Development of energy flow models to predict sound fields for sound absorbing structures in medium-to-high frequency ranges

Abstract

Energy flow analysis (EFA) model for the treatment of sound absorbing structures is developed to predict sound fields of engineering systems in medium-to-high frequency ranges. Thus far, EFA for acoustic models has been developed only for low damping media, such as air and water. A new energy flow governing equation is derived in this study by identifying a relationship between energy density gradient and intensity and acoustic energy dissipation for sound absorbing materials. With the developed EFA model, dispersive wave, and loss factors are identified using complex acoustic properties, and they are investigated to make ensure that they reflect the properties of sound absorbing materials. By solving the governing equation, acoustic energy density, and intensity distributions in sound absorbing materials are obtained, and noise analyses are performed for sound absorbing structures. They are compared with those obtained via a conventional method and experimental result for the verification, in which we confirmed that both results agreed well. Furthermore, various sound absorbing structures are analyzed using the developed EFA model to predict sound fields in medium-to-high frequency ranges. It is demonstrated that the developed EFA model is useful for medium-to-high frequency ranges.

Introduction

Owing to the increasing importance of addressing noise problems in various engineering fields, such as automobiles, ships, and aircraft, interest in noise prediction, and control has increased. Noise standards are regulated to maintain the appropriate noise level, and the levels are typically measured with A-weighting. A-weighting reflects the human ear’s sensitivity according to frequency ranges, with positive gains from 1 to 6 kHz, ¹ which indicates the importance of noise control in medium-to-high frequency ranges. Accordingly, sound absorbing materials have been widely used for building soundproofing systems, such as silencers, panels, and many types of insulators owing to their excellent performances in controlling noise in medium-to-high frequency ranges. ² At the design stage, it is important to predict noise by identifying sound fields with sound absorbing structures through reliable analysis methods suitable for medium-to-high frequency ranges. ³

Numerical methods have been developed to predict sound fields in sound absorbing structures. Finite element method (FEM) has been widely applied for noise analyses. Such a conventional approach discretizes analysis domains and boundaries according to frequencies and requires very high computational costs as the frequency increases. For sound absorbing materials, numerical costs are higher because more equations for describing sound absorbing materials are required compared with low damping media. ⁴ Therefore, the application of FEM to sound absorbing materials at high frequencies is limited due to unreasonable computational costs and inaccurate results.^5,6 Liu ⁷ applied a semi-analytical finite element for a rectangular model (0.1 × 0.1 × 0.05 m) and analyzed it up to 1 kHz to investigate sound fields in sound absorbing structures. Effects of the sound absorbing materials were studied for a cylinder shell model with a height of 1.125 m, diameter of 1.040 m, and thickness of 0.03 m by Li et al. ⁸ Sound pressure levels according to the changes in various parameters, such as tortuosity, porosity, and flow resistivity were analyzed with FEM up to 1 kHz. Also, Fard et al. ⁹ investigated the results of using and not using a sound absorbing material attached to a box-type structure and compared the experimental and FEM results. Rectangular sound absorbing structures with the dimension of 0.3 × 0.45 × 0.005 m and 0.3 × 0.45 × 0.02 m were analyzed and compared for frequencies up to 500 Hz.

To overcome the disadvantages of traditional deterministic methods, various methodologies have been developed. One of the methods for high-frequency analysis is statistical energy analysis (SEA). ¹⁰ SEA is a suitable method for cases where internal modal density is high. SEA classifies complex systems into multiple subsystems that have similar modes and uses the total energies in the subsystem as primary variables. From the difference in the modal energy level of each subsystem, a coupling relation can be derived with the assumption of proportionality, which is defined as the coupling loss factor. Results derived from SEA can yield single averaged energy density concerning time and space for each subsystem. However, detailed information about any energy density variation and energy flow path cannot be obtained using this method. Therefore, SEA cannot reflect the sound fields occurring inside a sound absorbing structure, hence, acoustic problems for structures are solved by indirectly reflecting the acoustic properties of sound absorbing materials. ¹¹

Energy flow analysis (EFA) was introduced by Belov et al. ¹² for medium-to-high frequency ranges. Whereas SEA pertains to global energies of finite subsystems, EFA is based on a local energy approach. EFA is derived from an energy equation analogous to the heat conduction equation in a steady state, of which the main quantities are energy density and intensity. Because this method is in the form of a parabolic partial differential equation, spatial variations in time, and local space averaged energy density and energy flow paths can be effectively predicted for a structure or an acoustic field. Nefske and Sung ¹³ applied finite element formulation of the EFA governing equation to derive the vibration response for an uncoupled and a coupled beam with harmonic excitations. Wohlever and Bernhad ¹⁴ conducted extensive studies on rods and Euler-Bernoulli beams. Also, Bouthier and Bernhard^15,16 derived vibrational and acoustic response characteristics for membranes, Kirchhoff plates, and acoustic cavities. Cho ¹⁷ studied derivation of the joint relationships for coupled structures by applying the concept of power transmission and reflection coefficient, and Bitsie ¹⁸ applied Cho’s research to solve coupled structural-acoustic problems using FEM to an EFA model as a numerical tool. LeBot ¹⁹ developed a response model for a multi-domain analysis of structures, such as plates and acoustic cavities. Recently, in the field of acoustics, Kwon et al.^20–22 applied an energy flow boundary element method to single and multi-domain acoustic problems to solve complicated structures, and performed acoustic radiation noise analysis for an underwater environment. Kim et al. ²³ improved the existing acoustic energy model by deriving an energy governing equation that included a near-field energy term. Thus far, EFA models for low damping media, such as air and water, have been developed. However, the effect of sound absorbing materials that are used widely for noise control cannot be fully reflected using the existing approach, which uses an absorption coefficient for indirectly reflecting sound absorbing materials. EFA model is required to analyze acoustic characteristics of high damping media.

