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The Formula of Sound Absorption Spectrums For Fibrous

Materials

Zhang Xin’an

Acoustics Institute, Tongji University ，ShangHai,China (200092)

key laboratory of functional fabric of Shaanxi Provrnce, Xi’an Polytechnic University,

Xi’an,China (710048)

E-mail ：zxafafa@http://www.wendangku.net/doc/576d7f0f4a7302768e993999.html

Abstract

The current sound absorption theory is still the Rayleigh model. But this model is just the theory model. It is not agreement with the practice even though it had been studied for over 100 years. Based of this theory, Zwikker and Kosten established the theory of acoustical effective density and the effective bulk modulus which is also the popular theory in acoustics literature to explain the sound absorption characteristics of fibrous materials. However, because of the complexity of these expressions, it is difficult to obtain physical insight into the acoustic behavior of the porous materials and to determine the dominant mechanism for sound absorption for a given material at a given frequency. Alternatively there are very simple expressions. Through testing the sound absorption coefficient of thick and thin fiber layers at different frequency with the back distance 5cm,10cm,20cm,30cm and 40cm,the author found the law of sound absorption at different frequency and also get the formula. With this formula, we can get the accurate relative sound absorption coefficient of fibrous materials from frequency. Keywords ：fibrous material, sound absorption coefficient, frequency, formula

1. Introduction

The current sound absorption theory is still the Rayleigh model. But this model is just the theory model. It is not agreement with the practice even though it had been studied for over 100 years.。while for there are not another suitable theory to explain the sound absorption properties of fibrous materials, most of the scientists accept it and use it as the theory to design the room sound absorption . Based of this theory, Zwikker and Kosten established the theory of effective density and the effective bulk modulus which is also the popular theory in acoustics literature to explain the sound absorption characteristics of fibrous materials .

)2,1()4,3()8,7,6,5()9(The basic equations of Zwikker and Kosten theory was reviewed below.

Let be the sound absorption coefficient ，a Z be the acoustical impedance ，then,

c

z c

z a ρρ+??

=1

Where ，c ρis the wave impedance 。 For the material with thickness D ，

??

?

??????

?+?++=)tan(1)tan(1coth )tan(1)tan(1B p B p j j n c D

j j j n

B

z δδγωδδγ

ρ

Where ρis effective density and B is the effective bulk modulus of the fluid in the pore space of porous material ，ωis the angular frequency, is structure factor,n γ is the thermal attenuation ，and

p δB δis the attenuation angular.

In practice ， ,n γ，and

p δB δhave a large variation range 。So ，it is extremely difficult

to get accurate result. In this theory, sound propagation in the porous materials is governed by the

)9(

effective density and the effective bulk modulus of the fluid in the pore space. These quantities are frequency-dependent, complex and non-linear. Because of the complexity of these expressions, it is difficult to obtain physical in sight into the acoustic behavior of the porous materials and to determine the dominant mechanism for sound absorption for a given material at a given frequency.

Recently, some person claimed that Zwikker and Kosten theory had been successfully applied to describe the principle and process of nonwovens. But the process to deduction the the equation is too complex which cost over ten pages to publish it and the equation is also critical complicated must be operate by his computer soft then get the result

.In addition, there are

another similar acoustical impedance formula and also very complicated 。

)

12,11,10()13(To solve this problem, many scientists had established many experimental model ,and some others study the flow resistance to get the useful formula .But the results is also not cheerful. We must still use the test methods to determine the SAC( as for simple, we will use the SAC denote the sound absorption coefficient below),while not the prediction formula

. So ，we can get a conclusion that scientist have not find the real relation

between sound absorption and the frequency. In the other words, the principle of sound absorption of fiber material has not been completely found yet.

)

18,17,16,15,14()

21,20,19()

22(In this paper, three kinds of thin and thick fiber layer had been tested for its SAC at different frequency with the back distance of 5cm,10cm,20cm,30cm and 40cm.By analyzing the spectrum of SAC with different frequencies, a formula to calculate the SAC from frequency was established. One of the popular figure applying for design the sound absorption materials is figure1,from which ,scientist get the conclusion that porous materials (fibrous material is one of it) absorb sound more at high frequency and less at lower frequency . But this is just an example of the figures discussed in this paper. As we will discuses below, figure 1 is the sound absorption spectrum of fibrous material with the back cave distance of 5cm and below.In another statement, for example, the back cave distance is 20cm,the SAC will also have the lower SAC even the frequency is high. So, a different idea was given that that high wave amplitude will cause high SAC and lower wave amplitude will cause lower SAC. The reason will be discussed in author’s another paper.

)

7,5( This paper’s formula is so important that people can use the frequency calculate the SAC with a too simple formula and which had never been founded in this paper.

2．The sound absorption spectrum

The SAC of three kinds of material at different frequency with different back distance had been list in the table 1.

From the table above, it can be seen that there are obvious law for the maximum and minimum SAC emerging. When D ≈λ1/4、3λ2/4、5λ3/4， …

or

λ≈4D/（2n-1） n=1,2,3, … There are maximum SAC. When D ≈

λ1/2、λ2、3λ3/2， …

or ，λ≈2D/n n=1,2,3, …

There are minimum SAC.

