Sage Journals: Discover world-class research

Abstract

Boundary element solutions of the Helmholtz integral equation for the sound attenuation of acoustically lined curved pipes are presented. The curved pipes are in the form of cylindrical and toroidal coordinates. The main objective of the study is a comparative analysis of sound attenuation characteristics of lined toroidal pipes. Although toroidal pipes with circular cross-section are used extensively in industrial and agricultural applications, they have been taken little attention in literature. One reason may be the impossibility of theoretical solution since wave equation is not separable in toroidal coordinates. A numerical method should be utilized in the solution; in fact, not many researchers are familiar with toroidal coordinates on the contrary of other coordinate systems, such as cylindrical coordinates. The use of toroidal coordinates and generation of a pipe in these coordinates are illustrated herein. Sound waves in the pipes are produced by the vibrating inlet surface and resulting sound pressure distribution on the interior surfaces of side walls and anechoic exit are computed. Sound transmission loss of the pipes is represented by the ratio of inlet to exit pressures, as sound attenuation spectra. The results of cylindrical and toroidal pipes with the same cross-sectional area are compared. The sound attenuation performance of lined toroidal pipes is compared in terms of bend sharpness as a measure of curvature. The comparison is extended to cover the hard-walled and lined toroidal pipes to conclude their common and/or different acoustic behaviours.

Keywords

Boundary element method curved pipe cylindrical pipe Helmholtz integral sound attenuation toroidal pipe

1. Introduction

The sound propagation in pipes and ducts is a prominent subject in both theoretical and applied acoustics. Straight pipe problems have become a classical topic; however, curved pipes have been relatively less endeavoured due to their complex geometries (Cummings, 1974; Rostafinski, 1974; Cabelli, 1980; Keefe and Benade, 1983; Firth and Fahy, 1984; Furnell and Bies, 1989; Campos and Serrão, 2003; Dequand et al., 2003; Kim et al., 2003; Bakkali et al., 2014; Beauvais et al., 2021). A basic problem in applied acoustics is to attenuate the sound output of pipes. Although a curved shape causes attenuation at certain level (Sarıgül, 1999), the most efficient application incontestably is to line the pipe with an acoustic material. The combination of acoustic lining with curvature magnifies the sound attenuation and complexity of the problem. In literature, there are a few distinguished studies on this field (Grigor’yan, 1970; Myers and Mungur, 1976; Ko and Ho, 1977; Ko, 1979; Rostafinski, 1982; Felix and Pagneux, 2004; Yang, 2019).

The wave equation is separable only in very few coordinates, and toroidal coordinates are not one of them. Since the theoretical solution for toroidal pipes has not been accomplished, a reliable numerical technique is required. It has been shown that the boundary element method (BEM) that bases on the interior surface Helmholtz integral equation is a perfect numerical tool to perform the solution in these coordinates (Sarıgül, 1999). The solution for hard-walled pipes is relatively straightforward. However, the modelling of lined pipes includes a lot of questions and steps because they are explicit to many alternatives. Therefore, a validation study is needed to determine the most suitable BEM modelling that exposes closest results to the analytical solutions. This pre-study has been carried out for the 45°-cylindrical pipe with rectangular cross-section and the best BEM configurations have been determined (Sarıgül, 2019).

This study is mainly devoted to lined toroidal pipes. Cylindrical pipe is solved only for the comparison of its acoustic behaviour with that of the toroidal pipe. Lined toroidal pipes are examined extensively to put forward their curvature and sound attenuation relation. The paper finalizes with a comparative analysis between sound attenuation characteristics of lined and hard-walled toroidal pipes.

2. Formulation of the problem

If the domain of interest is the interior acoustic field of a body with a boundary S and a unit outward normal n , the interior surface Helmholtz integral equation may be written as $C (x) p (x) = \int_{S} {p (y) [\partial G (R, k) / \partial n (y)] + i z_{o} k u_{n} (y) G (R, k)} d s (y),$ (1)where p is the sound pressure amplitude, u_n is the normal velocity amplitude of the fluid particle at the boundary, y is any point on S and x is any point in space, R = ∣ x - y ∣, i = √(-1). k denotes the wavenumber 2πf/c where f is the frequency, c is the speed of sound and exp(i2πft) time dependence is assumed. ds denotes a differential boundary element, and G(R,k) is the free-space Green function G = exp(-ikR)/R for the Helmholtz operator in three dimensions. z₀ is the characteristic impedance of the medium z₀ = ρ₀c where ρ₀ is the density of the medium at rest. For x in the interior of the body, C( x ) = 4π; for x in the exterior of the body, C( x ) = 0 and for x on S, C( x ) is given by Cheng and Seybert (1988) $C (x) = \int_{S} [\partial (1 / R) / \partial n (y)] d s (y) .$ (2)

