In the present paper, a methodology is proposed for obtaining empirical equations describing the sound absorption characteristics of an absorbing material obtained from natural fibers, specifically from coconut. The method, which was previously applied to other materials, requires performing measurements of airflow resistivity and of acoustic impedance for samples of the material under study. The equations that govern the acoustic behavior of the material are then derived by means of a leastsquares fit of the acoustic impedance and of the propagation constant. These results can be useful since they allow the empirically obtained analytical equations to be easily incorporated in prediction and simulation models of acoustic systems for noise control that incorporate the studied materials.
In recent years, building regulations and the growing social demand for sustainability, have led to the emergence of new absorbent materials. For these new materials to be usable in the design stage, it becomes necessary to establish simplified models to easily incorporate them into simulation programs. This is one reason why, for a while now, various strategies have been proposed based on empirical models of material behavior or other linear equations. The materials studied in this respect have been the rock wool and glass wool, which are the most widely used in engineering practice. One of the most known and used models in this line is that of Delany and Bazley (
In this work, a specific natural material is studied, namely coconut fiber. Coconuts, originated from the
These coconut fibers (known as coir) are abundant in many Asian countries such as India, SriLanka or Malaysia, and have been sometimes been regarded as a natural waste product. In recent years, sustainability concerns lead to an increasing interest in its application in construction, industrial and technological solutions, and raised the attention of both the industry and of the scientific community. Traditionally, these fibers were used in floor mats, doormats, brushes or mattresses; recently, technological industries, such as the automotive industry in Europe, have been increasingly using brown coir pads sprayed with rubber latex (bonding the fibers together) as upholstery padding. In the construction industry, its application has been studied in many forms, such as a reinforcing fiber in cement boards (see for example (
Due to their natural unprocessed origin, coir fibers are typically heterogeneous both in color, thickness and length, and, as natural untreated materials, they also exhibit performance limitations in some respects. For example in what concerns fire reaction, these fibers are a combustible material, and so their application without protective measures becomes limited. Due to this behavior, commercial products incorporating a fireproof treatment became available for use when the mattresses are to be used with some possibility of being exposed to fire. Two forms of industrially produced coconut fiber mattresses can thus be commonly found in western countries, corresponding to semirigid mattresses (usually higher density panels) with fireproof treatment and to flexible mattresses (usually rolls with lower density), without any synthetic treatment.
Studies regarding its applicability from the acoustic viewpoint jobs include Fouladi et al. (
a) Detail of coconut fibers. b) Photograph of the fiber without fireproof treatment taken using an electron microscopic. c) Idem with fireproof treatment.
This work has developed and implemented a method for obtaining an empirical model for the case of absorbent materials obtained from natural fibers, in particular coconut fiber, following a procedure similar to that described in (
The model presented in this paper aims to predict the acoustic behavior of the fibrous material under study using the fewest number of nonintrinsic physical parameters, thus mitigating some propagation errors people make when using parameters obtained experimentally. Basically, it comes to finding the coefficients
where α and β are the real and imaginary parts of the propagation constant Γ of the material,
To determine the specific acoustic impedance of a material, the commonly used transfer function method is used, which considers a planewave tube with a speaker mounted in one end. The sample of the material to be characterized is placed at the other end of the tube, at a distance
Experimental setup for measuring the impedance of a material using the method of the transfer function.
In the standards ASTM E1050 (
Driving the speaker with a random noise signal (allowing an assessment of the full frequency range) and recording the sound pressures in the two microphone positions, these pressures can be written as the sum of the incident and reflected sound waves. The complex transfer function
where
Thus, it is possible to obtain the specific acoustic impedance (real and imaginary) using the equation
being
From this reflexion coefficient, it becomes possible to compute an acoustic parameter of significant practical relevance, namely the sound absorption coefficient for normal incidence α_{
n
} as:
This technique has the disadvantage of requiring the calculation of the transfer functions using successive measurements with exchanged measurement channels, in order to allow for phase correcting mismatches between microphones. Otherwise large errors can occur in the impedance calculation.
The relation between this absorption coefficient for normal incidence and the acoustic impedance can be defined by the equation:
where
To allow fitting the material model, it becomes necessary to obtain experimental data concerning the airflow resistance and absorption coefficient under normal incidence.
The following quadratic error function for the iterative method used here can be defined as:
where α_{
n,i
} is the absorption coefficient for normal incidence, measured for the absorbing material under study, for the ith frequency, and
To obtain the coefficients that best describe the measured acoustic behavior of the samples, an iterative method of decrease of quadratic error function was used. As initial values (input values), different values proposed by different authors were tried, and all converged to the coefficients values as obtained and presented in this paper.
The experimental basis taken for adjusting the model consisted of:
Coconut fiber material without fireproof treatment with 2, 3 and 4 cm nominal thickness.
Idem with fireproof treatment 2 and 3 cm.
As for the distribution of fiber diameter, its analysis was conducted on a sample including 200 random fibers, leading to an average fiber diameter of 0.25 mm, with a size distribution between 1.17 and 0.48 mm.
The experimental determination of the airflow resistance has been carried out according to the procedure specified in ISO 9053 (
Left: Device used for measuring the flow resistance. Right: Samples prepared for the measurement.
In order to illustrate the homogeneity of the results,
Flow resistivity obtained for different air flow rates. S1, S2 and S3 correspond to samples of coconut fiber without fireproof treatment 2 cm thick, S4, S5 and S6 correspond to samples of coconut fiber without fireproof treatment 3 cm thick, S7, S8 and S9 correspond to samples of coconut fiber without fireproof treatment 4 cm thick.
Flow resistivity obtained for different air flow rates using samples with fireproof treatment. S1, S2 and S3 correspond to samples 2 cm thick, and S4, S5 and S6 to samples 3 cm thick.
Similarly,
As already indicated above, for each of the 10 samples measurements are performed. The plotted results in
As indicated in the reference standard, the value of the flow resistivity of each sample is the value at the origin of the horizontal axis of a linear law fitted to the performed measurements. The average values obtained from the three samples is the value taken as final. Here, the authors consider the uncertainty as the standard deviation of the tested samples. The results are shown in
Resistivity values obtained for samples 2, 3 and 4 cm thick without fireproof treatment
Coconut fiber without fireproof treatment 
Resistivity (


