A Novel Theoretical Modeling for Predicting the Sound Absorption of Woven Fabrics Using Modification of Sound Wave Equation and Genetic Algorithm

1 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)

Where the term a_a 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)

Where the term a_a is the sound absorption coefficient; Z_tot, 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)

Where the term α and Z₂ are the sound absorption coefficient and the acoustic impedance; Z_tot, the total acoustic surface impedance on the media in the unit (kg/m²s or rayl); ρ_o, the density of air; c_o, 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):

Where the terms r and m are the acoustic resistance and the acoustic reactance, ρ_o is the density of air, c_o 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):

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):

Where the term σ is flow resistivityin the unit (kg/m²s or rayl); ρ_o is the density of air, c_o 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)

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]:

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.

2 Materials and methods

2.1 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)

(14) Z=ρoco

Where the terms ρ_o and c_o 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)

(15) ∇.ρv=−∂ρ∂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)

(16) σij,j+fi=ρai

(17) ∇⋅σ↔+f¯=ρa¯

Where ρ is the volume mass density, σ↔ , is a tensor stress which has a general form

(18) σij=δijλe+2μeij=λtr(e↔)I+2μe↔

Where the terms σ^ij and e^ij 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) and the sound wave moves in the z direction, then the wave motion equation can be written as in Eq. (19) and Eq. (20)

(19) ∇⋅σ↔+f¯=ρa¯

(20) ∇⋅σ↔+ρg+f¯ext=ρa

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 f̅_ext (can be ignored), then it can be written as in Eq. (21) and Eq. (22)

(21) ∇¯(−P)=ρdvdt

(22) ∇¯(P)=−ρdvdt

Divergent of the two segments, we get Eq. (23) to Eq. (25)

(23) ∇¯⋅[∇¯(P)]=∇¯⋅[−ρdvdt]=[−ρd(∇¯⋅v)dt]

(24) ∇2P=−ddt(∇.ρv)

(25) ∇2P=ddt(∂ρ∂t)

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)

(26) CdT=−PdV

(27) NcvdT=−PdV

Where C is the total heat capacity, C = Nc_v and pressure, P=kBTV [30], Where the terms k_B, 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)

(28) cvdT=−kBTdVV

(29) ∫dTT =−kBcv∫dVV

(30a) cvkB∫dTT =−∫dVV

(30b) cvkBln(TTo)=ln(VoV)

To simplify the calculations, suppose that the ratio cvkB=32 [30], then it can be described as follows Eq. (31) and Eq. (32)

(31) 32ln(TTo)=ln(VoV)

(32) (TTo)3/2=VoV

To determine the form of P-V, it can be described as follows Eq. (33) and Eq. (34)

(33) (PVPoVo)3/2=VoV

(34) (PPo)3/2=(VoV)1+32

Where P=NkBTV=ρkBT , then the relationship between ρ and V is inversely proportional, so we get the following equation (Eq. (35) to Eq. (38))

(35) (PPo)3/2=(ρρo)1+32

(36) ρρo=(PPo)3/5

(37) ρρo=(PPo)γ

(38) ρ=ρo(PPo)γ

Where γ=11+(kBcv)=cvcv+kB . Suppose that Eq. (38) is differentiated twice with time, then it is obtained Eq. (39) and Eq. (40)

(39) 1ρo∂2ρ∂t2=(1Po)γγ(γ−1)Pγ−2(∂P∂t)2+(1Po)γγPγ−1∂2P∂t2

(40) ∂2ρ∂t2=ρo[(1Po)γγ(γ−1)Pγ−2(∂P∂t)2+(1Po)γγPγ−1∂2P∂t2]

Substitute Eq. (40) to Eq. (25) then we get Eq. (41)

(41) ∇2P=ρo[(1Po)γγ(γ−1)Pγ−2(∂P∂t)2+(1Po)γγPγ−1∂2P∂t2]

Suppose that (∂P∂t)2 is very small (there is no sound absorption), then Eq. (42) to Eq. (44)

(42) ∇2P=ρo[(1Po)γγPγ−1∂2P∂t2]

(43) ∇2P=ρoPo[(PPo)γ−1γ∂2P∂t2]

