Finite-Element Method for Calculating the Sound Field in a Tank with Impedance Boundaries

In this paper, a finite-element method for calculating the sound field in a water tank with impedance boundaries is proposed based on the theory of standing waves in a tube.*e equivalent acoustic impedance of the tank walls is calculated by establishing a threedimensional axisymmetric virtual standing-wave tube in finite-element software, whereupon boundaries with that impedance are used as the tank boundaries. Since the impedance is the property of the material itself, the calculated impedance value can be used for the calculation of the three-dimensional sound field.*e sound field due to a point source in a glass tank is calculated using the proposed method, the correctness of which is assessed experimentally. By comparing the experimental and numerical results, the proposed method is shown to be correct.


Introduction
Sound field in a rectangular enclosed space is an important and classical problem in acoustics and has applications to office buildings, production facilities, cars, ship cabins, and reverberant rooms, among others. Understanding this problem helps with predicting sound fields and controlling noise. In applications, it is usually more practical to consider a rectangular enclosed space with general impedance boundary conditions than one with ideal boundaries, but the former approach is yet to be studied fully in the field of acoustics. A nonanechoic tank is an important acoustic measuring device that is used widely for calibrating transducers [1,2] and measuring sound power [3,4] and the sound absorption coefficient of underwater acoustic materials [5,6]. Being able to predict the sound field in a nonanechoic tank reasonably and effectively will help understand the sound field characteristics and provide the necessary basis for acoustic measurements using nonanechoic tanks. e accuracy of predicting the sound field in a nonanechoic tank depends largely on the correctness with which the boundaries of the sound field are modeled. Compared with an ideal boundary, an impedance boundary is more in line with an actual sound field. An impedance boundary combines both the attenuation of the energy amplitude and the phase change of the response, and it has obvious advantages compared with an absorption boundary [7].
In 1944, Morse and Bolt [8] established a theoretical model of the sound field in a rectangular room. By combining sound pressure modes and impedance boundaries, they derived a nonlinear transcendental characteristic equation. When the impedance values of the sound-field boundaries are the same, the well-known Morse table is obtained by solving that characteristic equation. Maa [9] deduced the nonlinear transcendental characteristic equation of the sound field in a rectangular enclosed space when the impedance values of the sound-field boundaries differ. Because that characteristic equation has no analytical solution, it can be solved only numerically.
Classical numerical methods for solving nonlinear equations (e.g., Newton iteration) are not suitable for solving the aforementioned equations because they yield only one root for a single initial guess. Naka et al. [10] proposed an interval Newton/generalized bisection method to help solve such nonlinear equations. Bistafa and Morrissey [11] proposed a new numerical method in which the eigenvalue problem is posed as one of homotopic continuation from a nonphysical reference configuration in which all eigenvalues are known and obvious. Du et al. [12] proposed using Fourier series to analyze the acoustic field in a rectangular cavity with general impedance boundary conditions, thereby transforming the problem of finding the characteristic root of a nonlinear transcendental equation into that of finding a standard matrix eigenvalue. However, the aforementioned approaches require the sound field to be rectangular. Instead, Xie et al. [13] proposed an approach based on a weak variational principle to study the sound field inside an acoustic enclosure whose walls have arbitrary inclinations and impedance conditions.
Because the physical properties of air and water are quite different, the acoustic characteristics of a nonanechoic tank are quite different from those of a reverberation chamber in air [14]. It is more complicated to analyze the sound field in a water tank with impedance boundaries. To calculate an underwater sound field with impedance boundaries, Stotts and Koch [15] proposed a two-way coupled modal method that satisfies energy conservation.
Finite element method is one of the widely used numerical analysis techniques. Because it can be applied to almost all the continuous medium problems or field problems, the finite element method has attracted great attention in various fields of physics including acoustics.
anks to the advances in computer technology, the acoustic finite element method is nowadays widely used to solve the acoustic problems. Vorlander [16] investigated the concepts and uncertainties of computer simulations in room acoustics, and he believes that the reliability depends on the skills of choosing the correct input data of boundary conditions such as absorption and scattering. Aretz and Vorlander [17] investigates the influence of different boundary representations of porous absorbers on the simulated sound field in small rooms. ey analyzed the sound field of a scale model room with a well-defined geometry through experimental measurements and finite element methods. Naka et al. [18] combined the geometric methods and finite element method to calculate the sound field in rooms with realistic impedance boundary conditions.
All the methods mentioned above require the room to have a regular shape. In the present paper, a method is presented for using acoustic software to calculate an impedance-boundary sound field. e advantages of this method are that it is not limited by the shape of the sound field during the calculation and the impedance value can be arbitrary. ere is no specific requirement for the material of the tank wall. As long as the parameters of the material are known, the sound field prediction can be performed using this method. is method uses the principle of a standingwave tube (SWT) to measure the acoustic impedance of materials, and a virtual SWT is established in software to obtain the impedance value at the boundary of the sound field. With the obtained impedance value, the sound field in a water tank with the same impedance boundaries can be calculated. We use the method proposed herein to calculate the sound field due to a point source in a glass water tank. By comparing the numerical results with experimental results, we verify the accuracy of the method. We then calculate and compare the sound fields in a large nonanechoic pool with different impedance values and analyze how the boundary impedance values influence the sound field.

