1. bookTom 22 (2022): Zeszyt 1 (March 2022)
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2300-0929
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A Novel Theoretical Modeling for Predicting the Sound Absorption of Woven Fabrics Using Modification of Sound Wave Equation and Genetic Algorithm

Data publikacji: 29 Jan 2021
Tom & Zeszyt: Tom 22 (2022) - Zeszyt 1 (March 2022)
Zakres stron: 108 - 122
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2300-0929
Pierwsze wydanie
19 Oct 2012
Częstotliwość wydawania
4 razy w roku
Języki
Angielski
Abstract

Woven fabric in Indonesia is generally known as a material for making clothes and it has been applied as an interior finishing material in buildings, such as sound absorbent material. This study presents a new method for predicting the sound absorption of woven fabrics using a modification of the wave equations and using genetic algorithms. The main aim of this research is to study the sound absorption properties of woven fabric by modeling using a modification of the sound wave equations and using genetic algorithms. A new model for predicting the sound absorption coefficient of woven fabric (plain, twill 2/1, rips and satin fabric) as a function of porosity, the weight of the fabric, the thickness of the fabric, and frequency of the sound wave, was determined in this paper. In this research, the sound absorption coefficient equation was obtained using the modification of the sound wave equation as well as using genetic algorithms. This new model included the influence of the sound absorption coefficient phenomenon caused by porosity, the weight of the fabric, the thickness of fabric as well as the frequency of the sound wave. In this study, experimental data showed a good agreement with the model

Keywords

Introduction

The application of theoretical physics in textile science, especially in computational physics and material physics is widely found both experimentally and theoretically. One of the applications of physic in textile science is to produce woven fabrics that can be used as sound-absorbing materials. Researchers examined the application of woven fabrics as sound-absorbent materials both experimentally and theoretically. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The utilization of woven fabric as a material that has the potential to be used as a sound absorber material was reported by Shoshani and Rosen-house [1]. Several studies have been conducted to utilize woven fabrics as sound-absorbent materials, and there have some efforts to measure the sound absorption coefficient of the woven fabric [12, 13, 14]. Some researchers reported [12, 15, 16] that woven fabrics have several micro-holes (around 1 mm) where the textile fabric can be assumed as the micro-perforated panel (MPP) which has quite a lot of porosity. Several studies have shown that in the formation of textile woven fabrics as sound-absorbent materials, factors such as porosity, fabric thickness, fabric surface density, and frequency affect the sound absorption coefficient on the fabric [1, 12, 13, 14, 17, 18, 19]. Research on acoustic textiles both experimentally and theoretically has been done in the area of nonwoven acoustic fabrics, but in-depth studies of woven and knit fabrics are still very limited. Theory and modeling studies on the properties of weight, thickness, porosity, and frequency of sound waves on woven fabrics are still rarely found as a complete and structured formula [20, 21, 22, 23, 24]. Several researchers [1, 12, 13, 14, 17, 18, 19, 23] had investigated the relationship between the impedance and the absorption characteristics of fabric as well as the effects of thickness, weight, and fabric. Vassiliadis [25] proposed that the sound absorption coefficient can be formulated using the following Eq. (1) aa=1(Z2Z1Z2+Z1)2=1R2 {a_a} = 1 - {\left( {{{{Z_2} - {Z_1}} \over {{Z_2} + {Z_1}}}} \right)^2} = 1 - {R^2}

Where the term aa is the sound absorption coefficient; Z2, the acoustic surface impedance on the media 2 in the unit (kg/m2s or rayl); Z1, the acoustic surface impedance on the media 1 in the unit (kg/m2s or rayl); R, the reflection factor or pressure reflection coefficient. Another model to measure the total sound absorption coefficient was proposed by Yairi, Sakagami, Takebayashi, and Morimoto [26] and it can be expressed as Eq. (2) aa=4Re{Ztot}(1+Re{Ztot})2+(Img{Ztot})2 {a_a} = {{4Re\{ {Z_{tot}}\} } \over {{{\left( {1 + Re\{ {Z_{tot}}\} } \right)}^2} + {{\left( {Img\{ {Z_{tot}}\} } \right)}^2}}}

Where the term aa is the sound absorption coefficient; Ztot, the total acoustic surface impedance on the media in the unit (kg/m2s or rayl). According to Prasetiyo, et al. [27], the sound absorption coefficient can be written using the following Eq. (3) α=4Re{Ztot}(1+Re{Ztot})2=1(Z2ρ0c0Z2+ρ0c0)2 \alpha = {{4Re\{ {Z_{tot}}\} } \over {{{\left( {1 + Re\{ {Z_{tot}}\} } \right)}^2}}} = 1 - {\left( {{{{Z_2} - {\rho _0}{c_0}} \over {{Z_2} + {\rho _0}{c_0}}}} \right)^2}

Where the term α and Z2 are the sound absorption coefficient and the acoustic impedance; Ztot, the total acoustic surface impedance on the media in the unit (kg/m2s or rayl); ρo, the density of air; co, the speed of acoustic wave. According to Vassiliadis [25] and Na et al. [23], the sound absorption coefficient of fabric is influenced by several factors such as porosity, fabric thickness, and the type of fiber in the fabric. Some researchers [1, 12, 13, 14, 17, 18, 19, 23, 25, 26, 27, 28, 29] concluded that: 1) the sound absorption coefficient increases with a decrease in yarn diameter; 2) The shape of the structure of woven fabric affects the value of sound absorption coefficient, plain woven fabric has a better fabric structure in increasing the value of sound absorption coefficient compared to other types of woven fabric (Fabrics are woven with equal warp and weft density); 3) the higher the porosity, the lower is the sound absorption coefficient; 4) the higher the volume density of the fabric, the higher is the sound absorption coefficient. In some studies, the value of the sound absorption coefficient is influenced by several factors, such as the value of the speed of acoustic wave or speed of sound, the acoustic pressure, the angular frequency, the wavenumber, particle speed of the propagation medium which vibrates around the development of sound and also the acoustic impedance [1, 12, 13, 14, 17, 18, 19, 23, 25, 26, 27, 28, 29]. According to Atalla et al. [28] and Maa [4], the acoustic impedance of the sound absorber consists of real parts and imaginary parts (resistance impedance for real parts and reactance impedance for imaginary parts). The impedance of the sound absorber was formulated using the following Eq. (4) to Eq. (6): Z=Zresistance+Zreactance=r+jωm Z = {Z_{resistance}} + {Z_{reactance}} = r + j\omega m Z=r+jωm=32ηtρocod2p(1+x232+xd28t)+jωtpco(1+(9+x22)1/2+0.85dt) \matrix{ Z \hfill & { = r + j\omega m = {{32\eta t} \over {{\rho _o}{c_o}{d^2}p}}\left( {\sqrt {1 + {{{x^2}} \over {32}}} + {{xd\sqrt 2 } \over {8t}}} \right)} \hfill \cr {} \hfill & { + {{j\omega t} \over {p{c_o}}}\left( {1 + {{\left( {9 + {{{x^2}} \over 2}} \right)}^{ - 1/2}} + {{0.85d} \over t}} \right)} \hfill \cr } x=d2ωρη x = {d \over 2}\sqrt {{{\omega \rho } \over \eta }}

Where the terms r and m are the acoustic resistance and the acoustic reactance, ρo is the density of air, co is the speed of acoustic wave, ω is the angular frequency, t is the thickness of material, d is the diameter of the holes, p is the perforation ratio and η is the viscosity of air. Maa [5] reported that the impedance of the sound absorber could be written using the following Eq. (7): Z=jcot(ωDc) Z = - jcot\left( {{{\omega D} \over c}} \right)

Where the terms D and c are the cavity depth and the speed of acoustic wave. Vassiliadis [25] formulated the impedance of the sound absorber and the propagation wavenumber as a function of flow resistivity for fibrous porous material using the following Eq. (8): Z=ρoco(1+0.0571X0.754j0.087X0.732) Z = {\rho _o}{c_o}\left( {1 + 0.0571{X^{ - 0.754}} - j0.087{X^{ - 0.732}}} \right) k=ω/co(1+0.0978X0.700j0.189X0.595) k = \omega /{c_o}\left( {1 + 0.0978{X^{ - 0.700}} - j0.189{X^{ - 0.595}}} \right) X=ρofσ=ρofvdP X = {{{\rho _o}f} \over \sigma } = {{{\rho _o}fvd} \over P}

Where the term σ is flow resistivityin the unit (kg/m2s or rayl); ρo is the density of air, co is the speed of acoustic wave, ω is the angular frequency, d is the thickness of the absorbent, v is the mean steady flow velocity and P is pressure drop. Vassiliadis [25] has also formulated the impedance of the sound absorber, Z, in simple form using the following Eq. (11) Z=ρv Z = \rho v

Where the terms ρ and v are the density of sound in specific medium in the unit (kg/m3) and the speed of sound in specific medium in the unit (m/s2). The equation of sound wave can be formulated as, Eq. (12) to Eq. (13) [25]: 2P1v2d2Pdt2=0 {\nabla ^2}P - {1 \over {{v^2}}}{{{d^2}P} \over {d{t^2}}} = 0 P(z,t)=Pocos(k.rωt) {\boldsymbol {P}}\left( {z,t} \right) = {P_o}\cos ({\boldsymbol {k}}{\bf{.}}{\boldsymbol {r}} - \omega t) v=λf v = \lambda f

Where the terms P and v are the acoustic pressure of sound in specific medium in the unit(Pa) and the speed of sound in specific medium in the unit (m/s2), ω is the angular frequencyin the unit(Hz), λ is the wave length in uit (m) and f is the frequencyin the unit (Hz). Although there are several studies [25, 26, 27] on sound absorption coefficient both theoretically and experimentally, the models which show the relationship between thickness, porosity, weight of fabric and frequency of the value of sound absorption coefficient have not been formulated in more detail about these variables. In this study, the sound absorption coefficient equation was examined using the modification of the sound wave equation and using genetic algorithms with computation to get a better model. This new model included the influence of sound absorption coefficient phenomenon caused by porosity, the weight of the fabric, the thickness of fabric as well as the frequency of the sound wave.