(44) ∇2P=ρoPo[(TTo)γ−1γ∂2P∂t2]

Suppose that TTo≈1 (there is no significant temperature difference), then we get Eq. (45)

(45) ∇2P=ρoPo[γ∂2P∂t2]=1v2d2Pdt2

By remembering that P=NkTV=ρkT , hence Pρ=kBT we get Eq. (46) and Eq. (47)

(46) ∇2P=1kBTo[γ∂2P∂t2]

(47) ∇2P−γkBTo∂2P∂t2=∇2P−1v2d2Pdt2=0

Then the sound wave velocity is obtained as in Eq. (48)

(48) v=kBToγ=1ρoϵ=Poρoγ

With the solution of the wave equation is as follows Eq. (49) to Eq. (52)

(49) 1v2∂2P(z,t)∂t2=∇2P(z,t)

(50) 1v21P(t)∂2P(t)∂t2=1P(z)∂2P(z)∂z2=−k2

(51) 1v21P(t)∂2P(t)∂t2=−k2

(52a) 1P(t)∂2P(t)∂t2=−k2v2=−ω2

With the wave number can be written as

(52b) k2=ω2v2

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):

(53) ∂2P(z,t)∂z2=1v2d2P(z,t)dt2

(54) P(t)∂2P(z)∂z2=1v2P(z)d2P(z)dt2

(55) 1P(z)∂2P(z)∂z2=1P(t)1v2d2P(z)dt2

(56) ∂2P(z)∂z2=−k2P(z)

(57) P(z)=Pocos(kz)

Do the same for the function variable t, then after a little mathematical calculation, so we get Eq. (58)

(58) 1P(t)∂2P(t)∂t2=−ω2

Then obtained P (t) = P_o cos (−ωt), so we get the general function of the wave equation as follows Eq. (59) and Eq. (60)

(59) P(x,t)=Pocos(kz−ωt).

(60) P(z,t)=Pocos(2πλz−2πTt)=Pocos(2π(zλ−tT))=Pocos(2πλ(z−λtT))=Pocos(2πλ(z−vt))

For the case ∂P∂t is not ignored (there is an absorption of sound waves in medium), then obtained Eq. (61) and Eq. (62)

(61) ∇2P=ρo[(1Po)γγ(γ−1)Pγ−2(∂P∂t)2+(1Po)γγPγ−1∂2P∂t2]

(62) ∇2P=ρo[1Po2(PPo)γ−2γ(γ−1)(∂P∂t)2+(PPo)γ−1γPo∂2P∂t2]

If the ratio PPo=TTo≈1 , we get Eq. (63) and Eq. (64)

(63) ∇2P=[ρoPoγ1Po(γ−1)(∂P∂t)2+ρoγPo∂2P∂t2]

(64) ∇2P=[γ(γ−1)kBToPo(∂P∂t)2+γkBTo∂2P∂t2]≈[γ(γ−1)kBTo1Po(∂P∂t)2+γkBTo∂2P∂t2]

For case Poρo=kBTo and γ(γ−1)Po2=σ , we get Eq. (65) to Eq. (69)

(65) ∇2P−ρoγ(γ−1)Po2(∂P∂t)2−ρoγPo∂2P∂t2=0

(66) ∇2P−ρoσ(∂P∂t)2−ρoγPo∂2P∂t2=0

(67) ∇2P−ρoσ(∂P∂t)2−ρoϵ∂2P∂t2≈∇2P−ρoσ(1+(∂P∂t−1))2−ρoϵ∂2P∂t2=∇2P−ρoσ(1+2(∂P∂t−1))−ρoϵ∂2P∂t2=0

(68) ∇2P−ρoσ(1+2(∂P∂t))−ρoϵ∂2P∂t2=ρoσ

(69) ∇2P−2ρoσ∂P∂t−ρoϵ∂2P∂t2=ρoσ

For case P_o is very highand ρ_o is very small, then σ=γ(γ−1)Po2≈0 with the condition that σ ≤ 0, we get Eq. 70