Acoustic Description of Rectangular Enclosed Space with
Impedance Boundaries. A rectangular enclosed space of dimensions L x × L y × L z and the associated coordinate system are shown in Figure 1. For an enclosed space with uniform impedance on each of its walls, the Helmholtz equation can be written as where k � (2πf/c) is the driving wave number with frequency f and c is the speed of sound. e Helmholtz equation can be solved by separating the variables. Because the solutions in different directions have the same form, we take the x direction as the example here. e eigenfunction in the x direction can be written as where k x is the wave number in the x direction and A and B are constants. e boundary conditions in the x direction can be written as where n → is the outward normal unit vector on the walls, ρ is the density of the medium, and Z is the impedance of the walls. Substituting equation (2) into equation (3) gives For equation (4) to have a nonzero solution, the coefficient determinant must equal zero, giving e lth eigenfunction in the x direction is then obtained as 2 Mathematical Problems in Engineering where k xl is the lth root of equation (5). Equation (5) is nonlinear and can be solved only numerically. e eigenequation and eigenfunction in the y direction have the same forms as those in the x direction, and the eigenfunction can be written as where k yl is the lth root of the eigenequation in the y direction. Likewise, the eigenequation and eigenfunction in the z direction have the same forms as those in the x direction, and the eigenfunction can be written as where k zl is the lth root of the eigenequation in the z direction.
Since equation (5) is nonlinear, there is no way to analyze the sound field problem in the enclosed space of the impedance boundary by analytical calculation. is problem can only be solved by numerical calculation.

Standing-Wave Tube eory.
A wave tube is a common device with which the acoustic parameters of materials are measured. e sound absorption coefficient and acoustic impedance of the acoustic material can be measured by using the standing-wave ratio method and the transfer-function method in the wave tube. ese methods are standard test methods for which international standards have been established [19,20]. A tube of diameter D is shown schematically in Figure 2, and an acoustic material with a normal acoustic impedance of Z a � R a + X a j (or a normal acoustic impedance ratio R s ) is mounted at the end of the tube. When the tube wall is rigid, plane waves will be generated in the tube below the cutoff frequency f 0 , which is calculated by e sound field in the tube can be expressed as where p i is the incident wave and p r is the reflected wave. e reflected wave is caused by the acoustical material load at the tube end and not only differs in size from the incident wave but also may have a different phase. e relationship between p ai and p ar can be expressed as p where r p is the sound pressure reflection coefficient and σπ is the phase difference. Substituting equation (11) into equation (10) gives From equation (12), the velocity in the tube can be obtained as v � p ai ρc e − jkx + r p e j(kx+σπ) e jωt .
At the interface x � 0 between the medium and the material, the acoustic impedance can be obtained as erefore, the impedance of the acoustic material can be obtained when the sound field in the tube is known, and the sound field can also be calculated from the impedance of the material.

Finite-Element Solution
Acoustic finite element applications are nowadays widely used to predict the sound field in enclosed cavities and water tanks.
ere is many mature commercial acoustic finite element software with high computing efficiency and accuracy. When using acoustic finite element software to calculate the sound field in a closed space, the difficulty is an accurate description of the boundary conditions. e proposed numerical method for calculating the sound field in a water tank can be divided into two main steps. e first step is to use software to build a virtual SWT and calculate the acoustic impedance of the material of the tank walls. e second step is to calculate the sound field using the obtained impedance value. e acoustic calculation software used here is Actran. e sound field of a glass tank with 1.474 m inner length, 0.9 m inner width, 0.013 m thickness, and 0.6 m depth is calculated using this method. e proposed method can not only calculate the sound field of glass tanks but also be used for calculation as long as the material parameters of the tank wall are known.