Materials and methods
Modeling of Sound Absorption Coefficient on Fabric

Suppose two acoustic media 2 and 1 interfacing through a plane surface and characterized by the surface impendence Z2 and Z1 (Figure 1). The acoustic wave moves from medium 1 to medium 2. The sound wave moves from medium 1 is partly reflected and partly refracted (transmitted) as well as partly absorbed.

Figure 1

An acoustic wave moves from medium 1 to medium 2

If in the case the medium 1 is the air, its surface impedance can be written as Eq. (14) Z=ρoco Z = {\rho _o}{c_o}

Where the terms ρo and co are the density of the ambient air in the unit (kg/m3) and the speed of sound in the ambient air in the unit (m/s2). The continuity equation of sound wave propagation can be formulated as follows Eq. (15) .ρv=ρt \nabla .\rho v = - {{\partial \rho } \over {\partial t}}

Where the terms ρ and v are the density of sound in specific medium in the unit (kg/m3) and the speed of sound in specific medium in the unit (m/s2). With the Cauchy equation, it can be formulated as follows Eq. (16) and Eq. (17) σij,j+fi=ρai {\sigma ^{ij}}_{,j} + {f^i} = \rho {a^i} σ+f¯=ρa¯ \nabla \cdot \mathop {\boldsymbol \sigma} \limits^ \leftrightarrow + \overline {\boldsymbol {f}} = \rho \overline a

Where ρ is the volume mass density, σ \mathop {\sigma} \limits^ \leftrightarrow , is a tensor stress which has a general form σij=δijλe+2μeij=λtr(e)I+2μe {\sigma ^{ij}} = {\delta ^{ij}}\lambda e + 2\mu {e^{ij}} = \lambda tr\left( {\mathop e\limits^ \leftrightarrow } \right){\boldsymbol {I}} + 2\mu \mathop e\limits^ \leftrightarrow

Where the terms σij and eij are the stress tensor and the strain tensor. λ and μ are the elasticity constant. If the magnitude of the gravitational force can be ignored, then by defining that σ=¯(P) \nabla \cdot \mathop {\sigma} \limits^ \leftrightarrow = \overline \nabla ( - P) and the sound wave moves in the z direction, then the wave motion equation can be written as in Eq. (19) and Eq. (20) σ+f¯=ρa¯ \nabla \cdot \mathop \sigma \limits^ \leftrightarrow + \overline {\boldsymbol {f}} = \rho \overline a σ+ρg+f¯ext=ρa \nabla \cdot \mathop \sigma \limits^ \leftrightarrow + \rho {\boldsymbol {g}} + {\overline {\boldsymbol {f}} _{{\boldsymbol {ext}}}} = \rho a

Where the terms ρ and g are the density of sound in specific medium in the unit (kg/m3) and the constant of gravity in the unit (m/s2). If there is no external force ext (can be ignored), then it can be written as in Eq. (21) and Eq. (22) ¯(P)=ρdvdt \overline \nabla \left( { - P} \right) = \rho {{dv} \over {dt}} ¯(P)=ρdvdt \overline \nabla \left( P \right) = - \rho {{dv} \over {dt}}

Divergent of the two segments, we get Eq. (23) to Eq. (25) ¯[¯(P)]=¯[ρdvdt]=[ρd(¯v)dt] \overline \nabla \cdot \left[ {\overline \nabla \left( P \right)} \right] = \overline \nabla \cdot \left[ { - \rho {{dv} \over {dt}}} \right] = \left[ { - \rho {{d\left( {\overline \nabla \cdot v} \right)} \over {dt}}} \right] 2P=ddt(.ρv) {\nabla ^2}P = - {d \over {dt}}\left( {\nabla .\rho v} \right) 2P=ddt(ρt) {\nabla ^2}P = {d \over {dt}}\left( {{{\partial \rho } \over {\partial t}}} \right)

Adiabatic expansion process is a process in which there is no change in Q heat in the system on the environment. In the case of adiabatic can be formulated as follows Eq. (26) and Eq. (27) CdT=PdV CdT = - PdV NcvdT=PdV N{c_v}dT = - PdV

Where C is the total heat capacity, C = Ncv and pressure, P=kBTV P = {{{k_B}T} \over V} [30], Where the terms kB, T and V are Thermodynamic const, temperature in the unit (K) and volume of matter in the unit (m3), hence we get Eq. (28) to Eq. (30) cvdT=kBTdVV {c_v}dT = - {{{k_B}TdV} \over V} dTT=kBcvdVV \int {{dT} \over T}\; = - {{{k_B}} \over {{c_v}}}\int {{dV} \over V} cvkBdTT=dVV {{{c_v}} \over {{k_B}}}\int {{dT} \over T}\; = - \int {{dV} \over V} cvkBln(TTo)=ln(VoV) {{{c_v}} \over {{k_B}}}\ln \left( {{T \over {{T_o}}}} \right) = \ln \left( {{{{V_o}} \over V}} \right)

To simplify the calculations, suppose that the ratio cvkB=32 {{{c_v}} \over {{k_B}}} = {3 \over 2} [30], then it can be described as follows Eq. (31) and Eq. (32) 32ln(TTo)=ln(VoV) {3 \over 2}\ln \left( {{T \over {{T_o}}}} \right) = \ln \left( {{{{V_o}} \over V}} \right) (TTo)3/2=VoV {\left( {{T \over {{T_o}}}} \right)^{3/2}} = {{{V_o}} \over V}

To determine the form of P-V, it can be described as follows Eq. (33) and Eq. (34) (PVPoVo)3/2=VoV {\left( {{{PV} \over {{P_o}{V_o}}}} \right)^{3/2}} = {{{V_o}} \over V} (PPo)3/2=(VoV)1+32 {\left( {{P \over {{P_o}}}} \right)^{3/2}} = {\left( {{{{V_o}} \over V}} \right)^{1 + {3 \over 2}}}

Where P=NkBTV=ρkBT P = {{N{k_B}T} \over V} = \rho {k_B}T , then the relationship between ρ and V is inversely proportional, so we get the following equation (Eq. (35) to Eq. (38)) (PPo)3/2=(ρρo)1+32 {\left( {{P \over {{P_o}}}} \right)^{3/2}} = {\left( {{\rho \over {{\rho _o}}}} \right)^{1 + {3 \over 2}}} ρρo=(PPo)3/5 {\rho \over {{\rho _o}}} = {\left( {{P \over {{P_o}}}} \right)^{3/5}} ρρo=(PPo)γ {\rho \over {{\rho _o}}} = {\left( {{P \over {{P_o}}}} \right)^\gamma } ρ=ρo(PPo)γ \rho = {\rho _o}{\left( {{P \over {{P_o}}}} \right)^\gamma }

Where γ=11+(kBcv)=cvcv+kB \gamma = {1 \over {1 + \left( {{{{k_B}} \over {{c_v}}}} \right)}} = {{{c_v}} \over {{c_v} + {k_B}}} . Suppose that Eq. (38) is differentiated twice with time, then it is obtained Eq. (39) and Eq. (40) 1ρo2ρt2=(1Po)γγ(γ1)Pγ2(Pt)2+(1Po)γγPγ12Pt2 \matrix{ {{1 \over {{\rho _o}}}{{{\partial ^2}\rho } \over {\partial {t^2}}}} \hfill & { = {{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma \left( {\gamma - 1} \right){P^{\gamma - 2}}{{\left( {{{\partial P} \over {\partial t}}} \right)}^2}} \hfill \cr {} \hfill & { + {{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma {P^{\gamma - 1}}{{{\partial ^2}P} \over {\partial {t^2}}}} \hfill \cr } 2ρt2=ρo[(1Po)γγ(γ1)Pγ2(Pt)2+(1Po)γγPγ12Pt2] {{{\partial ^2}\rho } \over {\partial {t^2}}} = {\rho _o}\left[ {{{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma \left( {\gamma - 1} \right){P^{\gamma - 2}}{{\left( {{{\partial P} \over {\partial t}}} \right)}^2}} \right.\left. { + {{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma {P^{\gamma - 1}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right]

Substitute Eq. (40) to Eq. (25) then we get Eq. (41) 2P=ρo[(1Po)γγ(γ1)Pγ2(Pt)2+(1Po)γγPγ12Pt2] {\nabla ^2}P = {\rho _o}\left[ {{{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma \left( {\gamma - 1} \right){P^{\gamma - 2}}{{\left( {{{\partial P} \over {\partial t}}} \right)}^2}} \right.\left. { + {{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma {P^{\gamma - 1}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right]

Suppose that (Pt)2 {\left( {{{\partial P} \over {\partial t}}} \right)^2} is very small (there is no sound absorption), then Eq. (42) to Eq. (44) 2P=ρo[(1Po)γγPγ12Pt2] {\nabla ^2}P = {\rho _o}\left[ {{{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma {P^{\gamma - 1}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right] 2P=ρoPo[(PPo)γ1γ2Pt2] {\nabla ^2}P = {{{\rho _o}} \over {{P_o}}}\left[ {{{\left( {{P \over {{P_o}}}} \right)}^{\gamma - 1}}\gamma {{{\partial ^2}P} \over {\partial {t^2}}}} \right] 2P=ρoPo[(TTo)γ1γ2Pt2] {\nabla ^2}P = {{{\rho _o}} \over {{P_o}}}\left[ {{{\left( {{T \over {{T_o}}}} \right)}^{\gamma - 1}}\gamma {{{\partial ^2}P} \over {\partial {t^2}}}} \right]

