Micro-perforated absorbers with incompletely partitioned cavities

Applied Acoustics 126 (2017) 114–119

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Technical note

Micro-perforated absorbers with incompletely partitioned cavities Sibo Huang, Shengming Li, Xu Wang ⇑, Dongxing Mao ⇑ Institute of Acoustics, School of Physics Science and Engineering, Tongji University, No. 1239 Siping Road, Shanghai 200092, China

a r t i c l e

i n f o

Article history: Received 7 March 2017 Received in revised form 12 May 2017 Accepted 15 May 2017

Keywords: Micro-perforated panel Incomplete partitioning Sound absorption

a b s t r a c t Acoustic performance of a micro-perforated panel with an incompletely partitioned cavity (MPPIPC) is investigated. A MPPIPC is achieved by inserting separators, which are shorter than the depth of the cavity, periodically behind the micro-perforated panel (MPP). Due to these separators, the sound field in the cavity is significantly changed. This paper reveals these unique sound reflection modes among the separators, based on which a theoretical model is proposed to calculate the absorption coefficient of a MPPIPC in the diffusion field. Experiments verified the theoretical predictions. These results indicate that the appropriately arranged insertion can improve the performance of a MPP absorber effectively at low frequencies. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Compared to noise at middle and high frequencies, the adverse effects of low-frequency noise are of particular concern due to its pervasiveness of numerous sources, efficient propagation, and reduced efficacy of many structures (walls, noise barriers, and hearing-protection devices). A work on low-frequency noise has indicated that it causes great subjective reactions and to some extent physiological reactions on human beings [1]. Among those sound absorbers working at low frequencies, micro-perforated panels which possess excellent properties including fiber free, non-combustible, as well as aesthetically pleasing, are widely used in noise control engineering [2]. According to Maa’s theory [3,4], the cavity depth plays an important role in determining the MPPs’ sound absorption band. Aiming for improving sound absorption at lower frequencies, an increased cavity depth is required. However, in the case where space for the backed cavity is limited, conventional MPP absorbers cannot provide sufficient sound absorption at low frequencies. Therefore, various strategies treating the air cavity have been proposed. Yairi et al. [5] concluded that completely partitioning the air cavity with subwavelength interval was especially effective. This observation was further confirmed by Toyoda et al. [6] and Liu et al. [7,8]. To enhance low frequency absorption, flexible tube bundles [9,10] and mechanical impedance plates [11] were also incorporated into a MPP absorber.

⇑ Corresponding authors. E-mail addresses: [email protected] (S. Huang), [email protected] (S. Li), [email protected] (X. Wang), [email protected] (D. Mao). http://dx.doi.org/10.1016/j.apacoust.2017.05.016 0003-682X/Ó 2017 Elsevier Ltd. All rights reserved.

In this paper, an incomplete partition technique is proposed and applied to a MPP to improve its sound absorption at low frequencies. With appropriately arranged insertion, these separators could significantly elongate the equivalent depth of the air cavity. In this paper, Maa’s theory of MPPs in the diffusion field is discussed in Section 2. Section 3 reveals those unique sound reflection modes among the separators of MPPIPCs. In Section 4, theoretical predictions are compared with experiment results. Finally, conclusions are made in Section 5. 2. Maa’s theory of MPPs in the diffusion field The MPPs were initially proposed by Maa [3,4]. For normal incidence, the absorption coefficient is derived by

a¼

4r ð1 þ rÞ2 þ ðxm  cotðxD=cÞÞ2

;

ð1Þ

where r is the normalized specific acoustic resistance of the MPP, m is the normalized acoustic mass of the MPP, x is the angular frequency of incident wave, c is the speed of sound, and D is the cavity depth behind the MPP. For oblique incidence, a MPP itself provides a motion-constraint condition to the particles, leading to a one-dimensional sound field in each of MPP’s holes, which is also known as ‘‘locally reacting surface”. Then the acoustic wave can only propagate normally to the MPP, resulting in an unaltered specific normal acoustic impedance independent of the incident angle. According to Huygens principle, the sound wave transmitted through MPP’s holes will propagate in the same direction with the incident wave [3]. And for the air cavity, path difference between the incident and the

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reflected waves plays an important role in determining the transfer impedance. Proposed a plane wave impinges obliquely on the MPP’s surface, the specific normal transfer impedance can be derived by