We herein derive the EFA governing equation for sound absorbing materials by identifying a relationship between energy density gradient and intensity and acoustic energy dissipation to predict sound fields for sound absorbing structures. In the equation, a loss factor is newly defined, and dispersion relations are identified with acoustic parameters including imaginary parts that indicate high damping. Analyses are performed based on representative sound absorbing materials, and characteristics of the derived terms are identified according to material types. From the developed EFA model, energy density, and intensity distributions in sound absorbing materials can be predicted. Energy density and intensity distributions are compared with the conventional method to verify the developed EFA model. Furthermore, sound absorbing structures are verified by comparing experimental results. Various types of sound absorbing structures are analyzed in medium-to-high frequency ranges.

Derivation of energy governing equation for sound absorbing materials

A new energy governing equation for sound absorbing materials is derived in this section. Sound propagations in absorbing materials are dominated by attenuations owing to dissipative processes, such as viscosity and thermal conductivity. The propagations can be expressed using the characteristic impedance and wavenumber, which have been continuously studied through empirical ²⁴ and theoretical^4,25 approaches. The variables derived herein have a complex form characterized by periodic fluctuations and attenuations in a mathematical model.

When propagation constants and characteristic impedances of the sound absorbing materials are defined as $k_A$ and $k_z$ , respectively, each can be expressed in the following complex number form:

$$k_A=k_{A1}-j{k}_{A2},$$ (1)

$$Z_A=Z_{A1}-j{Z}_{A2}.$$ (2)

In this section, the energy governing equations for sound absorbing materials are derived by considering these acoustic properties.

Relationship between energy density gradient and intensity

The energy density gradient and intensity relation for sound absorbing materials can be derived from acoustic parameters defined above. When sound pressure is harmonic motion with frequency $\omega$ , a wave equation can be expressed as a modified Helmholtz equation.

$$({\nabla }^2+{k_A}^2)p=0.$$ (3)

In three-dimensional space, a wavenumber $k_A$ can be divided into three components as follows:

$${k_A}^2={k_x}^2+{k_y}^2+{k_z}^2.$$ (4)

Here, each element of the wavenumber can be expressed as follows:

$$\begin{array}{c}{k}_x=k_{x0}({\tilde{k}}_{A1}-j{\tilde{k}}_{A2}),\hfill \\ k_y=k_{y0}({\tilde{k}}_{A1}-j{\tilde{k}}_{A2}),k_z=k_{z0}({\tilde{k}}_{A1}-j{\tilde{k}}_{A2})\hfill \end{array}$$ (5)

where ${\left(\frac{\omega }{c_0}\right)}^2={k_0}^2={k_{x0}}^2+{k_{y0}}^2+{k_{z0}}^2.$ Expressing a general solution of the Helmholtz equation in the Cartesian coordinate system in a plane wave form as shown below,

$$\begin{array}{c}\multicolumn{1}{c}{p=(A_x{e}^{-j{k}_{x}x}+B_x{e}^{j{k}_{x}x})·(A_y{e}^{-j{k}_{y}y}+B_y{e}^{j{k}_{y}y})·}\\ (A_z{e}^{-j{k}_{z}z}+B_z{e}^{j{k}_{z}z}).\hfill \end{array}$$ (6)

Time-averaged energy density in an acoustic medium can be expressed as the sum of kinetic energy and potential energy as follows:

$${〈e〉}_A=\frac{1}{4}\mathrm{Re}({\rho }_{A}V·V^{*}+\frac{1}{{\rho }_A{c_A}^2}p{p}^{*}).$$ (7)

Here, the < > brackets denote time-averaged over a period, ${\rho }_A$ and $c_A$ indicate density and sound speed of sound absorbing materials, and $c_A=\frac{\omega }{\mathrm{Re}\left(k_A\right)}$ . From Euler equation,

$$V=\frac{j}{\omega {\rho }_A}\nabla p.$$ (8)

Directional components of the time-averaged acoustic intensity are as follows:

$$\begin{array}{c}\multicolumn{1}{c}{〈I_x〉=\frac{1}{2}\mathrm{Re}\left(p{V_x}^{*}\right),〈I_y〉=\frac{1}{2}\mathrm{Re}\left(p{V_y}^{*}\right),}\\ \multicolumn{1}{c}{〈I_z〉=\frac{1}{2}\mathrm{Re}\left(p{V_z}^{*}\right).}\end{array}$$ (9)

Here, the superscript * denotes complex conjugate, and the subscript denotes directional components of the intensity and particle velocity.