Suppose that TTo1 {T \over {{T_o}}} \approx 1 (there is no significant temperature difference), then we get Eq. (45) 2P=ρoPo[γ2Pt2]=1v2d2Pdt2 {\nabla ^2}P = {{{\rho _o}} \over {{P_o}}}\left[ {\gamma {{{\partial ^2}P} \over {\partial {t^2}}}} \right] = {1 \over {{v^2}}}{{{d^2}P} \over {d{t^2}}}

By remembering that P=NkTV=ρkT P = {{NkT} \over V} = \rho kT , hence Pρ=kBT {P \over \rho } = {k_B}T we get Eq. (46) and Eq. (47) 2P=1kBTo[γ2Pt2] {\nabla ^2}P = {1 \over {{k_B}{T_o}}}\left[ {\gamma {{{\partial ^2}P} \over {\partial {t^2}}}} \right] 2PγkBTo2Pt2=2P1v2d2Pdt2=0 {\nabla ^2}P - {\gamma \over {{k_B}{T_o}}}{{{\partial ^2}P} \over {\partial {t^2}}} = {\nabla ^2}P - {1 \over {{v^2}}}{{{d^2}P} \over {d{t^2}}} = 0

Then the sound wave velocity is obtained as in Eq. (48) v=kBToγ=1ρoϵ=Poρoγ v = \sqrt {{{{k_B}{T_o}} \over \gamma }} = {1 \over {\sqrt {{\rho _o}\epsilon } }} = \sqrt {{{{P_o}} \over {{\rho _o}\gamma }}}

With the solution of the wave equation is as follows Eq. (49) to Eq. (52) 1v22P(z,t)t2=2P(z,t) {1 \over {{v^2}}}{{{\partial ^2}P(z,t)} \over {\partial {t^2}}} = {\nabla ^2}P(z,t) 1v21P(t)2P(t)t2=1P(z)2P(z)z2=k2 {1 \over {{v^2}}}{1 \over {P(t)}}{{{\partial ^2}P(t)} \over {\partial {t^2}}} = {1 \over {P(z)}}{{{\partial ^2}P(z)} \over {\partial {z^2}}} = - {k^2} 1v21P(t)2P(t)t2=k2 {1 \over {{v^2}}}{1 \over {P(t)}}{{{\partial ^2}P(t)} \over {\partial {t^2}}} = - {k^2} 1P(t)2P(t)t2=k2v2=ω2 {1 \over {P(t)}}{{{\partial ^2}P(t)} \over {\partial {t^2}}} = - {k^2}{v^2} = - {\omega ^2}

With the wave number can be written as k2=ω2v2 {k^2} = {{{\omega ^2}} \over {{v^2}}}

The k value can be also called the propagation constant (wave propagation) and ω is the angular frequency of the wave with v called the velocity phase of the wave propagation. The sound pressure value P (z, t) can be described as follows Eq. (53) to Eq. (57): 2P(z,t)z2=1v2d2P(z,t)dt2 {{{\partial ^2}P(z,t)} \over {\partial {z^2}}} = {1 \over {{v^2}}}{{{d^2}P(z,t)} \over {d{t^2}}} P(t)2P(z)z2=1v2P(z)d2P(z)dt2 P(t){{{\partial ^2}P\left( z \right)} \over {\partial {z^2}}} = {1 \over {{v^2}}}P(z){{{d^2}P(z)} \over {d{t^2}}} 1P(z)2P(z)z2=1P(t)1v2d2P(z)dt2 {1 \over {P(z)}}{{{\partial ^2}P\left( z \right)} \over {\partial {z^2}}} = {1 \over {P\left( t \right)}}{1 \over {{v^2}}}{{{d^2}P(z)} \over {d{t^2}}} 2P(z)z2=k2P(z) {{{\partial ^2}P(z)} \over {\partial {z^2}}} = - {k^2}P(z) P(z)=Pocos(kz) P\left( z \right) = {P_o}\cos \left( {kz} \right)

Do the same for the function variable t, then after a little mathematical calculation, so we get Eq. (58) 1P(t)2P(t)t2=ω2 {1 \over {P(t)}}{{{\partial ^2}P(t)} \over {\partial {t^2}}} = - {\omega ^2}

Then obtained P (t) = Po cos (−ωt), so we get the general function of the wave equation as follows Eq. (59) and Eq. (60) P(x,t)=Pocos(kzωt). P\left( {x,t} \right) = {P_o}\cos (kz - \omega t). P(z,t)=Pocos(2πλz2πTt)=Pocos(2π(zλtT))=Pocos(2πλ(zλtT))=Pocos(2πλ(zvt)) \matrix{ {P\left( {z,t} \right)} \hfill & { = {P_o}\cos \left( {{{2\pi } \over \lambda }z - {{2\pi } \over T}t} \right)} \hfill \cr {} \hfill & { = {P_o}\cos \left( {2\pi \left( {{z \over \lambda } - {t \over T}} \right)} \right)} \hfill \cr {} \hfill & { = {P_o}\cos \left( {{{2\pi } \over \lambda }\left( {z - {{\lambda t} \over T}} \right)} \right)} \hfill \cr {} \hfill & { = {P_o}\cos \left( {{{2\pi } \over \lambda }\left( {z - vt} \right)} \right)} \hfill \cr }

For the case Pt {{\partial P} \over {\partial t}} is not ignored (there is an absorption of sound waves in medium), then obtained Eq. (61) and Eq. (62) 2P=ρo[(1Po)γγ(γ1)Pγ2(Pt)2+(1Po)γγPγ12Pt2] {\nabla ^2}{P} = {\rho _o}\left[ {{{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma \left( {\gamma - 1} \right){P^{\gamma - 2}}{{\left( {{{\partial P} \over {\partial t}}} \right)}^2}} \right.\left. { + {{\left( {{1 \over {{P_o}}}} \right)}^\gamma }\gamma {P^{\gamma - 1}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right] 2P=ρo[1Po2(PPo)γ2γ(γ1)(Pt)2+(PPo)γ1γPo2Pt2] {\nabla ^2}{P} = {\rho _o}\left[ {{1 \over {{P_o}^2}}{{\left( {{P \over {{P_o}}}} \right)}^{\gamma - 2}}\gamma \left( {\gamma - 1} \right){{\left( {{{\partial P} \over {\partial t}}} \right)}^2}} \right.\left. { + {{\left( {{P \over {{P_o}}}} \right)}^{\gamma - 1}}{\gamma \over {{P_o}}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right]

If the ratio PPo=TTo1 {P \over {{P_o}}} = {T \over {{T_o}}} \approx 1 , we get Eq. (63) and Eq. (64) 2P=[ρoPoγ1Po(γ1)(Pt)2+ρoγPo2Pt2] {\nabla ^2}{P} = \left[ {{{{\rho _o}} \over {{P_o}}}\gamma {1 \over {{P_o}}}\left( {\gamma - 1} \right){{\left( {{{\partial P} \over {\partial t}}} \right)}^2} + {{{\rho _o}\gamma } \over {{P_o}}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right] 2P=[γ(γ1)kBToPo(Pt)2+γkBTo2Pt2][γ(γ1)kBTo1Po(Pt)2+γkBTo2Pt2] \matrix{ {{\nabla ^2}{P}} \hfill & { = \left[ {{{\gamma \left( {\gamma - 1} \right)} \over {{k_B}{T_o}{P_o}}}{{\left( {{{\partial P} \over {\partial t}}} \right)}^2} + {\gamma \over {{k_B}{T_o}}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right]} \hfill \cr {} \hfill & { \approx \left[ {{{\gamma \left( {\gamma - 1} \right)} \over {{k_B}{T_o}}}{1 \over {{P_o}}}{{\left( {{{\partial P} \over {\partial t}}} \right)}^2} + {\gamma \over {{k_B}{T_o}}}{{{\partial ^2}P} \over {\partial {t^2}}}} \right]} \hfill \cr }

For case Poρo=kBTo {{{P_o}} \over {{\rho _o}}} = {k_B}{T_o} and γ(γ1)Po2=σ {{\gamma \left( {\gamma - 1} \right)} \over {{P_o}^2}} = \sigma , we get Eq. (65) to Eq. (69) 2Pρoγ(γ1)Po2(Pt)2ρoγPo2Pt2=0 {\nabla ^2}{\bf P} - {{{\rho _o}\gamma \left( {\gamma - 1} \right)} \over {{P_o}^2}}{\left( {{{\partial {\boldsymbol {P}}} \over {\partial t}}} \right)^2} - {{{\rho _o}\gamma } \over {{P_o}}}{{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}} = 0 2Pρoσ(Pt)2ρoγPo2Pt2=0 {\nabla ^2}{\bf P} - {\rho _o}\sigma {\left( {{{\partial {\boldsymbol {P}}} \over {\partial t}}} \right)^2} - {\rho _o}{\gamma \over {{P_o}}}{{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}} = 0 2Pρoσ(Pt)2ρoϵ2Pt22Pρoσ(1+(Pt1))2ρoϵ2Pt2=2Pρoσ(1+2(Pt1))ρoϵ2Pt2=0 \matrix{ {{\nabla ^2}{\bf P} - {\rho _o}\sigma {{\left( {{{\partial {\boldsymbol {P}}} \over {\partial t}}} \right)}^2} - {\rho _o}\epsilon {{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}}} \hfill \cr { \approx {\nabla ^2}{\bf P} - {\rho _o}\sigma {{\left( {1 + \left( {{{\partial {\boldsymbol {P}}} \over {\partial t}} - 1} \right)} \right)}^2} - {\rho _o}\epsilon {{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}}} \hfill \cr { = {\nabla ^2}{\bf P} - {\rho _o}\sigma \left( {1 + 2\left( {{{\partial {\boldsymbol {P}}} \over {\partial t}} - 1} \right)} \right) - {\rho _o}\epsilon {{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}} = 0} \hfill \cr } 2Pρoσ(1+2(Pt))ρoϵ2Pt2=ρoσ {\nabla ^2}{\bf P} - {\rho _o}\sigma \left( {1 + 2\left( {{{\partial {\boldsymbol {P}}} \over {\partial t}}} \right)} \right) - {\rho _o}\epsilon {{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}} = {\rho _o}\sigma 2P2ρoσPtρoϵ2Pt2=ρoσ {\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial {\boldsymbol {P}}} \over {\partial t}} - {\rho _o}\epsilon {{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}} = {\rho _o}\sigma