Before the numerical solution of the Helmholtz integral in equation (1), application of the boundary conditions is necessary. In general, if the normal velocity amplitude of the boundary is denoted by u, on acoustically hard walls $u_{n} (y) = u (y),$ (3)and on sound absorbing boundaries $u_{n} (y) = u (y) + β (y) p (y)$ (4)where β is the acoustic admittance of the absorbent material on the boundary (Suzuki, 1991).

Helmholtz integral equation (1) relates the sound pressure amplitude p and the normal velocity amplitude u subsequent to the application of the necessary boundary conditions. The numerical implementation of the interior surface Helmholtz integral equation by the boundary element method is performed sequentially applying the following steps:

1. The surface integral of the general body in equation (1) is represented by the numerical summation of the surface integrals of the discretized boundary elements. Boundary elements are quadrilateral and quadratic and have eight nodes.

2. Coordinates, pressures and normal velocities of all points on the elements are represented in terms of nodal values by using the same shape functions in accordance with the isoparametric element formulation.

3. Surface integrals of the boundary elements including singularity are handled by using the Gaussian quadrature technique. Sixteen collocation points are utilized for each element.

4. At the end of this procedure, a set of complex algebraic equations is obtained

[E - K (f)] p = H (f) u

(5)

Here, p and u are the nodal sound pressure and prescribed nodal normal velocity vectors, respectively. Matrices E , K ( f ) and H ( f ) are of size N times N, where N is the total number of nodes. E corresponding to the left-side coefficient of equation (1) is a diagonal matrix providing geometric data about the nodes. K ( f ) corresponding to the coefficient of p( y ) in equation (1) includes shape functions and nodal acoustic admittance values. H ( f ) corresponding to the coefficient of u_n( y ) in equation (1) includes shape functions.

The numerical solution is performed by an ‘in-house’ computer code which has the capability of generating equation (5) for arbitrarily shaped three-dimensional bodies possessing any type of boundary conditions. This code has been used to solve different interior and exterior problems (Dokumacı and Sarıgül, 1995; Sarıgül, 1999, 2019; Sarıgül and Kıral, 1999; Sarıgül and Seçgin, 2004; Seçgin and Sarıgül, 2010; Sarıgül et al., 2014).

3. Toroidal coordinates

Toroidal coordinates are three-dimensional orthogonal coordinate system that results from rotating two-dimensional bipolar coordinate system about the vertical axis that separates its two foci (Spiegel, 1968; Arfken, 1970). Figure 1 shows half of a torus with foci F₁ and F₂ located at distances in the x direction, and −e and e from the vertical axis. Here, R is the major radius and r is the minor radius. Toroidal coordinates are defined with parameters (u, v, ϕ), where u ϵ [0, 2π), v ϵ [0, ∞) and ϕ ϵ [0, 2π). u is the rotational angle about an axis passing through the centre of the circular section and parallel to the z axis. ϕ is the toroidal angle. v is given as (Arfken, 1970) $v = \ln \frac{ρ_{1}}{ρ_{2}}$ (6)where ρ₁ and ρ₂ are distances from a point on the torus to the two foci (Figure 2) $\begin{array}{l} ρ_{1}^{2} = {(x + e)}^{2} + y^{2} \\ ρ_{2}^{2} = {(x - e)}^{2} + y^{2} \end{array}$ (7)

Figure 1.

Representation of toroidal coordinates on a half torus.

Figure 2.

Distances ρ₁ and ρ₂ from a point on the torus to two foci.

The transformation from toroidal to Cartesian coordinates is given as (Spiegel, 1968; Arfken, 1970) $x = \frac{e s i n h v \cos ø}{c o s h v - c o s u},$ (8a) $y = \frac{e s i n h v \sin ø}{c o s h v - c o s u},$ (8b) $z = \frac{e s i n u}{c o s h v - c o s u} .$ (8c)

Some notes on toroidal coordinates may be gathered as follows:

1. Coordinate v is specific to a given torus and the same for all points on this torus.

2. Coordinate ϕ is specific to circular sections of the torus. For example, in Figure 1, the boundary points on the right circular section have ϕ = 0 and those on the left section have ϕ = π rad.