To derive the relationship between energy density gradient and intensity, sound pressure in equation (6) and particle velocity in equation (8) are substituted into equations (7) and (9) for energy density and intensity, respectively. Energy density and intensity terms become purely exponentially decreasing or increasing and spatially periodic, respectively. Time-averaged energy density and intensity are derived as a space average over one wavelength

$$〈\overline{e}〉=\frac{k_{x0}{k}_{y0}{k}_{z0}}{8{\pi }^3}\underset{x-\pi /k_{x0}}{\overset{x+\pi /k_{x0}}{\int }}\underset{y-\pi /k_{y0}}{\overset{y+\pi /k_{y0}}{\int }}\underset{z-\pi /k_{z0}}{\overset{z+\pi /k_{z0}}{\int }}〈e〉\mathrm{dzdydx},$$ (10)

$$〈\overline{\overrightarrow{I}}〉=\frac{k_{x0}{k}_{y0}{k}_{z0}}{8{\pi }^3}\underset{x-\pi /k_{x0}}{\overset{x+\pi /k_{x0}}{\int }}\underset{y-\pi /k_{y0}}{\overset{y+\pi /k_{y0}}{\int }}\underset{z-\pi /k_{z0}}{\overset{z+\pi /k_{z0}}{\int }}〈\overrightarrow{I}〉\mathrm{dzdydx}.$$ (11)

Here, $〈\overline{e}〉$ and $〈\overline{\overrightarrow{I}}〉$ are energy density and intensity averaged over time and space, respectively.

$$\begin{array}{c}\multicolumn{1}{c}{〈\overline{e}〉=\frac{1}{4}\mathrm{Re}\{(\frac{({\left|k_x\right|}^2+{\left|k_y\right|}^2+{\left|k_z\right|}^2)}{\omega {\rho }_A}+\frac{1}{{\rho }_A{c_A}^2})}\\ \times ({\left|A_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y+k_{z2}z)}\hfill \\ +{\left|A_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y-k_{z2}z)}\hfill \\ +{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y+k_{z2}z)}\hfill \\ +{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y+k_{z2}z)}\hfill \\ +{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y-k_{z2}z)}\hfill \\ +{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y-k_{z2}z)}\hfill \\ +{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y+k_{z2}z)}\hfill \\ +{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y-k_{z2}z)})\}\hfill \end{array}$$ (12)

$$\begin{array}{l}〈{\overline{I}}_x〉=\frac{1}{2}Re\{\left(\frac{k_x}{{\rho }_A\omega }\right)\\ \times ({\left|A_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y+k_{z2}z)}\\ +{\left|A_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y-k_{z2}z)}\\ +{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y+k_{z2}z)}\\ -{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y+k_{z2}z)}\\ +{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y-k_{z2}z)}\\ -{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y-k_{z2}z)}\\ -{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y+k_{z2}z)}\\ -{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y-k_{z2}z)}\},\end{array}$$ (13)

$$\begin{array}{c}\multicolumn{1}{c}{〈{\overline{I}}_y〉=\frac{1}{2}\mathrm{Re}\{\left(\frac{k_y}{{\rho }_A\omega }\right)}\\ \times ({\left|A_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y+k_{z2}z)}\hfill \\ +{\left|A_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y-k_{z2}z)}\hfill \\ -{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y+k_{z2}z)}\hfill \\ +{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y+k_{z2}z)}\hfill \\ -{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y-k_{z2}z)}\hfill \\ +{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y-k_{z2}z)}\hfill \\ -{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y+k_{z2}z)}\hfill \\ -{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y-k_{z2}z)}\},\hfill \end{array}$$ (14)

$$\begin{array}{c}\multicolumn{1}{c}{〈{\overline{I}}_z〉=\frac{1}{2}\mathrm{Re}\{\left(\frac{k_z}{{\rho }_A\omega }\right)}\\ \times ({\left|A_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y+k_{z2}z)}-{\left|A_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x+k_{y2}y-k_{z2}z)}\hfill \\ +{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y+k_{z2}z)}+{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y+k_{z2}z)}\hfill \\ -{\left|A_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(k_{x2}x-k_{y2}y-k_{z2}z)}-{\left|B_x\right|}^2{\left|A_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x+k_{y2}y-k_{z2}z)}\hfill \\ +{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|A_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y+k_{z2}z)}-{\left|B_x\right|}^2{\left|B_y\right|}^2{\left|B_z\right|}^2{e}^{-2(-k_{x2}x-k_{y2}y-k_{z2}z)}\}.\hfill \end{array}$$ (15)

To derive energy density gradient and intensity relationship, equation (12) is differentiated with respect to the $x$ -, $y$ -, and $z$ -direction, which yielded a relation for sound absorbing materials as follows:

$$〈\overline{\overrightarrow{I}}〉=-\frac{{c_A}^2{\tilde{k}}_{A1}}{2\omega {\tilde{k}}_{A2}}(\frac{\partial }{\partial x}\overrightarrow{i}+\frac{\partial }{\partial y}\overrightarrow{j}+\frac{\partial }{\partial z}\overrightarrow{k})〈\overline{e}〉.$$ (16)

This can be expressed as a general form with a loss factor of the diffusion equation used in EFA.

$$〈\overline{\overrightarrow{I}}〉=-\frac{{c_A}^2}{\omega {\eta }_A}(\frac{\partial }{\partial x}\overrightarrow{i}+\frac{\partial }{\partial y}\overrightarrow{j}+\frac{\partial }{\partial z}\overrightarrow{k})〈\overline{e}〉,$$ (17)

$${\eta }_A=\frac{2{\tilde{k}}_{A2}}{{\tilde{k}}_{A1}}.$$ (18)

Here, loss factors for sound absorbing materials are newly defined and are used as the main variable that indicates the relationship between energy density and intensity in sound absorbing materials. As shown in equation (18), loss factors are determined by both the real and imaginary terms of acoustic properties in sound absorbing materials.