For case Po is very highand ρo is very small, then σ=γ(γ1)Po20 \sigma = {{\gamma \left( {\gamma - 1} \right)} \over {{P_o}^2}} \approx 0 with the condition that σ ≤ 0, we get Eq. 70 2P2ρoσPtρoϵ2Pt2=2P2ρoσPt1v22Pt20 \matrix{ {{\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial {\boldsymbol {P}}} \over {\partial t}} - {\rho _o}\epsilon {{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}}} \hfill \cr { = {\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial {\boldsymbol {P}}} \over {\partial t}} - {1 \over {{v^2}}}{{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}}} \hfill \cr { \approx 0} \hfill \cr }

For example the solution is P = Poei(kz−ωt), then substitute to the wave equation 2P2ρoσPtρoϵ2Pt2=0 {\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial {\boldsymbol {P}}} \over {\partial t}} - {\rho _o}\epsilon {{{\partial ^2}{\boldsymbol {P}}} \over {\partial {t^2}}} = 0 , so we get Eq. (71) and Eq. (72) k22iωρoσ+ω2ρoϵ=0 - {k^2} - 2i\omega {\rho _o}\sigma + {\omega ^2}{\rho _o}\epsilon = 0 k2=ω2ρoϵ2iωρoσ {k^2} = {\omega ^2}{\rho _o}\epsilon - 2i\omega {\rho _o}\sigma

Suppose that σ* = 2σ, we get Eq. (73) and Eq. (74) k2=ω2ρoϵ+2iωρoσ=ω2ρoϵ(12iσωϵ)=ω2ρoϵ(1iσ*ωϵ) \matrix{ {{k^2}} \hfill & { = {\omega ^2}{\rho _o}\epsilon + 2i\omega {\rho _o}\sigma = {\omega ^2}{\rho _o}\epsilon \left( {1 - 2{{i\sigma } \over {\omega \epsilon }}} \right)} \hfill \cr {} \hfill & { = {\omega ^2}{\rho _o}\epsilon \left( {1 - {{i{\sigma ^*}} \over {\omega \epsilon }}} \right)} \hfill \cr } where wavenumber can be written as in Eq. (74) k=ωρoϵ(1iσ*ωϵ)1/2 k = \omega \sqrt {{\rho _o}\epsilon } {\left( {1 - {{i{\sigma ^*}} \over {\omega \epsilon }}} \right)^{1/2}}

Suppose that = αiβ=ωρoϵ(1iσ*ωϵ)12 \alpha - i\beta = \omega \sqrt {{\rho _o}\epsilon } {\left( {1 - {{i{\sigma ^*}} \over {\omega \epsilon }}} \right)^{{1 \over 2}}} , we get Eq. (75) k=ωv(1iσ*ωϵ)1/2=ωv(1iσ*2πfϵ)1/2=ωv(1i.0,16σ*fϵ)1/2 \matrix{ k \hfill & { = {\omega \over v}{{\left( {1 - {{i{\sigma ^*}} \over {\omega \epsilon }}} \right)}^{1/2}} = {\omega \over v}{{\left( {1 - {{i{\sigma ^*}} \over {2\pi f\epsilon }}} \right)}^{1/2}}} \hfill \cr {} \hfill & { = {\omega \over v}{{\left( {1 - i.0,16{{{\sigma ^*}} \over {f\epsilon }}} \right)}^{1/2}}} \hfill \cr }

Therefore we get Eq. (76) k2=(αiβ)(αiβ)=α2β22iαβ=ω2ρoϵiωρoσ* \matrix{ {{k^2}} \hfill & { = \left( {\alpha - i\beta } \right)\left( {\alpha - i\beta } \right) = {\alpha ^2} - {\beta ^2} - 2i\alpha \beta } \hfill \cr {} \hfill & { = {\omega ^2}{\rho _o}\epsilon - i\omega {\rho _o}{\sigma ^*}} \hfill \cr } where α2β2 = ω2ρoɛ and −2iαβ = −iωρoσ* where is αβ=ωρoσ*2 \alpha \beta = {{\omega {\rho _o}{\sigma ^*}} \over 2} . The solution of the sound pressure can be obtained as in Eq. (77) P=Poei(kzωt)=Poei((αiβ)zωt)=Poe((iαβ)ziωt)=Poeβzei(αzωt) \matrix{ {\boldsymbol {P}} \hfill & { = {{\boldsymbol {P}}_o}{e^{i(kz - \omega t)}} = {{\boldsymbol {P}}_o}{e^{i(\left( {\alpha - i\beta } \right)z - \omega t)}} = {{\boldsymbol {P}}_o}{e^{(\left( {i\alpha - \beta } \right)z - i\omega t)}}} \hfill \cr {} \hfill & { = {{\boldsymbol {P}}_o}{e^{ - \beta z}}{e^{i(\alpha z - \omega t)}}} \hfill \cr }

For a case α = β, then β=ωρoσ*2 \beta = \sqrt {{{\omega {\rho _o}{\sigma ^*}} \over 2}} β=ωρoσ*2α=vρoσ*2 \beta = {{\omega {\rho _o}{\sigma ^*}} \over {2\alpha }} = {{v{\rho _o}{\sigma ^*}} \over 2}

For the case for α = β, then with further mathematical calculations in Eq. (76), Eq. (78b) is obtained. k=αiβ=ωρoϵ2(1+(1Q)2+1)1/2iωρoϵ2(1+(1Q)21)1/2 \matrix{ k \hfill & { = \alpha - i\beta = \omega \sqrt {{{{\rho _o}\epsilon } \over 2}} {{\left( {\sqrt {1 + {{\left( {{1 \over Q}} \right)}^2}} + 1} \right)}^{1/2}}} \hfill \cr {} \hfill & { - i\omega \sqrt {{{{\rho _o}\epsilon } \over 2}} {{\left( {\sqrt {1 + {{\left( {{1 \over Q}} \right)}^2}} - 1} \right)}^{1/2}}} \hfill \cr } where Q=ωϵσ* Q = {{\omega \epsilon } \over {{\sigma ^*}}} . If it is defined that the thickness of the absorption δ=1β=2vρoσ* \delta = {1 \over \beta } = {2 \over {v{\rho _o}{\sigma ^*}}} which is the thickness when the sound waves disappear, then for ω=2πT \omega = {{2\pi } \over T} and v=λT v = {\lambda \over T} , we get Eq. (79) δ=1β=2vρoσ*=2λTρo(2σ)=2(2π)λωρo(2σ)=2πλωρoσ=1λfρoσ \matrix{ \delta \hfill & { = {1 \over \beta } = {2 \over {v{\rho _o}{\sigma ^*}}} = {2 \over {{\lambda \over T}{\rho _o}\left( {2\sigma } \right)}} = {{2\left( {2\pi } \right)} \over {\lambda \omega {\rho _o}(2\sigma )}} = {{2\pi } \over {\lambda \omega {\rho _o}\sigma }}} \hfill \cr {} \hfill & { = {1 \over {\lambda f{\rho _o}\sigma }}} \hfill \cr } the impedance can be written as Z=ρv=ρωk Z = \rho v = \rho {\omega \over k} . By knowing the wavenumber, k, then the impedance value Z. Large β can be determined related to the energy dissipation (loss of energy) in the sound wave into the form of heat energy. Large impedance values show that in the real part there is an acoustic resistance, while in the imaginary part there is an acoustic reactant or a phase change or energy storage mechanism. The value of reflection and transmission without any absorption event can be described as follows for incident wave pressure, reflection pressure and transmittance pressure as shown in Eq. (80) to Eq. (82) Pi(z,t)=Picos(k1zωt) {{\boldsymbol {P}}_{\boldsymbol {i}}}\left( {z,t} \right) = {P_i}\cos ({k_1}z - \omega t) PR(z,t)=PRcos(k1zωt) {{\boldsymbol {P}}_{\boldsymbol {R}}}\left( {z,t} \right) = {P_R}\cos ( - {k_1}z - \omega t) PT(z,t)=PTcos(k2zωt) {{\boldsymbol {P}}_{\boldsymbol {T}}}\left( {z,t} \right) = {P_T}\cos ({k_2}z - \omega t)

To get the PR and PT values, it can be determined from the boundary conditions, which are as follows, for when z = 0, we get Eq. (83) to Eq. (85) Pi(0,t)+PR(0,t)=PT(0,t) {{\boldsymbol {P}}_{\boldsymbol {i}}}\left( {0,t} \right) + {{\boldsymbol {P}}_{\boldsymbol {R}}}\left( {0,t} \right) = {{\boldsymbol {P}}_{\boldsymbol {T}}}\left( {0,t} \right) Picos(k10ωt)+PRcos(k10ωt)=PTcos(k20ωt) {P_i}\cos ({k_1}0 - \omega t) + {P_R}\cos ( - {k_1}0 - \omega t) = {P_T}\cos ({k_2}0 - \omega t) Pi+PR=PT {P_i} + {P_R} = {P_T} differentiated with z, then Eq. (83) can be written as shown in Eq. (86) to Eq. (88) z(Pi(0,t)+PR(0,t))=PT(0,t)z {\partial \over {\partial z}}\left( {{{\boldsymbol {P}}_{\boldsymbol {i}}}\left( {0,t} \right) + {{\boldsymbol {P}}_{\boldsymbol {R}}}\left( {0,t} \right)} \right) = {{\partial {{\boldsymbol {P}}_{\boldsymbol {T}}}\left( {0,t} \right)} \over {\partial z}} k1Pik1PR=k2PT {k_1}{P_i} - {k_1}{P_R} = {k_2}{P_T} PiPR=k2k1PT {P_i} - {P_R} = {{{k_2}} \over {{k_1}}}{P_T}