(70) ∇2P−2ρoσ∂P∂t−ρoϵ∂2P∂t2=∇2P−2ρoσ∂P∂t−1v2∂2P∂t2≈0

For example the solution is P = P_oe^i(kz−ωt), then substitute to the wave equation ∇2P−2ρoσ∂P∂t−ρoϵ∂2P∂t2=0 , so we get Eq. (71) and Eq. (72)

(71) −k2−2iωρoσ+ω2ρoϵ=0

(72) k2=ω2ρoϵ−2iωρoσ

Suppose that σ^* = 2σ, we get Eq. (73) and Eq. (74)

(73) k2=ω2ρoϵ+2iωρoσ=ω2ρoϵ(1−2iσωϵ)=ω2ρoϵ(1−iσ*ωϵ)

where wavenumber can be written as in Eq. (74)

(74) k=ωρoϵ(1−iσ*ωϵ)1/2

Suppose that = α−iβ=ωρoϵ(1−iσ*ωϵ)12 , we get Eq. (75)

(75) k=ωv(1−iσ*ωϵ)1/2=ωv(1−iσ*2πfϵ)1/2=ωv(1−i.0,16σ*fϵ)1/2

Therefore we get Eq. (76)

(76) k2=(α−iβ)(α−iβ)=α2−β2−2iαβ=ω2ρoϵ−iωρoσ*

where α² − β² = ω²ρ_oɛ and −2iαβ = −iωρ_oσ^* where is αβ=ωρoσ*2 . The solution of the sound pressure can be obtained as in Eq. (77)

(77) P=Poei(kz−ωt)=Poei((α−iβ)z−ωt)=Poe((iα−β)z−iωt)=Poe−βzei(αz−ωt)

For a case α = β, then β=ωρoσ*2

(78a) β=ωρoσ*2α=vρoσ*2

For the case for α = β, then with further mathematical calculations in Eq. (76), Eq. (78b) is obtained.

(78b) k=α−iβ=ωρoϵ2(1+(1Q)2+1)1/2−iωρoϵ2(1+(1Q)2−1)1/2

where Q=ωϵσ* . If it is defined that the thickness of the absorption δ=1β=2vρoσ* which is the thickness when the sound waves disappear, then for ω=2πT and v=λT , we get Eq. (79)

(79) δ=1β=2vρoσ*=2λTρo(2σ)=2(2π)λωρo(2σ)=2πλωρoσ=1λfρoσ

the impedance can be written as Z=ρv=ρωk . 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)

(80) Pi(z,t)=Picos(k1z−ωt)

(81) PR(z,t)=PRcos(−k1z−ωt)

(82) PT(z,t)=PTcos(k2z−ωt)

To get the P_R and P_T values, it can be determined from the boundary conditions, which are as follows, for when z = 0, we get Eq. (83) to Eq. (85)

(83) Pi(0,t)+PR(0,t)=PT(0,t)

(84) Picos(k10−ωt)+PRcos(−k10−ωt)=PTcos(k20−ωt)

(85) Pi+PR=PT

differentiated with z, then Eq. (83) can be written as shown in Eq. (86) to Eq. (88)

(86) ∂∂z(Pi(0,t)+PR(0,t))=∂PT(0,t)∂z

(87) k1Pi−k1PR=k2PT

(88) Pi−PR=k2k1PT

Can be eliminated P_i + P_R = P_T and Pi−PR=k2k1PT , we get Eq. (89)

(89) PT=2k1k1+k2Pi

With v=ωk=zρ , for a medium that has more pivot, it can be assumed that ρ₁ ≈ ρ₂, then we get Eq. (90) and Eq. (91)

(90) PT=2k1k1+k2Pi

(91) PT=2v2v1+v2Pi

Due to Z = ρv, we get Eq. (92)

(92) PT=2Z2Z1+Z2Pi

do the same calculation to find the value of P_R, then Eq. (93) to Eq. (94) are obtained

(93) PR=v2−v1v2+v1PI

(94) PR=Z2−Z1Z2+Z1PI

It is defined that the wave intensity is as in Eq. (95) [25]

(95) I=PpowerA(kgs3)

or it can be written as shown in Eq. (96) [25]

(96) I=Ppressure2Z(kgs3)