Acoustic energy dissipation

Energy dissipation can be expressed from linear acoustic equations that consider the effect of viscosity and thermal conductivity in acoustic spaces, as follows:

$${\pi }_{\mathrm{diss}}=\frac{1}{2}\mu \sum _{\mathrm{ij}}{{\varphi }_{\mathrm{ij}}}^2+\frac{\kappa }{T_0}T\prime \nabla T\prime ,$$ (19)

where $\mu$ is the shear viscosity coefficient, ${\varphi }_{\mathrm{ij}}$ the rate of shear tensor, $\kappa$ the thermal conductivity, $T_0$ the ambient temperature, and $T\prime$ the temperature perturbation. Using the plane sound wave assumption and the linear relationship of the variables, equation (19) can be expressed as follow:

$${\pi }_{\mathrm{diss}}\approx [\frac{4}{3}\mu +\frac{(\gamma -1)\kappa }{c_p}]\frac{{\left(\frac{\partial p}{\partial t}\right)}^2}{{\left({\rho }_0{c_0}^2\right)}^2},$$ (20)

where $\gamma$ is specific heat ratio, and $c_p$ is specific heat at constant pressure. With time averaging, ${\left(\frac{\partial p}{\partial t}\right)}^2$ is expressed as ${\omega }^2{\left(p^2\right)}_{\mathrm{av}}$ or ${\omega }^2{\rho }_{0}c{I}_{\mathrm{av}}$ , where $I_{\mathrm{av}}$ is intensity in the direction of propagation. Thus, time-averaged energy dissipation is expressed as:

$${\pi }_{\mathrm{diss}}\approx 2{\alpha }_{\mathrm{cl}}{I}_{\mathrm{av}},$$ (21)

$${\alpha }_{\mathrm{cl}}=\frac{{\omega }^2{\delta }_{\mathrm{cl}}}{{c_0}^3},{\delta }_{\mathrm{cl}}=\frac{\mu }{2{\rho }_0}(\frac{4}{3}+\frac{\gamma -1}{\mathrm{Pr}}).$$ (22)

Here, $\mathrm{Pr}$ is the Prandtl number, which is expressed as $\mu c_p/\kappa$ .

In the equations above, ${\alpha }_{\mathrm{cl}}$ can be derived using experimental measurements or empirical equations for air and water media. However, it is difficult to derive ${\alpha }_{\mathrm{cl}}$ for sound absorbing materials because parameters in equation (22) cannot be measured. Therefore, ${\alpha }_{\mathrm{cl}}$ must be expressed as a formula for measurable variables. If sound waves are averaged over space and time, then intensity has the following form

$$I=I_{x=0}{e}^{-2k_{A2}x}.$$ (23)

Energy conservation-dissipation theorem can be expressed as $\frac{d{I}_{\mathrm{av}}}{\mathrm{dx}}=-{\pi }_{\mathrm{diss}}$ for plane wave propagating in the $x$ -direction, and the intensity is expressed as

$${\pi }_{\mathrm{diss}}\cong 2k_{A2}〈\overline{\overrightarrow{I}}〉.$$ (24)

Assuming isotropy, the above equation can be generalized in all directions. The $2{\alpha }_{\mathrm{cl}}$ term in equation (21) can be expressed as $2k_{A2}$ , which shows that energy dissipations owing to internal damping can be expressed through an imaginary part of the wavenumber for sound absorbing materials.

Energy dissipation in equation (24) should be expressed in terms of energy density instead of intensity to derive the energy governing equation. Plane wave assumption satisfies the following relation:

$$I_{\mathrm{av}}=c_A{e}_{\mathrm{av}}=\frac{\omega }{\mathrm{Re}\left(k_A\right)}{e}_{\mathrm{av}}.$$ (25)

Therefore, equation (24) can be expressed as follow in terms of energy density:

$${\pi }_{\mathrm{diss}}\cong 2k_2\frac{\omega }{k_{A1}}〈\overline{e}〉=\frac{2\omega {\tilde{k}}_{A2}}{{\tilde{k}}_{A1}}〈\overline{e}〉.$$ (26)

If the expression above is generalized including a loss factor, the following expression is obtained:

$${\pi }_{\mathrm{diss}}=\omega {\eta }_A〈\overline{e}〉.$$ (27)

Energy governing equation

When a steady state is considered, acoustic power injected into the space of a medium is equal to the sum of the power flowing out through its boundaries, and energy dissipated in the medium. A steady state energy balance equation of the system is as follows:

$${\pi }_{\mathrm{diss}}+\nabla ·\overrightarrow{I}={\pi }_{\mathrm{in}}.$$ (28)

Inserting the previously derived energy density gradient and intensity relationship and energy dissipation into equation (28), we can newly define the governing equations for sound absorbing materials as shown below:

$$-\frac{{c_A}^2}{\omega {\eta }_A}{\nabla }^2〈\overline{e}〉+\omega {\eta }_A〈\overline{e}〉={\pi }_{\mathrm{in}}$$ (29)

$$\mathrm{or}{\nabla }^2〈\overline{e}〉-k^2〈\overline{e}〉={{\pi }_{\mathrm{in}}}^{\prime },$$ (30)

where $k=\frac{\omega {\eta }_A}{c_A}$ and ${{\pi }_{\mathrm{in}}}^{\prime }=-\frac{\omega {\eta }_A}{{c_A}^2}{\pi }_{\mathrm{in}}$ .