Can be eliminated Pi + PR = PT and PiPR=k2k1PT {P_i} - {P_R} = {{{k_2}} \over {{k_1}}}{P_T} , we get Eq. (89) PT=2k1k1+k2Pi {P_T} = {{2{k_1}} \over {{k_1} + {k_2}}}{P_i}

With v=ωk=zρ v = {\omega \over k} = {z \over \rho } , for a medium that has more pivot, it can be assumed that ρ1ρ2, then we get Eq. (90) and Eq. (91) PT=2k1k1+k2Pi {P_T} = {{2{k_1}} \over {{k_1} + {k_2}}}{P_i} PT=2v2v1+v2Pi {P_T} = {{2{v_2}} \over {{v_1} + {v_2}}}{P_i}

Due to Z = ρv, we get Eq. (92) PT=2Z2Z1+Z2Pi {P_T} = {{2{Z_2}} \over {{Z_1} + {Z_2}}}{P_i} do the same calculation to find the value of PR, then Eq. (93) to Eq. (94) are obtained PR=v2v1v2+v1PI {P_R} = {{{v_2} - {v_1}} \over {{v_2} + {v_1}}}{P_I} PR=Z2Z1Z2+Z1PI {P_R} = {{{Z_2} - {Z_1}} \over {{Z_2} + {Z_1}}}{P_I}

It is defined that the wave intensity is as in Eq. (95) [25] I=PpowerA(kgs3) I = {{{P_{power}}} \over A}\left( {{{kg} \over {{s^3}}}} \right) or it can be written as shown in Eq. (96) [25] I=Ppressure2Z(kgs3) I = {{{P_{pressure}}^2} \over Z}\left( {{{kg} \over {{s^3}}}} \right)

It is defined that the reflection coefficient, transmission coefficient and absorption coefficient are as follows as Eq. (96) aR=IRIi=PR2Z1Pi2Z1=PR2Pi2=(Z2Z1Z2+Z1)2=R2 {a_R} = {{{I_R}} \over {{I_i}}} = {{{{{P_R}^2} \over {{Z_1}}}} \over {{{{P_i}^2} \over {{Z_1}}}}} = {{{P_R}^2} \over {{P_i}^2}} = {\left( {{{{Z_2} - {Z_1}} \over {{Z_2} + {Z_1}}}} \right)^2} = {R^2} if Z1 is the impedance in ambient air, then Z1 = ρov, we get Eq. (98) aR=(Z2Z11Z2Z1+1)2=(Z2ρov1Z2ρov+1)2=R2 {a_R} = {\left( {{{{{{Z_2}} \over {{Z_1}}} - 1} \over {{{{Z_2}} \over {{Z_1}}} + 1}}} \right)^2} = {\left( {{{{{{Z_2}} \over {{\rho _o}v}} - 1} \over {{{{Z_2}} \over {{\rho _o}v}} + 1}}} \right)^2} = {R^2}

We get that Z2ρov=1+R1R {{{Z_2}} \over {{\rho _o}v}} = {{1 + R} \over {1 - R}} and also we can found Eq. (99) aT=ITIi=PT2Z2Pi2Z1=PT2Pi2Z1Z2=(2Z2Z1+Z2)2Z1Z2=4Z2Z1(Z1+Z2)2=T2 \matrix{ {{a_T}} \hfill & { = {{{I_T}} \over {{I_i}}} = {{{{{P_T}^2} \over {{Z_2}}}} \over {{{{P_i}^2} \over {{Z_1}}}}} = {{{P_T}^2} \over {{P_i}^2}}{{{Z_1}} \over {{Z_2}}} = {{\left( {{{2{Z_2}} \over {{Z_1} + {Z_2}}}} \right)}^2}{{{Z_1}} \over {{Z_2}}}} \hfill \cr {} \hfill & { = {{4{Z_2}{Z_1}} \over {{{\left( {{Z_1} + {Z_2}} \right)}^2}}} = {T^2}} \hfill \cr }

aT is called the reflection factor or the pressure reflection coefficient. Sound absorption coefficient can be defined as Eq. (100) aTaa=IaIi=PA2Pi2Z1Z2=A2 {a_T}{a_a} = {{{I_a}} \over {{I_i}}} = {{{P_A}^2} \over {{P_i}^2}}{{{Z_1}} \over {{Z_2}}} = {A^2}

In this study, there are two types of equations in the event of the propagation of sound waves in a medium, namely the absorption and the absence of absorption in the medium. in the absence of absorption, it can be written that aR + aT = 1, while in the event of absorption by the medium, it can be written aR + aT + aA = 1, so the sound absorption coefficient can be written as follows aa = 1 − aRaT, then obtained Eq. (101) and Eq. (102) aa+aT=1aR=1IiIR=1Pi2Z1PR2Z1=1PR2Pi2=1(Z2Z1Z2+Z1)2 \matrix{ {{a_a} + {a_T}} \hfill & { = 1 - {a_R} = 1 - {{{I_i}} \over {{I_R}}} = 1 - {{{{{P_i}^2} \over {{Z_1}}}} \over {{{{P_R}^2} \over {{Z_1}}}}} = 1 - {{{P_R}^2} \over {{P_i}^2}}} \hfill \cr {} \hfill & { = 1 - {{\left( {{{{Z_2} - {Z_1}} \over {{Z_2} + {Z_1}}}} \right)}^2}} \hfill \cr } aa+aT=4Z2Z1(Z1+Z2)2=4Z2Z1(Z1+Z2)2 {a_a} + {a_T} = {{4{Z_2}{Z_1}} \over {{{\left( {{Z_1} + {Z_2}} \right)}^2}}} = {{4{Z_2}{Z_1}} \over {{{\left( {{Z_1} + {Z_2}} \right)}^2}}}

With Z = ρv, which is the speed on the material has the relationship as written v=Kρ=Kρ v = \sqrt {{K \over \rho }} = \sqrt {K\rho } with K is the modulus of elasticity in the unit (N/m2) and ρ (kg/m3) is the volume density. The relationship of the parameters can be written as vSolid > vliquid > vgas, then ZSolid > Zliquid > Zgas. if Z2 > Z1 and Z1Z2=n {{{Z_1}} \over {{Z_2}}} = n , then we get Eq. (103) aa+aT=4Z2Z1(Z1+Z2)2=4Z2Z1(Z1+Z2)2=4Z2Z1(Z2(1+Z1Z2))2=4Z2Z1Z12(1+Z1Z2)2=4Z1Z2(1+Z1Z2)2=4n(1+n)2 \matrix{ {{a_a} + {a_T}} \hfill & { = {{4{Z_2}{Z_1}} \over {{{\left( {{Z_1} + {Z_2}} \right)}^2}}} = {{4{Z_2}{Z_1}} \over {{{\left( {{Z_1} + {Z_2}} \right)}^2}}}} \hfill \cr {} \hfill & { = {{4{Z_2}{Z_1}} \over {{{\left( {{Z_2}\left( {1 + {{{Z_1}} \over {{Z_2}}}} \right)} \right)}^2}}} = {{4{Z_2}{Z_1}} \over {{Z_1}^2{{\left( {1 + {{{Z_1}} \over {{Z_2}}}} \right)}^2}}}} \hfill \cr {} \hfill & { = {{4{Z_1}} \over {{Z_2}{{\left( {1 + {{{Z_1}} \over {{Z_2}}}} \right)}^2}}} = {{4n} \over {{{\left( {1 + n} \right)}^2}}}} \hfill \cr }

Suppose that Z1 = 1, then aa depends on Z2, by remembering that Z = ρv, then we get Eq. (104) to Eq. (106) aa+aT=4Z2(1+Z2)2=4v2(1+v2)2 {a_a} + {a_T} = {{4{Z_2}} \over {{{\left( {1 + {Z_2}} \right)}^2}}} = {{4{v_2}} \over {{{\left( {1 + {v_2}} \right)}^2}}} aa+aT=4ω2/k2(1+ω2k2)2=42πf2/k2(1+2πf2k2)2=4f2/Jnm(1+f2/Jnm)2=4m(1+m)2 \matrix{ {{a_a} + {a_T}} \hfill & { = {{4{\omega _2}/{k_2}} \over {{{\left( {1 + {{{\omega _2}} \over {{k_2}}}} \right)}^2}}} = {{42\pi {f_2}/{k_2}} \over {{{\left( {1 + {{2\pi {f_2}} \over {{k_2}}}} \right)}^2}}}} \hfill \cr {} \hfill & { = {{4{f_2}/{J_{nm}}} \over {{{\left( {1 + {f_2}/{J_{nm}}} \right)}^2}}} = {{4m} \over {{{\left( {1 + m} \right)}^2}}}} \hfill \cr } aa=4m(1+m)2aT {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T}

Which requires that the value is m=f2Jnm m = {{{f_2}} \over {{J_{nm}}}} . with Jnm is a constant depending on the material, f2 is the frequency

Results and Discussion

In this study, the value of the sound absorption coefficient was related to a constant variable on woven fabric. The types of woven fabric in this study were commercial woven fabrics, such as plain, rips, twill 2/1 and satin fabrics and we used polyester fabrics (purchased in Bandung, Indonesia). The fabric thickness was measured at a pressure of 5 gr/cm2 using a standard compression tester. The fabric density and porosity were measured with the standard tester (Textile Research Center, Bandung, Indonesia). Porosity is defined as the ratio of the void space within the material to its total displacement volume [29]. In this study, The fabric characteristic was shown in Table 1 and Figure 2. The measurement of the sound absorption coefficient of the samples was conducted using the impedance tube (physics laboratory, Bandung Institute of Technology, Indonesia). The device had absorption frequency ranging from 64 Hz to 6300 Hz. The schematic diagram of the test can be shown in Figure 3

Figure 2

Type of woven fabric used in this study: a) plain; b)satin; c)twill 2/1;d) rips

Figure 3

The schematic diagram of the test (Physics Laboratory, Institut Teknologi Bandung (Bandung Institute of Technology))

Fabric structural properties (physics evaluation laboratory, Politeknik STTT Bandung and Textile Research Center Bandung, Indonesia)

Type of Fabric Fabric Weight (g/m2) Fabric Thickness (mm) Porosity (%)
Plain 160 0.51 0.76
Satin 148 0.59 0.82
Twill 2/1 154 0.53 0.79
Rips 151 0.56 0.81

Based on experimental data for plain fabric, the correlation between sound absorption coefficient and frequency values was obtained as shown in Table 2.