3. Different boundary points on any circular section have different u coordinates.

4. For any torus, focal distance e and v coordinate may be calculated by equating the transformations from polar and toroidal coordinates to Cartesian coordinates.

-Writing x coordinate of point A on the right section given in Figure 3 (ϕ = 0, u_A = 0), $x_{A} = R + r = \frac{e s i n h v}{c o s h v - 1}$ (9)

Figure 3.

Right circular section of the half torus.

-Writing x coordinate of point F (ϕ = 0, u_F = π rad), $x_{F} = R - r = \frac{e s i n h v}{c o s h v + 1}$ (10)

Solving equations (9) and (10) simultaneously, focal distance e and coordinate v are calculated as functions of the miner radius r and bend sharpness r/R $e = \frac{(\frac{R}{r} + 1) r}{\sqrt{{(\frac{R}{r})}^{2} - 1}},$ (11a) $v = \cos h^{- 1} (\frac{R}{r}) .$ (11b)

Taking bend sharpness of the torus r/R = 0.5, unknown values are found as $e = 1.7322 r$ (12a) $v = 1.3169$ (12b)

5. u coordinates of the other boundary points on the right circular section in Figure 3 may be calculated again by equating transformations from polar and toroidal coordinates to Cartesian coordinates. For example, for point B $x_{B} = R + r c o s \frac{π}{8} = \frac{e \sqrt{{(\frac{R}{r})}^{2} - 1}}{\frac{R}{r} - \cos u_{B}}$ (13)

For r/R = 0.5, it is found that u_B = 0.228682 rad. In a similar manner, u coordinates of the other points in Figure 3 are calculated as u_C = 0.469488 rad, u_D = 0.736365 rad and u_E = 1.047198 rad.

6. Toroidal coordinates of other boundary points located on the subsequent sections are given as follows:

-v coordinates of all points are the same.

-u coordinate of each point on the continuation of any specific point shown is the same.

-ϕ coordinates of points on the continuation of any specific point shown increase sequentially from ϕ = 0 to ϕ = π/2 rad. The increment of ϕ angles is selected in accordance with the required boundary element discretization.

4. Sound attenuation in pipes

No fluid flow is present in the pipes that have an entrance section vibrating with constant normal velocity amplitude like a piston to produce planar sound waves. The exit section behaves as if an anechoic surface representing no reflection condition. This section possesses an acoustic admittance of β = 1/z₀, where z₀ is the acoustic impedance of air. The walls of the pipes are lined with fibrous sheet mounted on a locally reacting core with impervious backing. Acoustic admittance is a function of geometry and frequency and a new value is computed for each frequency and each geometry change of the pipes. Acoustic admittance is given by Ko (1979) $β = \frac{1}{z_{0} [R_{*} (1 + i f_{*} / f_{0 *}) - i c o t (2 π f_{*} d_{*})]}$ (14)where $R_{*}$ = R’/z_o is the specific acoustic resistance ratio, $f_{*}$ = fD/c is the non-dimensional frequency, D = 2a is the diameter of the outer wall, f_o* = f_oD/c is the non-dimensional characteristic frequency, $d_{*}$ = d/D is the non-dimensional lining core depth and R’ and f_o are respective dimensional quantities. Ko (1979) has presented equation (14) for a cylindrical pipe with rectangular cross-section shown in Figure 4. Here, a is the radius of the outer wall, b is the radius of the inner wall, d is the lining core depth and ι is one-half of the pipe width.

Figure 4.

Geometry of 45°-cylindrical pipe with rectangular cross-section.

Since the cross-sectional area of the pipes is constant, sound reduction characteristics are represented by Sound Attenuation Level, that is measured by the ratio of the inlet and exit pressures $Sound Attenuation Level = 20 \log (\frac{p_{i n l e t}}{p_{e x i t}}) d B .$ (15)

Sound pressure is constant throughout the inlet surface; however, nodal pressures of the exit have quite different values. The average value is used for the exit pressure as performed by Sarıgül (1999, 2019).

In the present study, pipes are handled in three headings:

(1) 90°-cylindrical pipe with rectangular cross-section.