Energy density for sound absorbing materials is determined according to the defined wavenumber in equation (30). We investigated characteristics of the wavenumbers according to types of the sound absorbing materials. Sound absorbing materials have different acoustic performances with flow resistivity, which represents the resistance per unit thickness when air flows in them, and it ranges from $2\mathrm{X}{10}^3$ to $2\mathrm{X}{10}^5\mathrm{Pa}·s/m^2$ . ²⁶ Figure 1 shows loss factors for three sound absorbing materials using equation (18). Here, the materials are used, those of 10, 50, and 100  $\mathrm{kPa}·s/m^2$ flow resistivity and 50  $\mathrm{kg}/m^3$ density, and acoustic properties are derived from Delany and Bazley model. ²⁴ As shown, loss factors have different tendencies according to frequencies and types of sound absorbing materials. Figure 2 shows wavenumbers in equation (30) for three types of sound absorbing materials. Dispersive waves occurred because of acoustic characteristics in sound absorbing materials. In the case of air, it shows almost constant value depending on the frequency, but for sound absorbing material, wavenumbers increase gradually up to a certain frequency range and then steeply. Different tendencies for material types indicate that acoustic characteristics are expected to be demonstrated from EFA in medium-to-high frequency ranges.

Figure 1.

Loss factors for sound absorbing materials.

Figure 2.

Wavenumbers of energy flow model for sound absorbing materials.

Verification of the EFA governing equation for sound absorbing structures

Energy flow solutions for the EFA model

The derived governing equations are verified by comparing results obtained from numerical analysis. We obtain energy density distributions in sound absorbing materials by solving equation (30) using a Fourier series technique. An acoustic source is located in sound absorbing materials to show energy density attenuation as distances from the source increased. When a spherical sound source exists at $x=x_0,$ $y=y_0,$ $z=z_0$ in a rectangular domain, the energy governing equation for sound absorbing materials is expressed as

$$-\frac{{c_A}^2}{\omega {\eta }_A}{\nabla }^2〈\overline{e}〉+\omega {\eta }_A〈\overline{e}〉={\pi }_{\mathrm{in}}\delta (x-x_0)\delta (y-y_0)\delta (z-z_0),$$ (31)

where $\delta$ is Dirac delta. When sound absorbing materials are surrounded by plates, no energy flows out on the boundaries, in which partial derivatives of the energy density in the normal direction are zero on the boundaries. A general solution is expressed as follows to satisfy the boundary condition:

$$〈\overline{e}〉=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{e}_{\mathrm{nml}}\cos \left(\frac{n\pi x}{L_x}\right)\cos \left(\frac{m\pi y}{L_y}\right)\cos \left(\frac{l\pi z}{L_z}\right),$$ (32)

where $e_{\mathrm{nml}}$ indicates $(n,m,l)$ mode coefficient; and $L_x,$ $L_y$ and $L_z$ are dimensions of the rectangular sound absorbing material. Furthermore, $e_{\mathrm{nml}}$ can be obtained by substituting equation (32) into equation (31) as follows:

$$\begin{array}{l}{e}_{nml}=\frac{\omega {\eta }_A}{{c_A}^2}\\ \frac{E_{nml}}{\left[{\left(\raisebox{1ex}{$n\pi $}\!\left/ \!\raisebox{-1ex}{$L_x$}\right.\right)}^2+{\left(\raisebox{1ex}{$m\pi $}\!\left/ \!\raisebox{-1ex}{$L_y$}\right.\right)}^2+{\left(\raisebox{1ex}{$l\pi $}\!\left/ \!\raisebox{-1ex}{$L_z$}\right.\right)}^2+{\left(\raisebox{1ex}{$\omega {\eta }_A$}\!\left/ \!\raisebox{-1ex}{$c_A$}\right.\right)}^2\right)]},\end{array}$$ (33)

where $E_{\mathrm{nml}}$ and $\epsilon$ are expressed as follows:

$$E_{\mathrm{nml}}=\frac{\epsilon {\pi }_{\mathrm{in}}}{L_x{L}_y{L}_z}\cos \left(\frac{n\pi x_0}{L_x}\right)\cos \left(\frac{m\pi y_0}{L_y}\right)\cos \left(\frac{l\pi z_0}{L_z}\right),$$ (34)

$$\epsilon ={\epsilon }_n{\epsilon }_m{\epsilon }_l,{\epsilon }_i=\{\begin{array}{c}1(i=0)\hfill \\ \multicolumn{1}{c}{2(i\ne 0).}\end{array}$$ (35)

Input power applied in this study is derived from a relationship between pressure and velocity in a free field. If a sphere of radius $a$ is pulsated at frequency $\omega$ , then pressure fields in sound absorbing materials are as follows:

$$p(r,t)=\frac{1}{1+j{k}_{A}a}\frac{j\omega {\rho }_A\tilde{Q}}{4\pi r}\exp \left[j(\omega t-k_A(r-a))\right]),$$ (36)

where $\tilde{Q}$ is the complex amplitude of the volume flow of the source. Equation (36) can be restated when the point source is applied as follows:

$$p(r,t)=\frac{j\omega {\rho }_A\tilde{Q}}{4\pi r}\exp \left[j(\omega t-k_{A}r)\right]).$$ (37)

Time averaged intensity and acoustic power are derived with pressure in equation (37) and velocity in equation (8) as follows:

$$I=\frac{1}{2}\mathrm{Re}\left\{p{V_r}^{*}\right\}=\mathrm{Re}\left\{\frac{{\rho }_A{\omega }^2{\tilde{Q}}^2}{32{\pi }^2{c}_A{r}^2}\right\},$$ (38)

$$\pi =4\pi r^{2}I=\mathrm{Re}\left\{\frac{{\rho }_A{\omega }^2{\tilde{Q}}^2}{8\pi c_A}\right\}.$$ (39)

The term $\frac{j\omega {\rho }_A\tilde{Q}}{4\pi r}$ in equation (37) is used for the input pressure when FEM is applied, and equation (39) is applied as the input power for EFA.