Zn ¼

p

vn

¼

pi þ pr

ð2Þ

v i cos hi þ v r cos hr

where pi is the incident wave, pr is the reflected wave, v i and v r are the particle velocity of air generated by incident and reflected waves respectively, hi and hr denote the angles of incidence and reflection. The analytical model for calculating cavity’s specific transfer impedance is shown in Fig. 1(a). Assume that point A is the origin, point E located at (X 0 , 0), the cavity wall is rigid, and the cavity depth D ¼ D0 . Hence, substituting these coordinates into Eq. (2), finally the specific normal transfer impedance of the air cavity can be expressed as

Z cn ¼ j

qc cotðxD0  cos hi =cÞ: cos hi

ð3Þ

Notice that the difference of air cavity’s specific transfer impedance between normal and oblique incidences is that the cavity depth D0 is replaced by D0 cos hi in the cotangent function. As illustrated in Fig. 1(a), points E and C belong to the same wave front. Then for the incident and reflected waves at E, the sound path difference is CD + DE. Since BC = EF = EF0 , this path difference equals to BD + DF0 (=2D0 cos hi ). In another word, the term D0cos hi is half of the path difference between incident and reflected waves at E, which is regarded as the ‘‘equivalent depth” of the cavity. Herein, the ‘‘equivalent depth” of the cavity is used to calculate the specific normal transfer impedance of air cavity under oblique incidence. With the obtained specific normal acoustic impedance of the MPP and its backed cavity, the absorption coefficient of MPPs can be calculated by




Z n;hi cos hi  qc
2
; Z cos h þ qc


ahi ¼ 1 



n;hi

ð4Þ

i

where

Z n;hi ¼ qcðr þ jxmÞ  j 

qc

cos hi

 cot

xD0 cos hi c

 :

ð5Þ

Here Z n;hi is the specific normal acoustic impedance of MPPs at incident angle hi . Substituting Eq. (5) into Eq. (4) yields

ahi ¼

4r cos hi ð1 þ r cos hi Þ þ ðxm cos hi  cotðxD0 cos hi =cÞÞ2 2

:

ð6Þ

This result is accord with Maa’s theory. Furthermore, considering a diffusion field where waves impinge on a MPP from all angles, then the random incident sound absorption coefficient as is given by

as ¼

Z p=2 0

ahi sin 2hi dhi :

ð7Þ

3. Sound reflection modes among the separators of a MPPIPC In Section 2, the Maa’s theory of MPPs in the diffusion field has been introduced, and the sound absorption coefficient at incident angle hi has been deduced. Herein, by inserting short separators behind the MPP periodically to partition the cavity incompletely, the original ways that sound rays undergo behind the MPP will be altered drastically. Accordingly, path differences of these rays will be changed and hence result in a variety of the normal acoustic impedance of a MPPIPC. To obtain the sound path differences, those sound reflection modes among the separators of a MPPIPC should be investigated systematically. As discussed above, according to the relation between the sound path difference and the ‘‘equivalent depth” of air cavity, the specific normal acoustic impedance of the proposed MPPIPCs can be calculated. In this section, the possible sound reflection modes in the cavity of a MPPIPC are analysed and classified into five categories. Those modes between two separators (the left panel I and the right panel II) are illustrated in Fig. 1(b)–(e). Fig. 1(b) represents the most simple mode that all the rays avoid the separators, leading to the same path difference with the case of a non-partitioned cavity. Fig. 1(c) represents the rays that first hit the bottom of the cavity and then are reflected by the separator II before leaving the MPPIPC. Fig. 1(d) represents the rays which first hit the separator panel II and then are reflected by the cavity bottom before leaving the MPPIPC. While Fig. 1(e) represents the rays that hit, in turn, separator II, cavity bottom, separator I, and finally leave the MPPIPC. Actually these four modes account for the majority proportion of incident rays. Only in the case with an incident angle greater than a

Fig. 1. (a): Analytical diagram to calculate the specific normal transfer impedance of the cavity; (b)–(e): demonstration of sound reflection modes between two separators; (f): demonstration of the reflection mode under large-angle incidence.