Sound Field in Standing-Wave Tube.
Although it is complicated to calculate a three-dimensional sound field with impedance boundaries, the sound field in an SWT with an impedance boundary at the end of the tube is easy to calculate. To verify the ability of the finite-element (FE) software Actran and to calculate a sound field with impedance boundaries, the sound field in an SWT with an impedance boundary at the end of the tube is calculated by the FE method and analytically. e analytical calculation is based on Section 2.2, with the impedance value set to Z s � 1.5 × 10 6 + 1.5 × 10 6 j. e FE calculation model is shown in Figure 3, for which a tube with a rigid wall is built. e tube is 1 m in length and 0.1 m in diameter. e sound velocity of the water is 1500 m/s and the density is 1000 kg/m 3 . A velocity source of 1 m/s was added to one end of the virtual SWT as a sound source. e size of the acoustic grid is 0.05 meters, which can satisfy more than 10 grids per wavelength at 2000 Hz. A velocity source of 1 m/s is imposed at one end of the tube, and an impedance boundary with an impedance value of Z s � 1.5 × 10 6 + 1.5 × 10 6 j is imposed at the other end. Field points are placed in the tube to sample the sound pressure. e sound field in the tube at 2,000 Hz is calculated, and the results are shown in Figure 4. Because the dimensional amplitudes of the sound field differ between the analytical and numerical calculations, the results are normalized by dividing by the maximum value. e origin of the coordinate system is at the impedance boundary.
As shown in Figure 4, the normalized results of the analytical and numerical calculations are consistent. us, the accuracy of using the FE software Actran to calculate a sound field with an impedance boundary is verified.

Obtaining Material Impedance Using Virtual Standing-
Wave Tube. When using impedance boundaries to simulate the boundaries of a nonanechoic tank, the specific value of the impedance should first be determined. Based on the actual situation, a three-dimensional axisymmetric virtual SWT is established in Actran, as shown schematically in Figure 5. Because the calculation model is axisymmetric, a two-dimensional rotation model is established to save computing time. e length of the water-filled virtual SWT is 5 m, from equation (7), the diameter of the virtual SWT is selected as 0.2 m, and the corresponding cut-off frequency is 4,400 Hz. e sound velocity of the water is 1500 m/s and the density is 1000 kg/m 3 . A velocity source of 1 m/s was added to one end of the virtual SWT as a sound source. A glass with a thickness of 0.013 m, Young's modulus of 6 × 10 10 + 1.2 × 10 9 j N/m 2 , Poisson's ratio of 0.23, and a density of 2500 kg/m 2 is added to the other end of the standing-wave tube. To obtain a more realistic impedance value, an air region, of which the sound speed is 340 m/s and the density is 1000 kg/m 3 , and a sound absorption boundary is added at the outermost end of the virtual SWT. To obtain a plane wave in the tube, a rigid boundary is included at the boundary of the virtual SWT. To measure the material impedance using an SWT, the sound pressure in the tube must be measured. However, in the virtual SWT, the measuring point can be placed directly at the interface between the material and the medium to obtain the sound pressure and vibration velocity. A field point is added to the interface of water and glass to obtain the complex sound pressure and the velocity of the complex point at the interface.
After performing the calculation, we extract the complex pressure and the complex velocity at the field point and calculate the impedance of the material at different frequencies according to where Z s is the impedance, p is the complex sound pressure, and v is the complex velocity. e results are given in Table 1. Figure 6 shows the sound-field distribution at 1,300 Hz inside the SWT; a standing wave is formed in the tube, and the nodes and antinodes can be seen clearly. e sound field in the SWT with the calculated equivalent impedance at the tube end is also calculated to see whether it is consistent with the sound field with glass at the   tube end. Figure 7 shows the distributions of sound pressure in the axial direction obtained with the two models at 1,500 Hz. e results calculated with the two models are indeed consistent, thereby verifying the feasibility of the proposed method.