2 cm  2.60±0.15 
3 cm  1.94±0.15 
4 cm  1.20±0.11 
Resistivity values obtained for samples 2 and 3 cm thick with fireproof treatment
Coconut fiber with fireproof treatment 
Resistivity ( 

2 cm  0.69±0.04 
3 cm  0.85±0.11 
At this point, it is important to comment on the uncertainty associated with the measurement of the flow resistance. One of the most common problems in the treatment of absorbent materials in practice is that the sample thickness varies. In effect, all processes from manufacturing to final assembly, as well as the process for conducting measurements in an impedance tube or flow resistance, where the technician has to place the sample between racks and / or probes, the material is compressed. This compression with the resulting change in the thickness leads to greater uncertainty in predicting realistic values of flow resistance.
To complement this comment, one may perform the following mathematical reasoning. The following expression, from Bies and Hansen (
On the other hand, if
This means that the relative variation in the normalized frequency (
Applying Napierian logarithms in [
As it can be seen in the literature,
Analyzing [
Therefore, it can be inferred that the uncertainty associated with the flow resistivity data is very large, and thus the presented data must be interpreted as a confidence interval.
Detail of the measurement equipment used to determine the absorption coefficient, including a sample of coconut fiber.
The following tables summarize the results of the experimental measurements and adjustments. To perform the measurements, the transfer function method has been applied (
The adjustment is made without considering the extreme values at 100 Hz and 3150 Hz. The corresponding constants are shown in
Constants calculated for the implemented model
C_{1}  C_{2}  C_{3}  C_{4}  C_{5}  C_{6}  C_{7}  C_{8} 

0.0713  0.8749  0.1216  0.4520  0.2129  0.4857  0.0997  0.5988 
Absorption coefficient as a function of frequency for samples of untreated coir. Results of experimental measurements and adjustments (full line: experimental data, dashed: numerical setting) for samples 2, 3 and 4 cm thick.
Absorption coefficient as a function of frequency for samples of treated (fireproof treatment) coir. Results of experimental measurements and adjustments (full line: experimental data, dashed: numerical setting) for samples 2 and 3 cm thick.
This paper described the process for obtaining empirical equations for the acoustic behavior of an absorbent material made natural fibers, particularly coconut fibers. The procedure, which has been successfully tested in other materials, involves performing impedance and airflow resistance measurements for samples of the material under study. The results are useful since the availability of empirically derived analytical equations, simplifies the incorporation of these materials into simulation programs in order to make predictions of their acoustic behavior when used as part of noise control devices. Note that there are different types of coconut fiber and, obviously, not all have the same degree of absorption.