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
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 soundabsorbing materials. Researchers examined the application of woven fabrics as soundabsorbent 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 Rosenhouse [1]. Several studies have been conducted to utilize woven fabrics as soundabsorbent 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 microholes (around 1 mm) where the textile fabric can be assumed as the microperforated panel (MPP) which has quite a lot of porosity. Several studies have shown that in the formation of textile woven fabrics as soundabsorbent 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 indepth 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; Z_{2}, the acoustic surface impedance on the media 2 in the unit (kg/m^{2}s or rayl); Z_{1}, the acoustic surface impedance on the media 1 in the unit (kg/m^{2}s 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/m^{2}s 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 Z_{2} are the sound absorption coefficient and the acoustic impedance; Z_{tot}, the total acoustic surface impedance on the media in the unit (kg/m^{2}s 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/m^{2}s 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/m^{3}) and the speed of sound in specific medium in the unit (m/s^{2}). 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/s^{2}), 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 Z_{2} and Z_{1} (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.
If in the case the medium 1 is the air, its surface impedance can be written as Eq. (14)
Where the terms ρ_{o} and c_{o} are the density of the ambient air in the unit (kg/m^{3}) and the speed of sound in the ambient air in the unit (m/s^{2}). The continuity equation of sound wave propagation can be formulated as follows Eq. (15)
Where the terms ρ and v are the density of sound in specific medium in the unit (kg/m^{3}) and the speed of sound in specific medium in the unit (m/s^{2}). With the Cauchy equation, it can be formulated as follows Eq. (16) and Eq. (17)
Where ρ is the volume mass density,
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
Where the terms ρ and g are the density of sound in specific medium in the unit (kg/m^{3}) and the constant of gravity in the unit (m/s^{2}). If there is no external force f̅_{ext} (can be ignored), then it can be written as in Eq. (21) and Eq. (22)
Divergent of the two segments, we get Eq. (23) to Eq. (25)
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)
Where C is the total heat capacity, C = Nc_{v} and pressure,
To simplify the calculations, suppose that the ratio
To determine the form of PV, it can be described as follows Eq. (33) and Eq. (34)
Where
Where
Substitute Eq. (40) to Eq. (25) then we get Eq. (41)
Suppose that
Suppose that
By remembering that
Then the sound wave velocity is obtained as in Eq. (48)
With the solution of the wave equation is as follows Eq. (49) to Eq. (52)
With the wave number can be written as
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):
Do the same for the function variable t, then after a little mathematical calculation, so we get Eq. (58)
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)
For the case
If the ratio
For case
For case P_{o} is very highand ρ_{o} is very small, then
For example the solution is P = P_{o}e^{i}^{(kz−ωt)}, then substitute to the wave equation
Suppose that σ^{*} = 2σ, we get Eq. (73) and Eq. (74)
where wavenumber can be written as in Eq. (74)
Suppose that =
Therefore we get Eq. (76)
where α^{2} − β^{2} = ω^{2}ρ_{o}ɛ and −2iαβ = −iωρ_{o}σ^{*} where is
For a case α = β, then
For the case for α = β, then with further mathematical calculations in Eq. (76), Eq. (78b) is obtained.
where
the impedance can be written as
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)
differentiated with z, then Eq. (83) can be written as shown in Eq. (86) to Eq. (88)
Can be eliminated P_{i} + P_{R} = P_{T} and
With
Due to Z = ρv, we get Eq. (92)
do the same calculation to find the value of P_{R}, then Eq. (93) to Eq. (94) are obtained
It is defined that the wave intensity is as in Eq. (95) [25]
or it can be written as shown in Eq. (96) [25]
It is defined that the reflection coefficient, transmission coefficient and absorption coefficient are as follows as Eq. (96)
if Z_{1} is the impedance in ambient air, then Z_{1} = ρ_{o}v, we get Eq. (98)
We get that
a_{T} is called the reflection factor or the pressure reflection coefficient. Sound absorption coefficient can be defined as Eq. (100)
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)
With Z = ρv, which is the speed on the material has the relationship as written
Suppose that Z_{1} = 1, then a_{a} depends on Z_{2}, by remembering that Z = ρv, then we get Eq. (104) to Eq. (106)
Which requires that the value is
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/cm^{2} 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
Type of Fabric  Fabric Weight (g/m^{2})  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_{aexp}  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 J_{nm} = 1000 and we evaluate f_{2} 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
Based on experimental data for satin, the correlation between sound absorption coefficient and frequency values was obtained as shown in Table 3.
a_{aexp}  f (Hz) 

0.1  210 
0.223  500 
0.3  710 
0.325  1000 
0.223  1210 
0.222  1500 
0.125  1710 
For J_{nm} = 650 and f_{2} 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
Based on experimental data for Rips fabric, the correlation between sound absorption coefficient and frequency values was obtained as shown in Table 4.
a_{aexp}  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 J_{nm} = 730 and we evaluate f_{2} 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
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_{aexp}  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 J_{nm} = 760 and we evaluate f_{2} 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
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
Type of fabric  Weight, w, (g/m^{2})  Thickness, d, (mm)  ϕ Porosity (%)  J_{nm}  a_{T} 

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)
If Eq. (123) is squared, it will produce Eq. (124) as shown below
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)
a^{T}X^{T}X = y^{T}X with a^{T} = (X^{T}X)^{−1}y^{T}X
Based on Equation (130) above, with the matrix X, therefore we obtained Eq. (131) to Equation (134) below
by using MATLAB programming, we get Eq. (133) with a_{o} = e^{Ao} and a_{1} = A_{1}, a_{2} = A_{2}, we get a_{T} = e^{0.3778}d^{−11}ϕ^{−42.7}. Do the same for J_{nm} as a function of weight, then we get J_{nm} = 5.39w
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)
a_{a} is the sound absorption coefficient, ϕ (%) is the porosity, d (mm) is the thickness of fabric, f (Hz) is the frequency and w (g/m^{2}) 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 properties of sound  Model  Literature 

Wave equation 


Sound pressure  P = P_{o}e^{−βz}e^{i(αz−ωt)}  P (z, t) = P_{o} cos (k.r − ωt) [25] 
Impedance 

Z = ρ_{o}c_{o} (1 + 0.0571X^{−0.754} − j0.087X_{−0.732}) [25]

Wavenumber 
where 
k = ω/c_{o}(1 + 0.0978X^{−0.700} − j0.189X^{−0.595}) [25] 
Sound absorption coefficient 


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.
_{aexp}  f (Hz)  α_{model} 

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 
α_{aexp}  f (Hz)  α_{model} 

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 
α_{aexp}  f (Hz)  α_{model} 

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 
α_{aexp}  f (Hz)  α_{model} 

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 R^{2} value obtained was 0.8399 with the relationship between the model and the experimental results shown in Figure 8
The R^{2} value was obtained 0.2815, with the relationship between the model and the experimental results shown in Figure 9
The R^{2} value obtained was 0.909, with the relationship between the model and the experimental results shown in Figure 10
The R^{2} value was 0.8303, with the relationship between the model and the experimental results shown in Figure 11
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 R^{2} value above 0.8, while for satin, the R^{2} 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.
4 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
Acknowledgment
We would like to express our gratitude to Politeknik STTT Bandung and Universitas Nusa Cendana as the research funders as well as to the contribution of colleagues who helped us during the research and discussion.
Conflict of Interest:
All authors declare no conflicts of interest regarding the content and implications of this manuscript
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