A sound absorption relationship Coefficient of the frequency of plain fabric

aaexp f (Hz)
0.15 210
0.425 500
0.525 710
0.55 1000
0.475 1210
0.425 1500
0.2 2000

For a case Jnm = 1000 and we evaluate f2 in range of 50 Hz to 2000 Hz with aT = 0.4, we get aa. By using Eq. (106), then we get Eq. (107) aa=4m(1+m)2aT=4(f2Jnm)(1+(f2Jnm))2aT {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \right)}^2}}} - {a_T}

Using curve fitting and genetic algorithm, then we get Eq. (108) aa=4m(1+m)2aT=4(f21000)(1+(f21000))20.4 {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {1000}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {1000}}} \right)} \right)}^2}}} - 0.4

The graph results between experiments and models can be shown in Figure 4

Figure 4

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for plain fabric

Based on experimental data for satin, the correlation between sound absorption coefficient and frequency values was obtained as shown in Table 3.

A sound absorption relationship Coefficient of the frequency of Satin fabric

aaexp f (Hz)
0.1 210
0.223 500
0.3 710
0.325 1000
0.223 1210
0.222 1500
0.125 1710

For Jnm = 650 and f2 analyzed for 50 Hz to 2000 HZ and aT = 0.65, it was found that the magnitude of aa is as follows. By using Eq. (106), Eq. (109) is obtained. aa=4m(1+m)2aT=4(f2Jnm)(1+(f2Jnm))2aT {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \right)}^2}}} - {a_T}

By using curve fitting and genetic algorithm, we get Eq. (110) aa=4m(1+m)2aT=4(f2650)(1+(f2650))20.65 {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {650}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {650}}} \right)} \right)}^2}}} - 0.65

The graph results between experiments and models can be shown in Figure 5

Figure 5

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for Satin fabric

Based on experimental data for Rips fabric, the correlation between sound absorption coefficient and frequency values was obtained as shown in Table 4.

A sound absorption relationship Coefficient of the frequency of Rips fabric

aaexp f (Hz)
0.12 210
0.26 500
0.38 710
0.41 1000
0.31 1210
0.25 1500
0.21 1710

For a case Jnm = 730 and we evaluate f2 in range of 50 Hz to 2000 Hz with aT = 0.6, we get aa. By using Eq. (106), then we get Eq. (111) aa=4m(1+m)2aT=4(f2Jnm)(1+(f2Jnm))2aT {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \right)}^2}}} - {a_T}

Using curve fitting and Genetic Algorithm, then we get Eq. (112) aa=4m(1+m)2aT=4(f2730)(1+(f2730))20.6 {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {730}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {730}}} \right)} \right)}^2}}} - 0.6

The graph results between experiments and models can be shown in Figure 6

Figure 6

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for Rips fabric

Based on experimental data for Twill 2/1 fabric, the correlation between sound absorption coefficient and frequency values was obtained as shown in Table 5.

A sound absorption relationship coefficient of the frequency of Twill 2/1 fabric

aaexp f (Hz)
0.14 210
0.38 500
0.48 710
0.45 1000
0.4 1210
0.33 1500
0.22 1710

For a case Jnm = 760 and we evaluate f2 in range of 50 Hz to 2000 Hz with aT = 0.52, we get aa. By using Eq. (106), then we get Eq. (113) aa=4m(1+m)2aT=4(f2Jnm)(1+(f2Jnm))2aT {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \right)}^2}}} - {a_T}

Using curve fitting and Genetic Algorithm, then we get Eq. (114) aa=4m(1+m)2aT=4(f2760)(1+(f2760))20.52 {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {760}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {760}}} \right)} \right)}^2}}} - 0.52

The graph results between experiments and models can be shown in Figure 7

Figure 7

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for twill 2/1 fabric

Based on the calculation results, the plain fabric specifications in the first model follow the following formula Eq. (115) aa=4(f21000)(1+(f21000))20.4 {a_a} = {{4\left( {{{{f_2}} \over {1000}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {1000}}} \right)} \right)}^2}}} - 0.4

The specifications of the satin fabric in the second model follow the following formula Eq. (116) aa=4(f2650)(1+(f2650))20.65 {a_a} = {{4\left( {{{{f_2}} \over {650}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {650}}} \right)} \right)}^2}}} - 0.65

The specifications of the Rips fabric in the second model follow the following formula Eq. (117) aa=4(f2730)(1+(f2730))20.6 {a_a} = {{4\left( {{{{f_2}} \over {730}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {730}}} \right)} \right)}^2}}} - 0.6

The specifications of the Twill2/1 fabric in the second model follow the following formula Eq. (118) aa=4(f2760)(1+(f2760))20.52 {a_a} = {{4\left( {{{{f_2}} \over {760}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {760}}} \right)} \right)}^2}}} - 0.52

In curve fitting using Genetic Algorithm, there are three constants that was connected with fabric parameters, such as weight, thickness and porosity (Table 4).

To determine the relationship between weight, thickness, porosity, Jnm and aT, therefore we used in a simple form using the following Eq. (119) to Eq. (122). The value aT had a relationship that is directly proportional to thickness and porosity, where as Jnm had a relationship that is directly proportional to weight i=1ny^i=ao+a1xi1+a2xi2++akxik \sum\limits_{i = 1}^n {\hat y_i} = {a_o} + {a_1}\sum {x_{i1}} + {a_2}\sum {x_{i2}} + \cdots + {a_k}\sum {x_{ik}} y^1=ao+a1x11+a2x12++a3x13++akx1ky^2=ao+a1x21+a2x22++a3x23++akx2ky^n=ao+a1xn1+a2xn2++a3xn3++akxnk \matrix{ {{{\hat y}_1} = {a_o} + {a_1}{x_{11}} + {a_2}{x_{12}} + + {a_3}{x_{13}} + \cdots + {a_k}{x_{1k}}} \hfill \cr {{{\hat y}_2} = {a_o} + {a_1}{x_{21}} + {a_2}{x_{22}} + + {a_3}{x_{23}} + \cdots + {a_k}{x_{2k}}} \hfill \cr {{{\hat y}_n} = {a_o} + {a_1}{x_{n1}} + {a_2}{x_{n2}} + + {a_3}{x_{n3}} + \cdots + {a_k}{x_{nk}}} \hfill \cr } (y^1:y^n)=(1x1k1:1xnk)(a0:ak) \left( {\matrix{ {{{\hat y}_1}} \cr : \cr {{{\hat y}_n}} \cr } } \right) = \left( {\matrix{ 1 & \cdots & {{x_{1k}}} \cr 1 & \ddots & : \cr 1 & \cdots & {{x_{nk}}} \cr } } \right)\left( {\matrix{ {{a_0}} \cr : \cr {{a_k}} \cr } } \right) y^i=xikak {\hat y_i} = {x_{ik}}{a_k} y^=Xa \hat y = Xa

A parameter of plain fabric and satin fabric

Type of fabric Weight, w, (g/m2) Thickness, d, (mm) ϕ Porosity (%) Jnm aT
Plain 160 0.51 0.77 1000 0.4
Satin 148 0.59 0.82 650 0.65
Twill 2/1 154 0.53 0.79 760 0.52
Rips 151 0.56 0.81 730 0.6

The difference between experimental data and predictive modeling data is referred to as error ɛ and has a value of Eq. (123) i=1n(yiy^i)=ϵ \sum\limits_{i = 1}^n \left( {{y_i} - {{\hat y}_i}} \right) = \epsilon

If Eq. (123) is squared, it will produce Eq. (124) as shown below ϵTϵ=L {\epsilon ^T}\epsilon = L

To find the value of a, optimization can be done through the squared differential Equation (124) with respect to a, hence we get Eq. (125) to Eq. (130) dLda=dda(yTyyTXa(yTXa)T+aTXTXa)=2yTX+2aTXTX=0 \matrix{ {{{dL} \over {da}}} \hfill & { = {d \over {da}}\left( {{y^T}y - {y^T}Xa - {{\left( {{y^T}Xa} \right)}^T} + {a^T}{X^T}Xa} \right)} \hfill \cr {} \hfill & { = - 2{y^T}X + 2{a^T}{X^T}X = 0} \hfill \cr } 2yTX=2aTXTX 2{y^T}X = 2{a^T}{X^T}X aTXTX = yTX with aT = (XTX)−1yTX (aTXTX)T=(yTX)T {\left( {{a^T}{X^T}X} \right)^T} = {\left( {{y^T}X} \right)^T} (XTX)a=XTy \left( {{X^T}X} \right)a = {X^T}y a=(XTX)1XTy a = {\left( {{X^T}X} \right)^{ - 1}}{X^T}y y^=Xa=X(XTX)1XTy \hat y = Xa = X{\left( {{X^T}X} \right)^{ - 1}}{X^T}y