(2) 90°-toroidal pipe with circular cross-section with a definite curvature.

(3) 90°-toroidal pipes with circular cross-section and varying curvatures.

For all pipes, the boundary element modelling features proposed for lined cylindrical pipes by Sarıgül (2019) are used. Basically, this paper focuses on the third heading. The numerical solution of toroidal pipes is essential since they have no analytical solution on the contrary of cylindrical coordinates. In the study, 90° pipes with a shape of quarter torus are examined due to their common use in industry as right-angle bends. The paper basically aims to obtain the relation between the bend sharpness and the sound attenuation characteristics of lined toroidal pipes. This relation has been put forward for the same hard-walled pipes by Sarıgül (1999). The question to be answered here is that ‘the trend is similar or some new characteristics are gained with lining?’

4.1. 90°-cylindrical pipe with rectangular cross-section

A 90°-cylindrical pipe with rectangular cross-section is shown in Figure 5. The discretization is the same as the model with 384 elements and 1186 nodes applied for the 45°-pipe by Sarıgül (2019). In this model, side walls have 256 elements and 800 nodes; each of the inlet and outlet surfaces has 64 elements and 193 nodes. 90°-pipe model is constructed by extending the side elements of the 45° pipe. Also, in the analyses, the same numerical parameters are used (a = 10’’, b = 7”, ι = 1.5’’, d = 1’’, f_o = 15000 Hz, c = 1120 ft/s, d* = 0.05, f_o* = 22.32, R* = 1.5).

Figure 5.

90°-cylindrical pipe with rectangular cross-section.

Using the same configuration of the selected model by Sarıgül (2019) (A3), the sound attenuation spectrum of this pipe is obtained and presented in Figure 6 with the spectra of the 45° pipe. Spectrum of the 90° pipe continues smoothly and effectively up to almost f * = 10, whereas f * = 6.5 has been possible for 45° pipe as discussed by Sarıgül (2019). This is the result of two contradicting phenomena: Coarser discretization of the side walls and longer lined path for the acoustical waves, which is more dominant factor. On the other hand, this pipe has almost twice of the maximum attenuation of the 45° pipe.

Figure 6.

Sound attenuation spectra of cylindrical pipes with rectangular cross-section (- 45° analytical, - - 45° BEM and . . 90° BEM).

4.2. 90°-toroidal pipe with circular cross-section with a definite curvature

A 90°-toroidal pipe with circular cross-section is seen in Figure 7 and its geometry is given in Figure 8. Here, r is the pipe radius that includes side nodes and R is the median arc radius. For this pipe, d = 1’’ is taken as in the previous pipes; and R = 8.5’’ and r = 2.2568’’ are calculated. The pipe is modelled with the same cross-sectional area of the 90°-cylindrical pipe with rectangular cross-section. The pipe curvature, defined as bend sharpness, is r/R = 0.2655.

Figure 7.

90°-toroidal pipe with circular cross-section.

Figure 8.

Geometry of 90°-toroidal pipe with circular cross-section.

The pipe is modelled by using 256 elements and 800 nodes on the side walls again and 60 elements and 205 nodes on each of the inlet and outlet surfaces. Therefore, 376 elements and 1210 nodes are used in the numerical implementation. The discretization of inlet–outlet surfaces is shown in Figure 9. The attenuation spectrum of this pipe is presented in Figure 10 comparatively with that of the 90°-cylindrical pipe with rectangular cross-section. No apparent difference is observed between the two spectra in trend, in the position and level of the maximum attenuation. However, the cylindrical pipe exposes smoother variation throughout the entire spectrum. Although the discretization sensitivity of both types is quite similar, the spectrum of the toroidal pipe has some imperfections. These fluctuations may be considered as small for lower frequencies, whereas large for higher frequencies. Therefore, this spectrum is extended up to $f_{*}$ = 8. It is seen that a finer discretization of the toroidal pipe is necessary particularly for the inlet–outlet surfaces.

Figure 9.

Discretization of the inlet–outlet surfaces of 90°-toroidal pipe with circular cross-section (60 elements and . 205 nodes).

Figure 10.

Sound attenuation spectra of 90°-pipes (. . cylindrical with rectangular cross-section and - - toroidal with circular cross-section).