Finite-Element Method.
Using the obtained equivalent impedance, the sound field in the glass tank can be calculated. e current idea [5] of simulating the boundaries of glass or iron tanks in air is to simulate them as pressurerelease boundaries. Because the characteristic impedances of glass and water are close and are far bigger than that of air, this approximation is reasonable. To compare the difference, both boundary conditions are used and the results are compared with the test results. e FE model with a length of 1.5 m, a width of 0.9 m, and a depth of 0.6 m is shown in Figure 8, and the coordinate system is defined according to Figure 1. A point source of 1 Pa is placed at the coordinates (1.3 m, 0.15 m, and 0.3 m), and 100 field points are spaced equally in the x direction with (y, z) coordinates of (0.45 m and 0.4 m). Since the impedance is the property of the material itself, the calculated impedance values are directly used for the calculation of the three-dimensional sound field.
When simulating the tank boundaries using the obtained equivalent impedance, the impedance is imposed at the surface of the tank except at z � 0.6 m. e surface z � 0.6 m is treated as a water surface given that the characteristic impedance of air is much smaller than that of water and instead a pressure-release boundary is added at z � 0.6 m.

Test.
To verify the proposed method, we measured the sound field in a glass tank with the same dimensions as those of the FE model, the thickness of the glass is 0.013 m. e test system is shown in Figure 9. A spherical sound source driven by a power amplifier was placed at the coordinates (1.3 m, 0.15 m, and 0.3 m) and transmitted single-frequency signals. A hydrophone array (type B&K 8103) was placed in the glass tank and moved along the x direction to measure the sound field. e hydrophone (type B&K 8103) used in the test is 50 mm in length and 9.5 mm in diameter. e size of the hydrophone and hydrophone holder is much smaller than the wavelength at 2000 Hz, and the sound scattering of the hydrophone can be ignored. e sound energy in enclosed space reaches dynamic equilibrium when the sound energy radiated by the sound source is consistent with the acoustic energy absorbed by the wall and the medium. e steady state sound field is obtained during the simulation calculation. In order to obtain the steady state sound field during the experiment, data acquisition is performed 10 seconds after the sound source starts transmitting.

Results of Finite-Element Method and Tests
For the sound field in the closed space, as the frequency increases, the number of acoustic modes in the sound field increases and the distribution of the sound field becomes more and more complicated. At low frequencies, due to the interference, the sound field will have a distinct distribution. When the number of modes is sufficient, the sound field tends to diffuse. At low frequencies, in addition to considering the absorption of the boundary, it is also necessary to consider the phase of the reflected wave in order to accurately calculate the sound field. At high frequencies, since the sound field tends to diffuse, it is of little significance to consider the distribution of the sound field, mainly to calculate the acoustic energy density in the tank. In order to verify the accuracy of the proposed method, the calculation results and experimental results of the proposed method at low frequencies are mainly compared.
In Figure 10 It can be seen from Figure 10 that the sound field in the glass tank is simulated accurately by using the obtained equivalent impedance. When using pressure-release boundaries to simulate the tank boundaries, the sound field in the glass tank is also calculated accurately at 1,300 Hz. However, as the frequency is increased, the sound field simulated with pressure-release boundaries differs from the actual sound field obtained by the test measurements, the sound field becoming increasingly sensitive to the phase and  Mathematical Problems in Engineering 5   Mathematical Problems in Engineering erefore, with increasing frequency, the sound field obtained by simulating the tank boundaries as impedance boundaries becomes more accurate.

Conclusions
In this study, the FE method was used to calculate the sound field in a tank with impedance boundaries. e accuracy of doing so was verified by comparing the sound field in an SWT calculated by the FE method and analytical method. To obtain the equivalent normal impedance of the tank walls, a virtual SWT was established in FE software and the wall material was imposed at one end of the tube. From the sound pressure and vibration velocity on the interface between the medium and the material, the equivalent normal impedance was obtained and was used in the calculation of the sound field. e sound field due to a point source in a glass tank was calculated by the proposed method. For comparison, the results of using pressure-release boundaries as the tank boundaries were also calculated. Experiments were carried out to verify the proposed method, and the results of simulating the tank boundaries as impedance boundaries agreed with the experimental results. e tank boundaries could be simulated as pressure-release boundaries only when the frequency was lower than the first mode frequency. e proposed method is suitable for calculating not only water tanks but also the sound field in rooms lined with acoustic materials.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.