Based on Equation (130) above, with the matrix X, therefore we obtained Eq. (131) to Equation (134) below aT=aoda1ϕa2 {a_T} = {a_o}{d^{{a_1}}}{\phi ^{{a_2}}} lnaT=lnao+a1ln(d)+a1ln(ϕ) \ln {a_T} = \ln {a_o} + {a_1}\ln (d) + {a_1}\ln (\phi ) Y=Ao+A1X1+A2X2 Y = {A_o} + {A_1}{X_1} + {A_2}{X_2} by using MATLAB programming, we get Eq. (133) with ao = eAo and a1 = A1, a2 = A2, we get aT = e0.3778d−11ϕ−42.7. Do the same for Jnm as a function of weight, then we get Jnm = 5.39w aa=4m(1+m)2aT=4(f2Jnm)(1+(f2Jnm))2aT=4(f25.39w)(1+(f25.39w))2(e0.3778d11ϕ42.7) \matrix{ {{a_a}} \hfill & { = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} = {{4\left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{J_{nm}}}}} \right)} \right)}^2}}} - {a_T}} \hfill \cr {} \hfill & { = {{4\left( {{{{f_2}} \over {5.39w}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {5.39w}}} \right)} \right)}^2}}} - \left( {{e^{0.3778}}{d^{ - 11}}{\phi ^{ - 42.7}}} \right)} \hfill \cr }

Then we get the general formula of sound absorption coefficient as a function of weight, thickness and porosity that it can be formulated using the following Eq. (135) aa=4(f2C1w)(1+(f2C1w))2(C2dC3ϕC4) {a_a} = {{4\left( {{{{f_2}} \over {{C_1}w}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{C_1}w}}} \right)} \right)}^2}}} - \left( {{C_2}{d^{ - {C_3}}}{\phi ^{ - {C_4}}}} \right)

aa is the sound absorption coefficient, ϕ (%) is the porosity, d (mm) is the thickness of fabric, f (Hz) is the frequency and w (g/m2) is the weight of fabric. The results showed that factors such as porosity, fabric thickness, fabrication, and frequency affect the value of the sound absorption coefficient, according to Eq. (135). Several studies have shown that in the formation of textile woven fabrics as sound absorbent materials, factors such as porosity, fabric thickness, fabric surface density, and frequency affect the sound absorption coefficient on the fabric [1, 12, 13, 14, 17, 18, 19, 23]. In some studies, the value of the sound absorption coefficient is influenced by several factors, such as the value of the speed of acoustic wave or speed of sound, the acoustic pressure, the angular frequency, the wavenumber, particle speed of the propagation medium which vibrates around the development of sound and also the acoustic impedance [1, 12, 13, 14, 17, 18, 19, 23, 25, 26, 27]. According to Atalla et al. [28] and Maa [4], the acoustic impedance of the sound absorber consists of real parts and imaginary parts (resistance impedance for real parts and reactance impedance for imaginary parts). In this study, several new equation results are found that are related to factors that affect the sound absorption coefficient. In Table 7. A new model equation was compared with some of the previous researchers’ models.

The Comparison between models and literature

The properties of sound Model Literature
Wave equation 2P2ρoσPtρoϵ2Pt20 {\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial P} \over {\partial t}} - {\rho _o}\epsilon {{{\partial ^2}P} \over {\partial {t^2}}} \approx 0 2P2ρoσPt1v22Pt2=0 {\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial P} \over {\partial t}} - {1 \over {{v^2}}}{{{\partial ^2}P} \over {\partial {t^2}}} = 0 2P1v2d2Pdt2=0 {\nabla ^2}{P} - {1 \over {{v^2}}}{{{d^2}P} \over {d{t^2}}} = 0 [25]
Sound pressure P = Poeβzei(αzωt) P (z, t) = Po cos (k.rωt) [25]
Impedance Z=ρv=μωk=μωα+iβ=μωα+iβαiβαiβ=μωαiμωβα2β2=ZrealJZimg Z = \rho v = \mu {\omega \over k} = {{\mu \omega } \over {\alpha + i\beta }} = {{\mu \omega } \over {\alpha + i\beta }}{{\alpha - i\beta } \over {\alpha - i\beta }} = {{\mu \omega \alpha - i\mu \omega \beta } \over {{\alpha ^2} - {\beta ^2}}} = {Z_{real}} - J{Z_{img}} Z = ρoco (1 + 0.0571X−0.754j0.087X−0.732) [25] Z=r+jωm=32ηtρocod2p(1+x232+xd28t)+jωtpco(1+(9+x22)1/2+0.85dt) Z = r + j\omega m = {{32\eta t} \over {{\rho _o}{c_o}{d^2}p}}\left( {\sqrt {1 + {{{x^2}} \over {32}}} + {{xd\sqrt 2 } \over {8t}}} \right) + {{j\omega t} \over {p{c_o}}}\left( {1 + {{\left( {9 + {{{x^2}} \over 2}} \right)}^{ - 1/2}} + {{0.85d} \over t}} \right) [4, 28]
Wavenumber k=αiβ=ωρoϵ2(1+(1Q)2+1)1/2iωρoϵ2(1+(1Q)21)1/2 k = \alpha - i\beta = \omega \sqrt {{{{\rho _o}\epsilon } \over 2}} {\left( {\sqrt {1 + {{\left( {{1 \over Q}} \right)}^2}} + 1} \right)^{1/2}} - i\omega \sqrt {{{{\rho _o}\epsilon } \over 2}} {\left( {\sqrt {1 + {{\left( {{1 \over Q}} \right)}^2}} - 1} \right)^{1/2}} where Q=ωϵσ* Q = {{\omega \epsilon } \over {{\sigma ^*}}} k = ω/co(1 + 0.0978X−0.700j0.189X−0.595) [25]
Sound absorption coefficient aa=1(Z2Z1Z2+Z1)2aT {a_a} = 1 - {\left( {{{{Z_2} - {Z_1}} \over {{Z_2} + {Z_1}}}} \right)^2} - {a_T} aa=4m(1+m)2aT {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} aa=4(f2C1w)(1+(f2C1w))2(C2dC3ϕC4) {a_a} = {{4\left( {{{{f_2}} \over {{C_1}w}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{C_1}w}}} \right)} \right)}^2}}} - \left( {{C_2}{d^{ - {C_3}}}{\phi ^{ - {C_4}}}} \right) aa=4Re{Ztot}(1+Re{Ztot})2+(Img{Ztot})2 {a_a} = {{4Re\{ {Z_{tot}}\} } \over {{{\left( {1 + Re\{ {Z_{tot}}\} } \right)}^2} + {{\left( {Img\{ {Z_{tot}}\} } \right)}^2}}} [26] α=4Re{Ztot}(1+Re{Ztot})2=1(Z2ρ0c0Z2+ρ0c0)2 \alpha = {{4Re\{ {Z_{tot}}\} } \over {{{\left( {1 + Re\{ {Z_{tot}}\} } \right)}^2}}} = 1 - {\left( {{{{Z_2} - {\rho _0}{c_0}} \over {{Z_2} + {\rho _0}{c_0}}}} \right)^2} [27]

From the results of prediction and also the validation of experiments, we obtained data as follows Table 8, Table 9, Table 10 and Table 11.

Relationship between sound absorption values Coefficient, α, on the frequency, f, of plain fabric experiment and model

aexp f (Hz) αmodel
0.15 210 0.229638
0.425 500 0.529071
0.525 710 0.590539
0.55 1000 0.594592
0.475 1210 0.571981
0.425 1500 0.527331
0.2 2000 0.442283

Relationship between sound absorption values Coefficient, α, on the frequency, f, of satin fabric experiment and model

αaexp f (Hz) αmodel
0.1 210 0.294277
0.22 500 0.345752
0.3 710 0.338852
0.32 1000 0.310553
0.22 1210 0.260441
0.21 1500 0.222064
0.13 1710 0.170229

Relationship between sound absorption values Coefficient, α, on the frequency, f, of twill 2/1 fabric experiment and model

αaexp f (Hz) αmodel
0.14 210 0.158499
0.38 500 0.43742
0.48 710 0.478843
0.45 1000 0.461405
0.4 1210 0.427821
0.33 1500 0.372787
0.22 1710 0.332071

Relationship between sound absorption values Coefficient, α, on the frequency, f, of Rips fabric experiments and models

αaexp f (Hz) αmodel
0.12 210 0.093979
0.26 500 0.365034
0.38 710 0.399807
0.4 1000 0.375642
0.31 1210 0.338782
0.25 1500 0.280774
0.21 1710 0.238686

The R2 value obtained was 0.8399 with the relationship between the model and the experimental results shown in Figure 8

Figure 8

The relationship between the model and the experimental results of plain fabric

The R2 value was obtained 0.2815, with the relationship between the model and the experimental results shown in Figure 9

Figure 9

The relationship between the model and the experimental results of satin fabric

The R2 value obtained was 0.909, with the relationship between the model and the experimental results shown in Figure 10

Figure 10

The relationship between the model and the experimental results of twill 2/1 fabric

The R2 value was 0.8303, with the relationship between the model and the experimental results shown in Figure 11

Figure 11

The relationship between the model and the experimental results of rips fabric

In this model, the sound absorption coefficient equation was obtained by modeling the sound wave equation and by curve fitting using genetic algorithms. This model included the influence of the sound absorption coefficient phenomenon caused by porosity, the weight of the fabric, the thickness of fabric as well as the frequency of the sound wave. In this study, experimental data showed a good agreement with the model. In this study, the results of the model and experimental validation show quite good prediction and we had got the general formula of sound absorption coefficient as a function of weight, thickness, and porosity. The results showed that for twill 2/1, rips, and plain fabrics had good accuracy with an R2 value above 0.8, while for satin, the R2 value was 0.2815. The weakness of this model is that the structural equation of the fabric geometry had not been calculated in detail (using topology concepts and mechanical geometric formulations), but this model provided a good enough analysis to predict the sound absorption coefficient with good results compared to previous methods [25, 26, 27]. In this study, we found that the shape of the structure of woven fabric affects the value of sound absorption coefficient, plain woven fabric had a better fabric structure in increasing the value of sound absorption coefficient compared to other types of woven fabric as said by some researchers [25, 26, 27, 28, 29]. The sound absorption coefficient for plain, satin, rips and twill fabrics was 0.525, 0.325, 0.41, and 0.48. We also found that the higher the porosity, the lower is the sound absorption coefficient as reported by some researchers [25, 26, 27, 28, 29] and with the porosity values of plain, satin, rips, and twill 2/1 fabrics were 0.77, 0.82, 0.79 and 0.81 where the sound absorption coefficient of plain fabric is the largest, while the absorption coefficient of satin is the smallest.