4.3. 90°-toroidal pipes with circular cross-section and different curvatures

These pipes have the same general shape as that of the second heading. In this section, the curvature r/R of the pipes is changed in a wide region. In order to examine the curvature–sound attenuation relation for the lined pipes, first of all, a more sensitive pipe model is generated. This model has finer meshes on input–output surfaces as shown in Figure 11. Each of these surfaces has 128 elements and 385 nodes. Therefore, total number of elements and nodes attain to 512 and 1570, respectively. By applying this model, different pipes with bend sharpness values from 0.1 to 0.9 with 0.1 increments in r/R ratio are constructed.

Figure 11.

Final discretization of the inlet–outlet surfaces of 90°-toroidal pipe with circular cross-section (128 elements and . 385 nodes).

In the generation of these geometries, two parameters are fixed and equal to those of the 90°-cylindrical pipe and toroidal pipe with r/R = 0.2655.

1. Median arc radius R = 8.5’’.

2. Relative depth of the lining (lining core depth/pipe radius) d/r = 0.4431.

The other geometrical and related admittance parameters change with the variation of bend sharpness, as listed in Table 1. It is obvious that since R is constant, increase in bend sharpness means increase in r.

Table 1.

The variation of pipe parameters with bend sharpness.

r/R	r	d	a	$d_{*}$	$f_{o *}$
0.1	0.85	0.3766	9.1617	0.0206	20.4502
0.2	1.70	0.7533	9.8234	0.0383	21.9272
0.3	2.55	1.1299	10.4850	0.0539	23.4041
0.4	3.40	1.5065	11.1467	0.0676	24.8811
0.5	4.25	1.8832	11.8084	0.0797	26.3581
0.6	5.10	2.2598	12.4701	0.0906	27.8350
0.7	5.95	2.6364	13.1318	0.1004	29.3120
0.8	6.80	3.0131	13.7935	0.1092	30.7890
0.9	7.65	3.3897	14.4551	0.2341	32.2659

The sound attenuation spectra with respect to non-dimensional frequency $f_{*}$ are presented in Figure 12 for pipes with different bend sharpness. Table 2 presents a numerical representation of the magnitude and position of peak attenuations and the computed frequency range of attenuations. Table 2 also includes non-dimensional wave number $k r (k r = f_{*} π r / a)$ equivalents of $f_{*}$ values since frequency axis of the attenuation spectra of hard-walled pipes has been given in kr values by Sarıgül (1999).

Figure 12.

Comparative sound attenuation spectra of toroidal pipes with different bend sharpness r/R.

Table 2.

The variation of sound attenuation with bend sharpness for lined toroidal pipes.

r/R	Max (dB)	Max $(f_{*})$	Max $(k r)$	Attenuation range $(f_{*}) (k r)$
0.1	67.49	6.29	1.83	0.60–10.00	0.18–2.92
0.2	53.01	4.22	2.29	0.60–10.00	0.33–5.44
0.3	43.15	3.16	2.41	0.60–8.08	0.46–6.17
0.4	37.67	2.35	2.25	0.70–6.53	0.67–6.26
0.5	32.77	1.80	2.04	0.70–5.60	0.79–6.33
0.6	30.39	1.48	1.90	0.70–4.93	0.90–6.33
0.7	29.83	1.27	1.81	0.10–4.41	0.14–6.28
0.8	33.60	1.17	1.81	0.10–3.84	0.15–5.95
0.9	31.28	1.06	1.76	0.10–3.69	0.17–6.14

The comparison among the spectra in Figure 12 puts forward the following features of lined toroidal pipes:

1. As the bend sharpness increases, the magnitude of the peak attenuation decreases.

2. This decrease is rapid for pipes with smaller bend sharpness and slow for larger ones.

3. The maximum attenuation occurs at higher non-dimensional frequencies for pipes with smaller bend sharpness.

4. As the bend sharpness increases, these frequencies shift to smaller values, rapidly at first, slowly afterwards.

5. Sound attenuation spectra of pipes with small bend sharpness have a smooth distribution. The attenuation capacity of these pipes also has a large frequency range from zero to ten.

6. As the bend sharpness increases, the upper limit of attenuation range decreases towards smaller frequencies.

Although Table 2 shows the aforementioned trends apparently, the peak attenuation values computed for pipes with high bend sharpness somewhat violate the decrease trend. This is due to the computational inaccuracy regarding to the increase in boundary element size.