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threshold, arctanðLx =Li Þ, the complex mode illustrated in Fig. 1(f) appears. Here Li and Lx represent the length of separators and the distance between two nearby separators (i.e. the period of insertion) respectively. For this extreme mode of large-angle incidence, the rays both undergo multiple reflections between two separator panels before and after they are reflected by the cavity bottom. Notice that all the rays limited in two nearby separators are illustrated by Fig. 1(b)–(e). And straightforwardly, by extending these four categories from two adjacent separators to those far departed and adding the last mode shown in Fig. 1(f), all the incident rays can be reckoned in. By using the simple geometrical relationship, the sound path differences of these categories can be calculated. Fig. 2(a) illustrates the schematic diagram for the path category shown in Fig. 1(c), and its geometrical relationship is given by

x þ Lx þ C 0 D0 ¼ 2D0 tan hi ;

ð8Þ

2C 0 D0 sin hi ¼ D0 E:

ð9Þ

Compared with the category shown in Fig. 1(a) and (b), which represents the case that a ray has a round trip inside the MPPIPC without any reflection on the separators (the path difference 2D0 cos hi ), the path difference for this category is changed from FB + BD0 to GB + BD0 , which is equivalent to adding a distance of GF (=D0 E). According to Eqs. (8) and (9), the length of D0 E can be expressed as 4D0 tan hi sin hi  2 sin hi ðx þ Lx Þ. Thus the corresponding sound path difference turns into

2ðD0 cos hi þ 2D0 tan hi  sin hi  x sin hi  Lx sin hi Þ:

ð10Þ

For simplicity, Fig. 2(a) demonstrates a ray leaves from the nearby period of the separators. In general, by taking the period that rays first reached as the first period, and replacing Lx in Eq. (10) by ðn  1ÞLx , the path difference for rays that leaves from nth period in this category can be expressed as

Fig. 2. (a)–(c): Schematic diagrams of the reflection modes; (d): three-dimensional view of a MPPIPC; (e): a photograph of the specimen.

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2ðD0 cos hi þ 2D0 tan hi  sin hi  x sin hi  ðn  1ÞLx sin hi Þ:

ð11Þ

Similarly, the path difference of the second mode shown in Fig. 2(b) and the third mode shown in Fig. 2(c) can be expressed by the following formulas respectively:

2ðD0 = cos hi þ D0 tan hi  sin hi  x sin hi Þ;

ð12Þ

2ðD0 = cos hi þ nLx sin hi  D0 tan hi  sin hi Þ:

ð13Þ

Moreover, assume an ideal case that a MPPIPC contains infinite separators mounted periodically along the x-axis. Then for the diffusion field, the averaged sound absorption coefficient in the x-z plane can be calculated by a double integral of x and hi (see Appendix A.1). For a MPPIPC in an ideal diffusion field, x is integrated from 0 to Lx , and hi is integrated from 0 to p=2. Furthermore, since different reflection modes depend on rays with different incident angles and positions, such an integral actually is piecewise. Therefore, the upper and lower limits of each subsectional integral should be determined by a further analysis. For instance, as the mode shown in Fig. 2(a), the integral upper and lower limits of x are determined by the following formulas:

x > Li tan hi ;

ð14Þ

x > 2D0 tan hi  ðn  1ÞLx  Li tan hi ;

ð15Þ

x < 2D0 tan hi  ðn  1ÞLx ;

ð16Þ

x < Lx :

ð17Þ

Likewise, by using a similar deviation procedure, the integral upper and lower limits for the second and third modes are given by

ð0; Li tan hi Þ \ ð2D0 tan hi  nLx ; 2D0 tan hi  ðn  1ÞLx  Li tan hi Þ; ð20Þ ð0; Li tan hi Þ \ ð2D0 tan hi  nLx  Li tan hi ; 2D0 tan hi  nLx Þ:

ð21Þ

For those rays with incident angles in the range from 0 to arctanðLx =Li Þ, the segmented averaged absorption coefficient can be calculated by using Eqs. (11)–(13), (19)–(21) (see Appendix A.1). However, in the case of extremely large-angle incidence, the mode shown in Fig. 1(f) appears. It is found that those rays with incident angles larger than arctanðLx =Li Þ perform little effect on determining the overall averaged absorption coefficient (see Appendix A.2). Then the absorption coefficients for the MPPIPC in the diffusion field (in x-z plane) can be expressed approximately as

as;xoz ¼

Z

Z

arctanðLx =Li Þ

Lx

sin 2hi dhi 0

0

ahi ;x dx



1 : 2 Lx sin ðarctanðLx =Li ÞÞ ð22Þ

4. Experiments and discussions

Eq. (14) prevents the rays from hitting the separator II; Eqs. (15) and (16) restrict the rays to exactly hit the separator III; while Eq. (17) indicates that the incident wave is limited in the first period. By solving the four equations above, the integral upper and lower limits of x for this subsectional integral are expressed as