These results are agreement with the test results of F.Ingerslev(Danish) and Y .Shaoshani

(Israel) who had tested the thin fabric and woven fabrics and also get the conclusion that when D ≈

λ/2, there are minimum SAC. while D ≈λ/4、3λ/4, there are maximum SAC

,and it

also had been discussed in another literature . In addition ，the test result that the maximum

SAC of thin fabric is 0.85 from F.Ingerslev is also agreement with the testing data of cotton fabric in this paper.

)

24,23()

5,1(3. The formula of sound absorption spectrum

According to the above discussion, it is known that there are maximum SAC at 4/λn D = (n=1,3,5….) and minimum SAC at 2/λn D =.This is also the characters of stand wave in the tube. So, it is reasonable to assume that the SAC of material in the stand wave tube is positive proportion with the wave amplitude. That means high wave amplitude will cause high SAC and lower wave amplitude will cause lower SAC.

Let y denote the wave amplitude, then

Y y =)sin(?ω+t

Where, Y is the maximum amplitude,ω is the angular frequency, let 0=? and KY A =

Then, =a A t ωsin For

ω=2πf , =c/f λ, (, c, f λ denote frequency, sound speed and wave length

respectively) Then,

=a A )/2sin(λπct

Let the back wall of the tube be the zero point, then, D ct =, So,

)2sin()2sin(

c

f

D

A D

A a πλ

π== ……………(1) This is the formula of sound absorption spectrum for fibrous material.

The spectrum of three kinds fiber layer at different frequency with the back distance of 5cm, 10cm, 20cm,30cm and 40cm had been show in the figure 2-6. The calculating results obtaining from the formula (let A=1) with a series frequency had also been showed in it.

In the spectrum, ther. represente the calculating result getting from the formula. It can be seen that the theory result are fair agreement with the testing result, especially for the thick materials.

In addition, the more thick and more grams per square meter bring higher SAC.

In another paper of the author, a new sound absorption theory will be established and the value A which is the function of material’s structure parameters will be derived.

4．Discussion

Some characters of sound absorption material are agreement with the conclusion from the formula:

A. The increase of the thickness is equal to the increase of cavity depth

In above discussion, D=thickness of material + cavity depth. So, the increase of thickness of material have the same meaning with increase cavity depth to the formula (1).If the material was touched with the back, then, the thickness is equal to the cave distance, the above conclusion will be naturally.

B. From the above figures, we can see that the theory spectrum will move to the low frequency with the increase of cavity depth which is also the character of sound absorption materials quoted in most of acoustics textbooks.

C. From the formula obtained in this paper, it can be concluded that high wave amplitude will cause high sound absorption coefficient and lower wave amplitude will cause lower sound absorption coefficient.

Here, I must explain that in stand wave tube, the increase of the wave amplitude of sound source will not cause the change of SAC tested. The reason is that the higher wave amplitude of sound source will cause the increase of both the maximum pressure and the minimum pressure in stand wave tube, but the value of SAC is determined by , so the SAC will not change. Generally speaking, the increase of the level of low frequency will not yield the change of SAC. While this need the further prove of experiment. But, if the amplitude of sound source is given, the SAC of materials will vary directly as the relative sound wave amplitude which has been proved by the analyzing of section 2 and 3 in this paper.

1p 2p 12/p p D. The longer the cavity depth, the higher sound absorption coefficient for low frequency sound. E . Just as showed in figure 2 to figure 6, the test result of thick materials is more accordance with calculating result from equation (1) and it nearly get the top SAC(theory spectrum tabbed by Ther. in figure 2-figure 6)). So, the increase of thickness yields higher SAC and relative wide frequency band of SAC. But as the conclusion in this paper and literature (1),(5) and (6),the SAC appear maximum value at D= and minimum value at D= is the law of sound absorption spectrum and the theory spectrum in figure 2 to figure 6will be the law of materials with highest SAC. So, the increase of the wide of frequency band of SAC will be limited in the given value which is determined by the arc of the spectrum curve of theory

F In stand wave tube, the increase of the wave amplitude of sound source will not cause the change of SAC tested. The reason is that the higher wave amplitude of sound source will cause the increase of both the maximum pressure and the minimum pressure in stand wave tube, but the value of SAC is determined by , so the SAC will not change. Generally speaking, the increase of the level of low frequency will not yield the change of SAC. While this need the further prove of experiment. But, if the amplitude of sound source is given, the SAC of materials will vary directly as the relative sound wave

ACKNOWLEDGMENT

It is a pleasure to thank professor Sheng shengwo who as my PHD professor let me have the opportunity to study the Acoustic science and given me a lot of knowledge of modern Acoustic science.

Reference

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Applied Acoustics V olume: 62, Issue: 4, April, 2001, pp. 447-459

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2 #1-nonwoven made of glass fiber, thickness 0.26mm，gram per square meter =56.4 g/m

2

#2- woven cotton fabric thickness =0.38mm，gram per square meter= 146.6 g/ m

2

#3-polyester fiber layer , thickness =20mm，gram per square meter =1170 g/ m

Figure 1.populer figure used to Design the sound absorption material.

Figure 2. 5cm cave

Figure 3. 10cm cave

Figure 4. 20cm cave

Figure 5. 30cm cave

Figure 6. 40cm cave

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