Specifically, since in the present study, increase in bend sharpness r/R means directly increase in pipe radius r, the previous inferences may be evaluated as the relations between pipe radius and sound attenuation.

Table 3 includes the numerical values of the sound attenuation spectra for the hard-walled pipes presented by Sarıgül (1999). In that study, the variation in bend sharpness has been provided again by changing r and taking R as constant. The comparisons between lined and hard-walled toroidal pipes in Table 2 and Table 3 put forward the following results:

1. Addition of lining greatly increases the attenuation capability of pipes both in level and range.

2. In the hard-walled pipes, as the bend sharpness increases, the peak attenuation level increases on the contrary of lined pipes. This means pipes with wider radii have better sound attenuation. This is a conflicting characteristic between lined and hard-walled pipes.

3. In the hard-walled pipes, the position of peak attenuation shifts to higher kr values, as the bend sharpness increases. For the lined pipes, there isn’t a uniform trend related to kr; however, there is an obvious opposite trend in terms of f_*.

4. For the hard-walled pipes, attenuation range increases and shifts towards the higher kr values, as the bend sharpness increases. There is no specific trend for lined pipes in terms of kr; however, an opposite trend is present with regard to f_*.

5. It is seen that the relation between pipe dimension and frequency is more sensitively represented by the non-dimensional frequency f_*, compared to the non-dimensional wavenumber kr. This is because f_* is related to a, radius of the outer wall with respect to global origin that is more slowly affected by changing the pipe radius r. kr being a linear function of the pipe radius shows a direct rapid change with r and therefore cannot reflect properly the frequency variation.

Table 3.

The variation of sound attenuation with bend sharpness for hard-walled toroidal pipes in Sarıgül (1999).

r/R	Max (dB)	Max $(k r)$	Attenuation range $(k r)$
0.1	—	—	—
0.2	—	—	—
0.3	—	—	—
0.4	0.56	1.950	1.90–1.96
0.5	2.65	1.995	1.00–2.15
0.6	4.62	2.090	1.20–2.35
0.7	6.19	2.210	1.90–2.65
0.8	7.18	2.340	1.90–2.90
0.9	—	—	—

5. Conclusion

The present study had the objective of examining sound attenuation characteristics of lined curved pipes, specifically 90°-toroidal pipes. The study put forward basic acoustical characteristics of lined toroidal pipes, the effects of dimension on sound attenuation and the difference of lined and hard-walled pipes in terms of sound attenuation behaviour. A brief comparative analysis on the effect of cross-sectional shape was also comprised. In order to expose these characteristics comparatively, solutions of a 90°-cylindrical pipe and many 90°-toroidal pipes with different dimensions leading to following inferences were accomplished.

1. The sound attenuation performance of toroidal pipes is almost the same as that of the cylindrical pipes with the same cross-sectional area. That is, there is almost no effect of the cross-sectional shape on the sound attenuation.

2. As the pipes get narrower, the sound attenuation performance of lined toroidal pipes increases both in level and range. Lining of a narrow pipe with an acoustical material would yield a better sound attenuation performance.

3. In fact, the attenuation in widest pipes lined with fibrous sheet mounted on a locally reacting core with impervious backing is almost 30 dB and may not be considered as low. However, the frequency range of this performance is quite limited and attention should be paid in practical applications.

4. Different lining materials may be applied to examine the effect of the impedance model. This application may alter the level of attenuation; however, the general behaviour of toroidal pipes would not be much affected.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

Ethical and consent statements

1. This research has been conducted in an ethical and responsible manner and complies with all relevant legislation.

2. The results are presented clearly,honestly and without fabrication,falsification or inappropriate data manipulation.

3. The submitted work is original,is not plagiarized and has not been published elsewhere.

ORCID iD

A Saide Sarigul

References

Arfken

(1970) Mathematical Methods for Physicists. 2nd edn. Academic Press.

Bakkali

Lhémery

Baronian

, et al. (2014) A modal formulation for the propagation of guided waves in straight and curved pipes and the scattering at their junction. Journal of Physics: Conference Series 498: 012010.

Beauvais

Pelat

Gautier

, et al. (2021) Inverse identification of the acoustic pressure inside a U-shaped pipe line based on acceleration measurements. In: Mechanical Systems and Signal Processing: Elsevier.