Validation for sound absorbing materials

For three-dimensional acoustic analyses, a rectangular model shown in Figure 3 is used. The rectangular dimension is $0.5m\times 0.5m\times 0.1m$ . To confirm energy density and intensity distributions along the $\mathrm{x}$ – $\mathrm{y}$ plane, an input source is placed on the center of the rectangular model, and volume flow of source $\tilde{Q}$ is $1m^3/s$ . All boundaries are set as rigid walls. The area of interest is defined for a plane with $Z=0.075m$ and centerline as in Figure 4. Figure 5 shows a finite element model used in FEM analysis. The meshes are divided into 1/12 wavelength sizes to ensure a reliable analysis. The total number of elements is 1620 for 500 Hz.

Figure 3.

Rectangular model for sound absorbing materials.

Figure 4.

Field points for investigation of energy density and intensity distributions: (a) mid-cross section and (b) centerline.

Figure 5.

FE mesh model for numerical analyses.

Numerical analyses are performed out at 250 and 500 Hz, and these frequencies are chosen to compare with FEM results in frequency ranges that guarantee reliability. Flow resistivity values of 20 and 100  $\mathrm{kPa}.s/m^2$ for sound absorbing materials are used for comparison to investigate results according to material type. Complex numbers of the characteristic impedance and wavenumber are obtained from bulk acoustic properties experimentally determined by Delany and Bazley ²⁴ to consider sound propagations in sound absorbing materials.

Figures 6 to 9 show spatial distributions of the energy density and intensity levels predicted from EFA and FEM. Energy density and intensity levels are distributed in the centerline in Figure. 4(b). EFA and FEM results show that energy density tends to decrease from a source in Figures 6 and 8. Performances of the sound absorbing materials according to frequency and flow resistivity can be confirmed through attenuations in energy density from the center position to the boundaries. For sound absorbing materials with the same flow resistivity, attenuation in energy density increases as frequency increases, which shows the general tendency of sound absorption coefficients for sound absorbing materials. In addition, attenuation in energy density increases with flow resistivity for the same frequency. This indicates that more energy density attenuates as the resistance of air passing through the sound absorbing material increases. In EFA, energy density levels without fluctuations are obtained, and the results are attributed to the EFA characteristics derived from spatially averaged acoustic variables.

Figure 6.

Comparisons of energy density distributions on centerline for FEM and EFA results when $R_f=20\mathrm{kPa}\mathrm{s}/{\mathrm{m}}^2$ : (a) $f=250\mathrm{Hz}$ and (b) $f=500\mathrm{Hz}$ .

Figure 7.

Comparisons of intensity distributions on centerline for FEM and EFA results when $R_f=20\mathrm{kPa}\mathrm{s}/{\mathrm{m}}^2$ : (a) $f=250\mathrm{Hz}$ , (b) $f=500\mathrm{Hz}$ .

Figure 8.

Comparisons of energy density distributions on centerline for FEM and EFA results when $R_f=100\mathrm{kPa}\mathrm{s}/{\mathrm{m}}^2$ : (a) $f=250\mathrm{Hz}$ and (b) $f=500\mathrm{Hz}$ .

Figure 9.

Comparisons of intensity distributions on centerline for FEM and EFA results when $R_f=100\mathrm{kPa}\mathrm{s}/{\mathrm{m}}^2$ : (a) $f=250\mathrm{Hz}$ and (b) $f=500\mathrm{Hz}$ .

Intensity levels can be derived in EFA using equation (17). In consideration of isotropy, it is confirmed that magnitudes of the intensity values flowing in all directions are the same, and only intensity levels in the $x$ -direction are compared with FEM. As shown in Figures 7 and 9, intensity levels spread from a noise source to the boundaries, and the opposite sign based on the source position indicates the energy flow flowing in opposite directions. Comparisons of the analyzed results indicate that EFA reflects locally averaged variation of FEM results accurately, and accordingly, verification of the developed EFA model is completed for sound absorbing materials.

Energy flow analysis for sound absorbing structures

Verification was conducted by applying the developed EFA model to a sound absorbing structure. The insertion loss of the structure with and without the sound absorbing material was derived with EFA, and the result was compared with experimental results. Also, various sound absorbing structures were analyzed to confirm the usefulness in designing sound absorbing materials through EFA, and energy distributions and energy flows were derived according to structure types.