It is defined that the reflection coefficient, transmission coefficient and absorption coefficient are as follows as Eq. (96)

(97) aR=IRIi=PR2Z1Pi2Z1=PR2Pi2=(Z2−Z1Z2+Z1)2=R2

if Z₁ is the impedance in ambient air, then Z₁ = ρ_ov, we get Eq. (98)

(98) aR=(Z2Z1−1Z2Z1+1)2=(Z2ρov−1Z2ρov+1)2=R2

We get that Z2ρov=1+R1−R and also we can found Eq. (99)

(99) aT=ITIi=PT2Z2Pi2Z1=PT2Pi2Z1Z2=(2Z2Z1+Z2)2Z1Z2=4Z2Z1(Z1+Z2)2=T2

a_T is called the reflection factor or the pressure reflection coefficient. Sound absorption coefficient can be defined as Eq. (100)

(100) aTaa=IaIi=PA2Pi2Z1Z2=A2

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 a_R + a_T = 1, while in the event of absorption by the medium, it can be written a_R + a_T + a_A = 1, so the sound absorption coefficient can be written as follows a_a = 1 − a_R − a_T, then obtained Eq. (101) and Eq. (102)

(101) aa+aT=1−aR=1−IiIR=1−Pi2Z1PR2Z1=1−PR2Pi2=1−(Z2−Z1Z2+Z1)2

(102) aa+aT=4Z2Z1(Z1+Z2)2=4Z2Z1(Z1+Z2)2

With Z = ρv, which is the speed on the material has the relationship as written v=Kρ=Kρ 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 v_Solid > v_liquid > v_gas, then Z_Solid > Z_liquid > Z_gas. if Z₂ > Z₁ and Z1Z2=n , then we get Eq. (103)

(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

Suppose that Z₁ = 1, then a_a depends on Z₂, by remembering that Z = ρv, then we get Eq. (104) to Eq. (106)

(104) aa+aT=4Z2(1+Z2)2=4v2(1+v2)2

(105) aa+aT=4ω2/k2(1+ω2k2)2=42πf2/k2(1+2πf2k2)2=4f2/Jnm(1+f2/Jnm)2=4m(1+m)2

(106) aa=4m(1+m)2−aT

Which requires that the value is m=f2Jnm . with J_nm is a constant depending on the material, f₂ is the frequency

3 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))

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

For a case J_nm = 1000 and we evaluate f₂ in range of 50 Hz to 2000 Hz with a_T = 0.4, we get a_a. By using Eq. (106), then we get Eq. (107)

Using curve fitting and genetic algorithm, then we get Eq. (108)

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, a_a, 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.

For J_nm = 650 and f₂ analyzed for 50 Hz to 2000 HZ and a_T = 0.65, it was found that the magnitude of a_a is as follows. By using Eq. (106), Eq. (109) is obtained.

By using curve fitting and genetic algorithm, we get Eq. (110)

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, a_a, 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.

For a case J_nm = 730 and we evaluate f₂ in range of 50 Hz to 2000 Hz with a_T = 0.6, we get a_a. By using Eq. (106), then we get Eq. (111)

Using curve fitting and Genetic Algorithm, then we get Eq. (112)

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, a_a, 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.

For a case J_nm = 760 and we evaluate f₂ in range of 50 Hz to 2000 Hz with a_T = 0.52, we get a_a. By using Eq. (106), then we get Eq. (113)

Using curve fitting and Genetic Algorithm, then we get Eq. (114)

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, a_a, for twill 2/1 fabric

Based on the calculation results, the plain fabric specifications in the first model follow the following formula Eq. (115)

The specifications of the satin fabric in the second model follow the following formula Eq. (116)

The specifications of the Rips fabric in the second model follow the following formula Eq. (117)

The specifications of the Twill2/1 fabric in the second model follow the following formula Eq. (118)

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, J_nm and a_T, therefore we used in a simple form using the following Eq. (119) to Eq. (122). The value a_T had a relationship that is directly proportional to thickness and porosity, where as J_nm had a relationship that is directly proportional to weight

The difference between experimental data and predictive modeling data is referred to as error ɛ and has a value of Eq. (123)