Conclusions

We have presented a new method for predicting the sound absorption of woven fabrics using modification of sound wave equations and curve fitting using genetic algorithms. A new model for predicting the sound absorption coefficient of woven fabric (plain, twill 2/1, rips and satin fabric) was presented in this article. In this study, the sound absorption coefficient equation was obtained by modeling the sound wave equation and the application of curve fitting using genetic algorithms. This model included the influence of sound absorption coefficient phenomenon caused by porosity, weight of fabric, thickness of fabric as well as frequency of the sound wave. In this study, the results of the model and experimental validation show quite good prediction and we had got the general formula of sound absorption coefficient as a function of weight, thickness, and porosity

Figure 1

An acoustic wave moves from medium 1 to medium 2
An acoustic wave moves from medium 1 to medium 2

Figure 2

Type of woven fabric used in this study: a) plain; b)satin; c)twill 2/1;d) rips
Type of woven fabric used in this study: a) plain; b)satin; c)twill 2/1;d) rips

Figure 3

The schematic diagram of the test (Physics Laboratory, Institut Teknologi Bandung (Bandung Institute of Technology))
The schematic diagram of the test (Physics Laboratory, Institut Teknologi Bandung (Bandung Institute of Technology))

Figure 4

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for plain fabric
Graphical results between the experiment and the model of the sound absorption coefficient, aa, for plain fabric

Figure 5

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for Satin fabric
Graphical results between the experiment and the model of the sound absorption coefficient, aa, for Satin fabric

Figure 6

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for Rips fabric
Graphical results between the experiment and the model of the sound absorption coefficient, aa, for Rips fabric

Figure 7

Graphical results between the experiment and the model of the sound absorption coefficient, aa, for twill 2/1 fabric
Graphical results between the experiment and the model of the sound absorption coefficient, aa, for twill 2/1 fabric

Figure 8

The relationship between the model and the experimental results of plain fabric
The relationship between the model and the experimental results of plain fabric

Figure 9

The relationship between the model and the experimental results of satin fabric
The relationship between the model and the experimental results of satin fabric

Figure 10

The relationship between the model and the experimental results of twill 2/1 fabric
The relationship between the model and the experimental results of twill 2/1 fabric

Figure 11

The relationship between the model and the experimental results of rips fabric
The relationship between the model and the experimental results of rips fabric

A sound absorption relationship Coefficient of the frequency of Satin fabric

aaexp f (Hz)
0.1 210
0.223 500
0.3 710
0.325 1000
0.223 1210
0.222 1500
0.125 1710

The Comparison between models and literature

The properties of sound Model Literature
Wave equation 2P2ρoσPtρoϵ2Pt20 {\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial P} \over {\partial t}} - {\rho _o}\epsilon {{{\partial ^2}P} \over {\partial {t^2}}} \approx 0 2P2ρoσPt1v22Pt2=0 {\nabla ^2}{\bf P} - 2{\rho _o}\sigma {{\partial P} \over {\partial t}} - {1 \over {{v^2}}}{{{\partial ^2}P} \over {\partial {t^2}}} = 0 2P1v2d2Pdt2=0 {\nabla ^2}{P} - {1 \over {{v^2}}}{{{d^2}P} \over {d{t^2}}} = 0 [25]
Sound pressure P = Poeβzei(αzωt) P (z, t) = Po cos (k.rωt) [25]
Impedance Z=ρv=μωk=μωα+iβ=μωα+iβαiβαiβ=μωαiμωβα2β2=ZrealJZimg Z = \rho v = \mu {\omega \over k} = {{\mu \omega } \over {\alpha + i\beta }} = {{\mu \omega } \over {\alpha + i\beta }}{{\alpha - i\beta } \over {\alpha - i\beta }} = {{\mu \omega \alpha - i\mu \omega \beta } \over {{\alpha ^2} - {\beta ^2}}} = {Z_{real}} - J{Z_{img}} Z = ρoco (1 + 0.0571X−0.754j0.087X−0.732) [25] Z=r+jωm=32ηtρocod2p(1+x232+xd28t)+jωtpco(1+(9+x22)1/2+0.85dt) Z = r + j\omega m = {{32\eta t} \over {{\rho _o}{c_o}{d^2}p}}\left( {\sqrt {1 + {{{x^2}} \over {32}}} + {{xd\sqrt 2 } \over {8t}}} \right) + {{j\omega t} \over {p{c_o}}}\left( {1 + {{\left( {9 + {{{x^2}} \over 2}} \right)}^{ - 1/2}} + {{0.85d} \over t}} \right) [4, 28]
Wavenumber k=αiβ=ωρoϵ2(1+(1Q)2+1)1/2iωρoϵ2(1+(1Q)21)1/2 k = \alpha - i\beta = \omega \sqrt {{{{\rho _o}\epsilon } \over 2}} {\left( {\sqrt {1 + {{\left( {{1 \over Q}} \right)}^2}} + 1} \right)^{1/2}} - i\omega \sqrt {{{{\rho _o}\epsilon } \over 2}} {\left( {\sqrt {1 + {{\left( {{1 \over Q}} \right)}^2}} - 1} \right)^{1/2}} where Q=ωϵσ* Q = {{\omega \epsilon } \over {{\sigma ^*}}} k = ω/co(1 + 0.0978X−0.700j0.189X−0.595) [25]
Sound absorption coefficient aa=1(Z2Z1Z2+Z1)2aT {a_a} = 1 - {\left( {{{{Z_2} - {Z_1}} \over {{Z_2} + {Z_1}}}} \right)^2} - {a_T} aa=4m(1+m)2aT {a_a} = {{4m} \over {{{\left( {1 + m} \right)}^2}}} - {a_T} aa=4(f2C1w)(1+(f2C1w))2(C2dC3ϕC4) {a_a} = {{4\left( {{{{f_2}} \over {{C_1}w}}} \right)} \over {{{\left( {1 + \left( {{{{f_2}} \over {{C_1}w}}} \right)} \right)}^2}}} - \left( {{C_2}{d^{ - {C_3}}}{\phi ^{ - {C_4}}}} \right) aa=4Re{Ztot}(1+Re{Ztot})2+(Img{Ztot})2 {a_a} = {{4Re\{ {Z_{tot}}\} } \over {{{\left( {1 + Re\{ {Z_{tot}}\} } \right)}^2} + {{\left( {Img\{ {Z_{tot}}\} } \right)}^2}}} [26] α=4Re{Ztot}(1+Re{Ztot})2=1(Z2ρ0c0Z2+ρ0c0)2 \alpha = {{4Re\{ {Z_{tot}}\} } \over {{{\left( {1 + Re\{ {Z_{tot}}\} } \right)}^2}}} = 1 - {\left( {{{{Z_2} - {\rho _0}{c_0}} \over {{Z_2} + {\rho _0}{c_0}}}} \right)^2} [27]

A sound absorption relationship Coefficient of the frequency of plain fabric

aaexp f (Hz)
0.15 210
0.425 500
0.525 710
0.55 1000
0.475 1210
0.425 1500
0.2 2000

A sound absorption relationship Coefficient of the frequency of Rips fabric

aaexp f (Hz)
0.12 210
0.26 500
0.38 710
0.41 1000
0.31 1210
0.25 1500
0.21 1710

Relationship between sound absorption values Coefficient, α, on the frequency, f, of twill 2/1 fabric experiment and model

αaexp f (Hz) αmodel
0.14 210 0.158499
0.38 500 0.43742
0.48 710 0.478843
0.45 1000 0.461405
0.4 1210 0.427821
0.33 1500 0.372787
0.22 1710 0.332071

Relationship between sound absorption values Coefficient, α, on the frequency, f, of satin fabric experiment and model

αaexp f (Hz) αmodel
0.1 210 0.294277
0.22 500 0.345752
0.3 710 0.338852
0.32 1000 0.310553
0.22 1210 0.260441
0.21 1500 0.222064
0.13 1710 0.170229

A sound absorption relationship coefficient of the frequency of Twill 2/1 fabric

aaexp f (Hz)
0.14 210
0.38 500
0.48 710
0.45 1000
0.4 1210
0.33 1500
0.22 1710

Fabric structural properties (physics evaluation laboratory, Politeknik STTT Bandung and Textile Research Center Bandung, Indonesia)

Type of Fabric Fabric Weight (g/m2) Fabric Thickness (mm) Porosity (%)
Plain 160 0.51 0.76
Satin 148 0.59 0.82
Twill 2/1 154 0.53 0.79
Rips 151 0.56 0.81

Relationship between sound absorption values Coefficient, α, on the frequency, f, of plain fabric experiment and model

aexp f (Hz) αmodel
0.15 210 0.229638
0.425 500 0.529071
0.525 710 0.590539
0.55 1000 0.594592
0.475 1210 0.571981
0.425 1500 0.527331
0.2 2000 0.442283

Relationship between sound absorption values Coefficient, α, on the frequency, f, of Rips fabric experiments and models

αaexp f (Hz) αmodel
0.12 210 0.093979
0.26 500 0.365034
0.38 710 0.399807
0.4 1000 0.375642
0.31 1210 0.338782
0.25 1500 0.280774
0.21 1710 0.238686

A parameter of plain fabric and satin fabric

Type of fabric Weight, w, (g/m2) Thickness, d, (mm) ϕ Porosity (%) Jnm aT
Plain 160 0.51 0.77 1000 0.4
Satin 148 0.59 0.82 650 0.65
Twill 2/1 154 0.53 0.79 760 0.52
Rips 151 0.56 0.81 730 0.6

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