Figure 10(a) shows an experimental model in the Ming and Pan ²⁷ and was analyzed by EFA for verification. The innermost box has a V-shape with a width of $1.25m$ , a length of $1.35m$ , and a height from $0.65m$ to $0.75m$ . This box is surrounded by a $0.05m$ rockwool layer, and the outermost is made of a $3\mathrm{mm}$ fibrous glass composite panel. The flow resistivity of rockwool is 32,960  $\mathrm{Pa}\cdots /m^2$ . In order to apply the methodology suggested in this paper, the number of eigenfrequencies in certain frequency bands for the analysis structure was derived. When at least 20 to 30 modes are formed in the considered bandwidth for a structure, it can be regarded as a diffuse field, that is, a high frequency range. ²⁸ For a rectangular structure, the number of eigenfrequencies is obtained by the following equation

$$\mathrm{\Delta }N\approx \frac{4\pi f^{2}V}{{c_0}^3}\mathrm{\Delta }f+\frac{\pi \mathrm{fS}}{2{c_0}^2}\mathrm{\Delta }f+\frac{L}{8c_0}\mathrm{\Delta }f$$ (40)

where $L$ is the sum of edge lengths, $S$ is a total wall area, and $V$ is a volume of the room. When deriving for the sound absorbing structure, it can be confirmed that at 500 Hz, approximately 18 modes are formed. We can expect more reliable results with the analysis above the frequency range.

Figure 10.

Geometry and mesh model for noise analysis of a sound absorbing structure: (a) geometry model and (b) mesh model.

EFA was performed by applying a boundary integral method with joint matrix at interfaces for multi-domain of structures with air and sound absorbing materials.^17,18 The boundary integration equation is expressed using equation (30) as follows

$$\begin{array}{c}\underset{\mathrm{V}}{\int }<\overline{\mathrm{e}}>\left(\overrightarrow{\mathrm{x}}\right)({\nabla }^2-{\mathrm{k}}^2)\mathrm{GdV}-\underset{\mathrm{V}}{\int }{\pi }_{\mathrm{in}}\left(\overrightarrow{\mathrm{z}}\right)\mathrm{GdV}\hfill \\ =\underset{\mathrm{S}}{\int }<\overline{\mathrm{e}}>(\overrightarrow{\mathrm{x}})\frac{\partial \mathrm{G}}{\partial \mathrm{n}}-\mathrm{I}(\overrightarrow{\mathrm{x}})\mathrm{GdS}\hfill \end{array}$$ (41)

$$G=-\frac{1}{4\pi r}\exp (-\mathrm{kr})$$ (42)

where $\overrightarrow{x}$ is the position vector from a discretized boundary point to a field point, and $\overrightarrow{z}$ indicates a location of the input power in the domain of interest. The boundary surfaces of the model are discretized as shown in Figure 10(b) to perform noise analysis. Joint matrix can solve energy discontinuity problems occurring at interfaces through power transmission and reflection coefficients, and group velocities of the two facing media. Joint matrix for air and sound absorbing material can be expressed as follows

$$\left[J\right]=[\mathrm{Id}-P]{[\mathrm{Id}+P]}^{-1}\left[C\right]$$ (43)

where $P=\left[\begin{array}{cc}{\gamma }^{11}\hfill & {\tau }^{21}\hfill \\ \multicolumn{1}{c}{{\tau }^{12}}& {\gamma }^{22}\hfill \end{array}\right],C=\left[\begin{array}{cc}{c}_0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& c_A\hfill \end{array}\right]$ . ${\gamma }^{\mathrm{ij}}$ and ${\tau }^{\mathrm{ij}}$ are the power reflection and transmission coefficients and $\mathrm{Id}$ is identity matrix. Acoustic properties for the sound absorbing material were derived through flow resistivity of the rockwool with Delany and Bazley model. ²⁴ Noise source power is 1  $W$ and is located at ( $0.625m$ , $0.35m$ , $0.675m$ ) point.

A mesh convergence test was performed to minimize the effects of the mesh generation on noise analysis. Generally, the mesh size criterion is selected through the wavelength for air corresponding to analysis frequency. Table 1 shows the mesh lengths and the number of meshes for the criterion at 2000 Hz. The error, an index indicating the degree of mesh convergence, is defined as follows

$$\mathrm{Error}(\%)=\frac{\sum ^{}E({\lambda }_0/i\mathrm{meshes})}{\sum ^{}E({\lambda }_0/8\mathrm{meshes})}\times 100$$ (44)

where $E$ is energy density $y=2m$ plane.

Table 1.

Mesh discretization criterion and numbers.

Case number	Mesh 1	Mesh 2	Mesh 3	Mesh 4	Mesh 5	Mesh 6
Criterion	${\mathit{\lambda }}_0$ /1	${\mathit{\lambda }}_0$ /2	${\mathit{\lambda }}_0$ /3	${\mathit{\lambda }}_0$ /4	${\mathit{\lambda }}_0$ /5	${\mathit{\lambda }}_0$ /6
Length ( $\mathrm{m}$ )	0.17	0.085	0.056	0.042	0.034	0.028
Number of mesh	544	2142	4900	8876	13600	19836

As shown in Figure 11, we confirmed that it is almost converged from Mesh 4. This tendency is different from conventional FE or BE analysis that can guarantee mesh convergence when it is less than 1/6 of the wavelength. In the case of EFA, as shown in equation (42), since the green function has an exponential decay function unlike FEM and BEM, convergence can be confirmed in more relaxed meshes. ¹³ We performed noise analyses with the criterion of Mesh 4 for all frequencies. Derived insertion loss from EFA is indicated in Figure 12. We confirmed that it is in good agreement with the experimental result.

Figure 11.

Convergence test for EFA analysis.

Figure 12.

Comparison of insertion loss for enclosure.

