4.

Distribution map of sound pressure when both sound sources are 1 Pa.

Different array forms

The phase conjugation methods were used to identify and locate sound source in the room. First, the sound pressure level of the array calculated by FEM and took the conjugate value then re-emitted into the sound field and focus at the position of the focus source.

Consider the planar array form

The distance between the planar array and sound source was 0.5 $λ$ , $λ$ , 2 $λ$ , and 3 $λ$ in X direction. The normal direction of the array was the X-axis positive direction. The length along the Y-axis and Z-axis direction was 3.6 m, 2.6 m. Array element number 266, the array element spacing: △ = 0.2 m = 0.18 $λ$ , It satisfied the requirement that the spacing of array elements in the phase conjugate method should be less than 0.5 $λ$ .

Consider the source location, that is, the intersection line of X = -1.9 m and plane Z = 0 plane. Figure 5 shows the sound amplitude field distribution of the source (dB), and Figure 6 shows the reconstructed sound field amplitude distribution by the planar array based on the phase conjugation method.

Figure 5.

The source pressure amplitude distribution at the cross line. between the plane X = −1.9 and Z = 0.

Figure 6.

The reconstructed pressure amplitude distribution with the planar array.

Consider the form of a line array

The line array is located on plane Z = 0. The distance between the linear array and the sound source in the X direction is also taken as 0.2 $λ$ , 0.5 $λ$ , 2 $λ$ , and 3 $λ$ , respectively. The length along the Y-axis is 3.6 m. The number of elements is 19, and the spacing between elements is: △ = 0.2 m = 0.18 $λ$ . Considering the location of the sound source, Figure 7 shows the reconstructed values (in dB) of the sound pressure amplitude at the intersection of plane X = −1.9 m and plane Z = 0.

Figure 7.

The reconstructed pressure amplitude distribution with the linear array.

The sound source location and identification could be obtained by using phase conjugation method as seen from the results in Figures 6 and 7, and the perfect reconstruction results were obtained at the distance between the array and source of 0.5 $λ$ and 2 $λ$ . The sound field reconstruction was complete failure when the distance between the sound source and array was 0.2 $λ$ as the phase conjugate sound field existed two extreme points. This is very different with the phenomenon³² to break the diffraction limit based on phase conjugation method in near field measurement and the spherical wave propagation attenuation inversely proportional to the distance in free field or semi-free field. This was due to that the normal mode wave was produced as the acoustic wave emitted by the sound source immediately reflected between the walls when the acoustic wavelength was the same order or less than the space dimensions roughly in the room not as the strength was inversely proportional to the square of the distance and the higher strength at the long distance point than the recent point. According to the interaction of the normal modes, so that the sound intensity at the source side was the strongest and leaving the sound source gradually reduced, but not far away increased with increasing distance from the endless undulating movements.³³

Wall laying of sound-absorbing materials

The above conclusions were obtained by assuming the six walls of the room to be rigid, and identifying and reconstructing the sound field distribution in an ideal state. In practical life, reinforced concrete walls have sound-absorbing properties, and materials with sound-absorbing properties are laid on ceiling panels and floors. Therefore, it is necessary to study the impact of phase conjugation method focusing characteristics when laying sound-absorbing materials on the wall surface. Considering the situation where the sound absorption coefficients of the six walls of the room are the same, the sound field distribution diagram is given in Figure 8 when the sound absorption coefficients are 0.2, 0.4, 0.6, 0.8, and 1, respectively. Figure 9 shows the relative error comparison curves of the reconstructed values of the sound pressure amplitude at the intersection of plane X = −1.9 m and plane Z = 0 using the phase conjugation method when the sound absorption coefficients are 0.2, 0.4, 0.6, 0.8, and 1, respectively. The array form is taken as a linear array, with a distance of 0.5 $λ$ from the sound source.

Figure 8.

Distribution map of sound pressure under different absorption coefficients.

Figure 9.

The relative error comparison curve source of the pressure amplitude distribution at the cross line between the plane X = −1.9 and Z = 0.

The relative error is defined as

ε = | \frac{p_{s o u r c e} - p_{i n v}}{p_{s o u r c e}} |

(17)

Among them, $p_{s o u r c e}$ is the radiated sound pressure value of the forward sound field calculated using the finite element method in the presence of a sound source; $p_{i n v}$ is the sound pressure value which reconstructed the sound field by using phase conjugation method.

From the results in the above figure, it can be seen that the closer the wall is to rigidity, that is, the smaller the sound absorption coefficient, the larger the surface sound pressure, and the smaller the relative error of the reconstructed value, resulting in a better reconstruction effect of the sound source. This is because the closer the wall is to the rigidity, the more complex the path of sound propagation, and the more it satisfies the principle of multi-path phase compensation using phase conjugation methods. When the sound absorption coefficient is set to 1, the sound field is a free field, and the sound waves emitted by the sound source are completely absorbed after propagating to the wall. At this point, the reconstruction relative error obtained is large. The phase conjugation method is very effective for solving the problem of multi-path propagation in sound fields.

Conclusions

This article uses finite element method based on the principle of phase conjugation to identify and locate sound sources in a steady-state sound field in a finite space. Two different array forms, namely, linear array and planar array, are used for sound field measurement and conjugation, forming a focus at the sound source. In addition, this article also explores the influence of the presence of sound-absorbing materials on the phase conjugation focusing characteristics of the wall. The numerical calculation results show that using the finite element method, both planar array and linear array based on phase conjugation method can fully achieve sound source identification and localization in a finite space steady-state sound field, while using the linear array form can identify the position of the sound source with fewer array elements. The sound source localization effect is best when the distance between the array and the sound source is 0.5 $λ$ and 2 $λ$ . The closer the wall surface is to rigidity, the smaller the sound absorption coefficient, and the more it meets the principle of multi-path phase compensation in the phase conjugation method, the better the reconstruction effect obtained.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Project of the National Natural Science Foundation of China: Research on the Separation and Identification of Multi target Noise Sources in Complex Sound Fields Based on Phase Conjugation Method (No. 51609037); Basic Research Business Fee Project of Dalian University of Technology: Research on Location and Imaging of Ship Radiated Noise Sources in Complex Marine Environment (No: DUT22GF206).

ORCID iD

Song Liu

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