Cabelli

(1980) The acoustic characteristics of duct bends. Journal of Sound and Vibration 68(3): 369–388.

Campos

LMBC

Serrão

PGTA

(2003) On helicoidal rectangular coordinates for the acoustics of bent and twisted tubes. Wave Motion 38: 53–66.

Cheng

CYR

Seybert

(1988) Recent applications of the boundary element method to problems in acoustics. SAE Paper No. 870997: 3.165-3.174.

Cummings

(1974) Sound transmission in curved duct bends. Journal of Sound and Vibration 35(4): 451–477.

Dequand

Hulshoff

Auregan

, et al. (2003) Acoustics of 90 degree sharp bends. Part 1: low-frequency acoustical response. Acta Acustica united with Acustica 89(6): 1025–1037.

Dokumacı

Sarıgül

Seçgin

Ertunç

, et al. (1995) Analysis of the near field acoustic radiation characteristics of two radially vibrating spheres by the Helmholtz Integral Equation Formulation and a critical study of the efficacy of the “CHIEF” overdetermination method in two-body problems. Journal of Sound and Vibration 187(5): 781–798.

10.

Felix Pagneux

(2004) Sound attenuation in lined bends. Journal of the Acoustical Society of America 116(4): 1921–1931.

11.

Firth

Fahy

(1984) Acoustic characteristics of circular bends in pipes. Journal of Sound and Vibration 97(2): 287–303.

12.

Furnell

Bies

(1989) Matrix analysis of acoustic wave propagation within curved ducting systems. Journal of Sound and Vibration 132(2): 245–263.

13.

Grigor’yan

(1970) Soundproofing by means of ducts with curved porous walls. Soviet Physics - Acoustics 16(2): 192–196.

14.

Keefe

Benade

(1983) Wave propagation in strongly curved ducts. Journal of the Acoustical Society of America 74(1): 320–332.

15.

Kim

Lee

Setoguchi

(2003) Study of the impulse wave discharged from the exit of a right-angle pipe bend. Journal of Sound and Vibration 259(5): 1147–1161.

16.

(1977) Sound attenuation in acoustically lined curved ducts in the absence of fluid flow. Journal of Sound and Vibration 53(2): 189–201.

17.

(1979) Three-dimensional acoustic waves propagating in acoustically lined cylindrically curved ducts without fluid flow. Journal of Sound and Vibration 66(2): 165–179.

18.

Myers

Mungur

(1976) Sound propagation in curved ducts. AIAA Progress in Astronautics and Aeronautics 44: 347–362.

19.

Rostafinski

(1974) Analysis of propagation of waves of acoustic frequencies in curved ducts. Journal of the Acoustical Society of America 56: 11–15.

20.

Rostafinski

(1982) Propagation of long waves in acoustically treated curved ducts. Journal of the Acoustical Society of America 71(1): 36–41.

21.

Sarıgül

(1999) Sound attenuation characteristics of right-angle pipe bends. Journal of Sound and Vibration 228(4): 837–844.

22.

Sarıgül

Kıral

(1999) Interior acoustics of a truck cabin with hard and impedance surfaces. Engineering Analysis with Boundary Elements 23: 769–775.

23.

Sarıgül

Seçgin

(2004) A study on the applications of the acoustic design sensitivity analysis of vibrating bodies. Applied Acoustics 65: 1037–1056.

24.

Sarıgül

(2014) Sound radiation in a half space with impedance surface. Journal of Vibration and Control 16: 2552–2560.

25.

Sarıgül

Seçgin

Ertunç

(2019) Boundary element modeling of sound attenuation in acoustically lined curved pipes. Journal of Theoretical and Computational Acoustics 27(3): 1850046.

26.

Seçgin

Sarıgül

(2010) An efficient sound source determination process based on half-space boundary element method (BEM). International Journal of Mechanics and Materials in Design 6(1): 17–25.

27.

Spiegel

(1968) Mathematical Handbook of Formulas and Tables (Schaum's Outline Series). Mc-Graw Hill Book Company.

28.

Suzuki

(1991) Applications in the Automotive industry. In: Ciskowski

Brebbia

(eds), Boundary Element Methods in Acoustics. Southampton: Computational Mechanics Publications. Chap.7.

29.

Yang

(2019) Acoustic attenuation of a curved duct containing a curved axial microperforated panel. Journal of the Acoustical Society of America 145(1): 501–511.