Three type models were studied to investigate the effects of the types for sound absorbing structures with EFA. The analyzed models are shown in Figure 13. Material properties were applied in the same way as the previous verification model. By performing analyses for each model, we derived energy distributions and energy flows in the model according to structure types.

Figure 13.

Types of sound absorbing structures.

For each model, energy density distributions and energy flow were derived and analyzed on the sections as shown in Figure 14. Figure 15 shows energy density levels performed at 5000 Hz. We confirmed that energy density distributions in sound absorbing structures vary depending on locations of the sound absorbing material. In each model, lower energy is formed in the direction where the sound absorbing material is installed, and energy density is concentrated in the area where the sound absorbing material is not installed. It can be easily confirmed through energy density levels derived through the Sec. 1 section, and energy density with the lowest level are distributed at the edge of the surface where the sound absorbing material is installed. Through a comparison of (A) model and (C) model, we confirmed differences in energy distributions according to the shape of the enclosure on which the sound absorbing material is installed. With energy density distributions for (B) model on Sec. 2, differences according to locations of the sound absorbing material installed on the sidewalls can be confirmed.

Figure 14.

Planes for energy distribution and flow.

Figure 15.

Energy density levels for three types of sound absorbing structures ( $\mathrm{dB}\mathrm{Ref}.{10}^{-12}J/{\mathrm{m}}^3$ ).

Figures 16 and 17 show the energy flows on Sec. 1 for different models. The direction of energy flows depending on installation locations of the sound absorbing material is well represented. In particular, it can be seen that in the case of the (A) and (C) models, a large difference appears in the trend of energy flow due to a different shape even though the arrangement is almost similar. When a sound absorbing material is installed on the V-shaped ceiling as in (A) model, the tendency of energy to escape to the sound absorbing material can be confirmed even more distinct compared to (C) model installed on a flat surface. We can confirm energy flow trends at the corner where sound absorbing material and panels are met by comparing (A) and (B) models. In the case of (A) model, it is clear that energy flows from side panels toward the ceiling with the sound absorbing material. For model (B), energy in the structure mainly escapes to the side.

Figure 16.

Energy flow for three types of sound absorbing structures.

Figure 17.

Details of energy flow according to sound absorbing structures attached to: (a) ceiling vs bottom and (b) ceiling vs sides.

Noise analyses were performed on the large room size based on the verified procedure. The room’s geometry and mesh for noise analysis are as shown in Figure 18. We used the previously performed model for the absorbing material, and the thickness was $10\mathrm{c}m$ . The dimension of the rectangular room was $10m\times 10m\times 3m$ . A noise source was located at the end of the room $(5m,1.5m,0.1m)$ with unit power, and we investigate energy density attenuation as distances from the noise source increased at 1000 and 5000 Hz.

Figure 18.

Geometry and mesh model for noise analysis of a large room: (a) geometry model and (b) mesh model.

Figure 19 shows the energy density distribution and energy flow in the large room at 5000 Hz. As shown in Sec 1, low energy density is distributed near the absorption material side, and energy flows in the direction of the absorption material side. On Sec. 2, the energy density attenuation and energy flow direction can be confirmed from the noise source toward the opposite boundary due to the effect of the absorption material. We indicated the relative energy attenuation at 1000 and 5000 Hz in Figure 20. The energy density is derived on $x=5m$ of Sec.2. As the distance from the noise source increases, the result of 5000 Hz has a higher attenuation. To analyze the sound energy attenuation corresponding to each frequency, we confirmed the energy absorption coefficients of the absorption material to be analyzed are 0.82 and 0.97 at 1000 and 5000 Hz, respectively. It has an energy absorption coefficient as high as 0.15 at 5000 Hz, and this effect seems to be well reflected through the noise analysis.

Figure 19.

Energy density and flow distributions for a large room: (a) energy density and flow on Sec. 1 and (b) energy density and flow on Sec. 2.

Figure 20.

Energy density attenuation at 1000 and 5000 Hz.

Conclusion

Herein, an EFA model for sound absorbing materials is derived, and acoustic properties are analyzed for different types of the materials in medium-to-high frequency ranges using the derived governing equation. To reflect the properties of sound absorbing materials, high damping terms expressed in imaginary acoustic variables must be considered. Through complex acoustic variables, relationship between energy density gradient and intensity is derived and energy dissipation in sound absorbing materials is identified. Loss factors, which are the main parameters in the EFA equation, are investigated according to frequencies to ensure that they reflect the acoustic characteristics of the sound absorbing materials. A dispersive wave is confirmed, and the acoustic characteristics according to type and frequency are analyzed.

To verify the derived EFA model, the energy density and intensity distributions in a rectangular sound absorbing model are investigated. The developed EFA model is verified by comparing the results obtained from FEM. The EFA results represent the locally averaged variation of whole systems, where energy density attenuates with reliable accuracy. Furthermore, a sound absorbing structure is verified by comparing with experimental results, and various types of sound absorbing structures are analyzed in medium-to-high frequency ranges with developed EFA model. From the analyses, energy density level distributions and energy flows according to the type were investigated, and the differences according to structures of the sound absorbing material and surface shapes could be confirmed. These results suggest that the method presented herein would be helpful for the predictions of sound fields in sound absorbing structures at the medium-to-high frequency ranges in the first design stage of mechanical systems. Especially in large systems, such as ships, submarines, and airplanes, the efficient modeling capability of the developed EFA model will be highly advantageous, and its efficiency and accuracy render it a promising method for noise analysis in these industries.