ðLi tan hi ; Lx Þ \ ð2D0 tan hi  Lx  Li tan hi ; 2D0 tan hi  Lx Þ:

ð18Þ

Expanding to the case of nth period yields

ðLi tan hi ; Lx Þ \ ð2D0 tan hi  ðn  1ÞLx  Li tan hi ; 2D0 tan hi  ðn  1ÞLx Þ:

ð19Þ

To validate the theoretical model established above, experiments were conducted in the reverberation room of Institute of Acoustics of Tongji University. The random incident sound absorption coefficients were measured according to standard ISO 354 [12]. Fig. 2(d) shows the schematic diagram of a MPPIPC in three dimensions, while the inset figure illustrates the sectional view and the assembly method of the separators. The picture of the specimen is given in Fig. 2(e), while the inset figure illustrates its sectional view. The geometries of the first tested MPPIPC were Li ¼ 50 mm, Lx ¼ 100 mm, D0 ¼ 150 mm, and for its MPP, which was made of aluminum, the diameter of holes d ¼ 0:3 mm, the perforation ratio r ¼ 2:8%, the panel thickness t ¼ 0:8 mm. The specimen area was 11:5 m2 , and the volume of the reverberation room is 268 m3 . The testing apparatus consisted of BSWA 490834 microphones, a

Fig. 3. (a) Experimental results (Lx = 100 mm) and the predictions by Maa’s theory; (b) performances of the MPPIPC and the honeycomb structure; (c) experimental and theoretical results of the MPPIPC with Lx = 100 mm; (d) experimental and theoretical results of the MPPIPC with Lx = 200 mm; (e) comparison of the MPPIPCs with Lx = 100 mm and Lx = 200 mm.

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multi-channel analysis system (ArtemiS and NI USB-9162), and a Norsonic power amplifier type Nor280. The measured results are shown in one-third octave band centre frequency in Fig. 3(a), which is compared to the absorption coefficients predicted by the Maa’s theory (for the diffusion field). As shown by Fig. 3(a), it is evident that, at low frequency (below 630 Hz), experimental results were superior to the predicted results, behaving much lower absorption band and possessing a higher peak. In other words, as expected, by periodically inserting short separators, the sound absorption at low frequencies were improved considerably without increasing the depth of the cavity. In the previous studies, MPPs backed by subwavelength honeycomb with various lengths leading to the air cavities from incompletely to fully partitioned, were also investigated [13,14]. Both their experimental and theoretical results showed that with the increased length of the inserted honeycomb, the absorption frequency band moved to lower frequency range constantly. This result indicates that fully partitioning the cavity may be the most effective treatment to move the absorption band to lower frequency range. Fig. 3(b) shows the comparison of the performance between the MPPIPC and the structure proposed by Yairi et al. [5], which is a honeycomb consisted of a same MPP and a fully partitioned cavity. The comparison indicates that the proposed MPPIPC provides very similar sound absorption to the honeycomb at low frequencies. However, for the fully partitioned one, severe performance fluctuation was observed in the range of 800– 1300 Hz; while the behavior of the MPPIPC was much more modest. Notice that, although the results from the proposed model for incompletely partitioned cavity and from the model proposed by Yairi for the fully partitioned one were very similar at low frequencies, their underlying working mechanisms were different. For the investigated MPPIPC, these separators inserted periodically along x-direction, while y-direction was free for sound wave propagation, i.e. no partition in y-direction. Moreover, rather curious fluctuation from Yairi’s structure is due to that, although under diffuse incidence, such a subwavelength-partitioned cavity only allows one-dimensional wave propagation, rendering it behaves similarly to a resonant tube. In fact, the proposed MPPIPCs are anisotropic structures. The above analysis limits the incident rays in x-z plane, i.e. a 2D diffusion field. However, in practice, as shown in Fig. 2(d), reflections with azimuth angle hp (the angle to x-axis in the x-y plane) are different from the situations discussed above, and this makes the theoretical modeling much more complicated. For a ray impinges on the MPPIPC with an azimuth angle hp, the interval length of the separators appears equivalently to be elongated from Lx to Lx/coshp. To estimate the performance of the MPPIPC in a 3D diffusion field, the numerical integral method is applied by considering the x-y plane to be made up of a large number of segments of angle dhp, and averaging the sound absorption coefficients from hp = 0 to p/2. By taking the azimuth angle hp into consideration, the absorption coefficients for MPPIPCs can be expressed as a triple integral:

as ¼

Z p=2

Z dhp

0

Z

arctanðLx =Li Þ

sin 2hi dhi 0

0

Lx

ahi ;x;hp dx



2

pLx  sin2 ðarctanðLx =Li ÞÞ

:

ð23Þ By integrating Eq. (23), the random incident sound absorption coefficients can be obtained. As shown in Fig. 3(c), the theoretical predictions agreed quite well with the experimental results in the whole frequency range of interest. Actually, the prediction by Eq. (23) is valid for those MPPIPCs with fairly large Lx =Li (see Appendix A.2) and with ample separators. With the declining number of inserted separators, the effect of cavity’s left and right boundaries (can be regarded as two fully partitioned separators)

plays a more important role in determining the overall absorption performance of a MPPIPC, by affecting some of the ray paths in their nearby periods. For the first MPPIPC specimen, there are 30 separators, thus the effect of the cavity’s boundaries is relatively little. The second specimen with larger separator spacing (Lx ¼ 200 mm) was then studied, while the rest parameters of this MPPIPC were kept the same with the previous specimen. Notice that this specimen only consists of 15 periods. The comparison of theoretical and experimental results is shown in Fig. 3(d), and they are also in acceptable accordance. But still, as expected, the deviation was observed at low frequencies due to the influence from cavity’s boundaries. This can be considered as the limitation of the developed theory that originally describes an ideal MPPIPC consisted of a large number of separators. Fig. 3(e) shows the comparison of the MPPIPCs with different insertion periods (Lx ¼ 200 mm and Lx ¼ 200 mm). Apparently, in order to achieve considerable absorption improvement at low frequencies, the appropriate arrangement of insertion is required. This will be further investigated systematically in future studies. 5. Conclusions In this paper, acoustic performance of a micro-perforated panel backed by an incompletely partitioned cavity is proposed. A theoretical model is further established to investigate the performances of the MPPIPCs in the diffusion field—the most common case in practice. Then experiments on MPPIPCs were conducted to verify the theoretical model. The theoretical predictions and experimental results were in good accordance, and both of them indicated that MPPIPCs provided considerable sound absorption performance at low frequencies, given that the insertion period and the length of separators were arranged appropriately. Additionally, the comparison of a MPPIPC and those fully partitioned structures was presented. The results showed that both the MPPIPC and the honeycomb possessed impressive absorption at low frequencies, but the MPPIPC exhibited modest performance in the middle to high frequency range. On the whole, rather than by increasing the depth of backed cavities, the proposed structures can enhance MPPs’ performance at low frequencies by inserting incomplete separators periodically. They are simple, easy to be fabricated, as well as economical in practical applications. Acknowledgements This work was financially supported in part by the National Natural Science Foundation of China under Grant No. 11304229 and the Fundamental Research Funds for the Central Universities. Appendix A A.1. Calculating the averaged absorption coefficient using Eqs. (11) (13), (19)(21) There are two methods to obtain the averaged absorption coefficient by using Eqs. (11)–(13), (19)–(21): mode analysis and computer programming. This section mainly introduces the former method. Provided the cavity depth D0 , separator length Li , and insertion period Lx , when 2D0 tan hi < Lx (i.e. hi < 18:43 ), sound rays are limited in one period, in other words, all the reflection modes are confined between two nearby separators. Therefore, in Eqs. (11)–(13), (19)–(21), n = 1. Then by substituting the sound path differences of the three modes given in Eqs. (11)–(13) into Eq. (3), the corresponding normal transfer impedance can be expressed as

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Z cn1 ¼ j

cos hi cotððD0 cos hi þ 2D0 tan hi  sin hi  x sin hi qc

 ðn  1ÞLx sin hi Þx=cÞ; Z cn2 ¼ j

A.2. Influence of rays with large-angle incidence

ðA1Þ

cos hi cotððD0 = cos hi þ D0 tan hi  sin hi  x sin hi Þx=cÞ; qc ðA2Þ

This section aims at demonstrating that rays with incident angles lager than arctanðLx =Li Þ perform little effect on determining the overall averaged absorption coefficient. For instance, suppose a material with a constant absorption coefficient aconst at any incident angle. Given that Li ¼ 50 mm and Lx ¼ 100 mm, as is expressed as

Z cos hi Z cn3 ðnÞ ¼ j cotððD0 = cos hi þ nLx sin hi  D0 tan hi  sin hi Þx=cÞ; qc ðA3Þ where the subscripts 1, 2 and 3 denote the categories of the analytical modes described in Fig. 2(a)–(c) respectively. Z cn0 represents the original mode (described by Eq. (3)) whose path avoids all the separators. Therefore, for incident angles in the range 0 < hi < 18:43 , by substituting n = 1 into Eqs. (19)–(21), the segmented averaged absorption coefficient is expressed as

Z

Z p18:43=180

Li tan hi

dh 0

U2 dx þ

Z

0

Z þ

2D0 tan hi

2D0 tan hi Li tan hi

U1;1 dx þ

Z

2D0 tan hi Li tan hi Li tan hi

Lx

2D0 tan hi

U0 dx

!

U0 dx

ðA4Þ

4r cos hi ð1 þ r cos hi Þ2 þ ðxm cos hi  jZ cn1 ðnÞ cos h=qcÞ 4r cos hi

ð1 þ r cos hi Þ2 þ ðxm cos hi  jZ cn2 cos h=qcÞ

U3;n ¼

2

p21:80=180

Z

þ

2D0 tan hi Lx

Li tan hi

2D0 tan hi Li tan hi Lx

U2 dx þ

0

U1;2 dx þ

Z

Lx

2D0 tan hi Lx

2

2

ðA10Þ

Notice that the contribution from the rays with incident angle 0 < h < arctanðLx =Li Þ (the first term on the left of equation) is 80%; while rays with large incident angles account for 20% of the final absorption coefficient. In practice, when sound wave impinges obliquely with azimuth angle hp to the x-z plane, the period of the separator appears equivalently to be elongated from Lx to Lx/coshp (noted as the equivalent period Lx0 ). This result indicates that, for the case with azimuth angle hp, the large-angle incidence with hi > arctanðL0x =Li Þ exists in a narrower range, and hence it performs less effect on determining the overall averaged absorption coefficient. Moreover, for case 2 (Li ¼ 50 mm, Lx ¼ 200 mm) and case 3 (Li ¼ 50 mm, Lx ¼ 300 mm), Eq. (A10) becomes

ðA6Þ

as ¼

ðA7Þ

2

:

ðA8Þ

Moreover, when 0 < 2D0 tan hi  Lx < Li tan hi (i.e. 18:43 < hi < 21:80 ), still, sound rays are limited in one period, but the integral upper and lower limits solved by Eqs. (19)–(21) are changed. Similarly, when Li tan hi < 2D0 tan hi  Lx < 2Li tan hi (i.e. 21:80 < hi < 26:57 ), the first and second analytical modes cover two periods, while the third analytical mode is still limited in one period. Then the segmented absorption coefficient is expressed as

Z

¼ aconst ðsin 63:43 Þ þ aconst ð1  sin 63:43 Þ ¼ aconst :

; 2

;

ð1 þ r cos hi Þ þ ðxm cos hi  jZ cn3 ðnÞ cos h=qcÞ

dh

aconst sin 2hdh

ðA5Þ

4r cos hi 2

Z p26:57=180

arctanðLx =Li Þ

;

ð1 þ r cos hi Þ2 þ ðxm cos hi  cotðxD0 cos hi =cÞÞ2

U2 ¼

Z p=2

these two expressions represent the contributions from those rays with large incident angles. Obviously, in case 2 and case 3, such an impact from large-angle incidence is negligible. Therefore, the averaged absorption coefficient can be expressed approximately as

4r cos hi

U1;n ¼

0

aconst sin 2hdh þ

aconst ðsin2 75:964 Þþ aconst ð1sin2 75:964 Þ ¼ 0:941aconst þ0:059aconst and 0:973aconst þ0:027aconst respectively. Herein, the last terms of

where

U0 ¼

arctanðLx =Li Þ

Z

!

Li tan hi

2D0 tan hi Li tan hi Lx

U0 dx :

U3;1 dx ðA9Þ

Following this algorithm, finally the averaged absorption coefficient of the rays with incident angles from 0 to arctanðLx =Li Þcan be calculated. Additionally, for the latter method—computer programming, works should be focused on obtaining these critical angles by using Eqs. (19)–(21), and then these piecewise integral functions are able to be built. Notice that the calculation expressions derived from computer programming method are in accord with what derived from mode analysis.

Z

arctanðLx =Li Þ

0

ah sin 2hdh



1 2

sin ðarctanðLx =Li ÞÞ

:

ðA11Þ

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