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This chapter begins with the threedimensional (3D) Helmholtz equation governing acoustic wave propagation in an infinite rigidwall elliptical cylindrical waveguide carrying a uniform mean flow. The 3D acoustic pressure field is obtained as modal summation of rigidwall transverse modes in terms of the angular and radial Mathieu functions and complex exponentials. A wellknown algorithm is presented for computing the expansion coefficients of even and odd angular Mathieu functions based on a set of infinite recurrence relations formulated as an algebraic eigenvalue problem. The modified or radial Mathieu functions are computed using a rapidly converging series of products of Bessel function. Rigidwall condition is imposed at the elliptical boundaries by setting the derivative of modified Mathieu functions to zero, and rootbracketing in conjunction with the bisection method is used to numerically compute its parametric zeros q. The corresponding nondimensional resonance frequencies of the transverse (rigidwall) radial, even and odd circumferential/crossmodes of the elliptical waveguide are tabulated for aspectratios ranging from \(D_{2} /D_{1} = 0.01\,{\text{to}}\,1.0\). The tables show that for a highly eccentric ellipse, i.e., \(e \to 1\) (small aspectratio), the resonance frequencies of the even modes are significantly smaller than those of its odd counterpart. With an increase in aspectratio, the resonance frequencies of odd modes gradually approach those of the even modes, and for ellipse with \(e \to 0{\text{ or }}D_{2} /D_{1} \to 1\) the even and odd modes coalesce to the circular duct mode. The mode shapes corresponding to the first few radial, even and odd modes are presented for a few aspectratios whereby one can visualize changes in modal pressure distribution as the ellipse approaches a circle. Development of interpolation polynomials which facilitates an easy evaluation of the resonance frequencies for a given mode type is a useful outcome in engineering acoustics.
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The nondimensional even and odd resonance frequencies of the vibration of a clamped elliptical membrane denoted by
\(\left. {\left( {k_{0} \displaystyle\frac{{D_{1} }}{2}} \right)} \right_{{(m,n){\text{ Even }}}}\) and
\(\left. {\left( {k_{0} \displaystyle\frac{{D_{1} }}{2}} \right)} \right_{{(m,n){\text{ odd }}}},\) respectively, were found by numerically solving Eqs. (
2.62) and (
2.63) for the parametric zeros
\(q_{m,n}\) and
\(\overline{q}_{mn} ,\) respectively, using the rootbracketing and bisection techniques explained in Sect.
2.5. Note that the parametric zeros computed here pertain to the Dirichlet boundary condition specified by Eqs. (
2.62) and (
2.63), and they are always interlaced with the parametric zeros corresponding to the Neumann or rigidwall duct condition specified by Eqs. (
2.49) and (
2.50). The interlacing of parametric zeros is demonstrated by Fig.
2.3b which shows the variation of the functions
\(Ce_{0} \left( {\xi = \xi_{0} ,q} \right)\) and
\(\frac{{\text{d}}}{{{\text{d}}\xi }}Ce_{0} \left( {\xi = \xi_{0} ,q} \right)\) with the parameter
q for
\(D_{2} /D_{1} = 0.6.\)
Table
2.29 presents the nondimensional resonance frequencies of the radial mode
\(Ce_{0} \left( {\xi ,q_{0,n} } \right)ce_{0} \left( {\eta ,q_{0,n} } \right)\) and the even circumferential modes
\(Ce_{m} \left( {\xi ,q_{m,n} } \right)ce_{m} \left( {\eta ,q_{m,n} } \right)\) for the first seven orders, i.e.,
\(m = 1,2, \ldots 7\) , while Table
2.30 presents the nondimensional resonance frequencies of the first eight odd circumferential modes up to four decimal places. For each mode type, the first four zeros are presented, i.e.,
\(n = 1,2,3,4\). Note that a complete range of aspectratio given by
is considered in Tables
2.29 and
2.30, and the nondimensionalization variable
D
_{1} is kept constant in these Tables.
For certain aspectratio, the nondimensional resonance frequencies shown in Tables
2.29 and
2.30 were compared with the values available in the literature for a clamped elliptical membrane [
25,
26,
31] whereby an excellent agreement was observed. The graphical variation of the resonance frequencies presented in Tables
2.29 and
2.30 with aspectratio is not included for brevity; however, it may be mentioned that the graphs for both even and odd circumferential and crossmodes as well as radial modes have a concave nature. As anticipated, the concavity is more pronounced for the odd mode graphs as compared to their even mode counterparts. Here, we present a 9th degree leastsquares interpolating polynomial using which one can accurately predict the nondimensional resonance frequency values for any aspectratio within the range
\(D_{2} /D_{1} = [0.3,1]\). It is given by
where
and the coefficients {
b
_{0},
b
_{1},
b
_{2}, …
b
_{9}} for the first few radial, even and odd modes are listed in Tables
2.31 and
2.32.
Figure
2.12a–c presents the mode shapes of the first three radial modes given by
\(Ce_{0} \left( {\xi ,q_{0,1} } \right)ce_{0} \left( {\eta ,q_{0,1} } \right),Ce_{0} \left( {\xi ,q_{0,2} } \right)ce_{0} \left( {\eta ,q_{0,2} } \right)\) and
\(Ce_{0} \left( {\xi ,q_{0,3} } \right)ce_{0} \left( {\eta ,q_{0,3} } \right),\) respectively, for the aspectratio
D
_{2}/
D
_{1} = 0.5. Note here that the fundamental (0, 1)
e mode exhibits a gradual variation across the elliptical crosssection and is characterized by a maximum at the center and zero displacement along the elliptical boundaries—indeed, this feature is in contrast with the fundamental or plane wave mode of a rigidwall elliptical waveguide for which the acoustic pressure field is uniform across the crosssection. Therefore, if the wall of an elliptical waveguide is not rigid, rather, if it is lined with a sound absorbent or dissipative material, then the plane wave mode does not exist. Rather, the fundamental mode exhibits variation across the cross section and has a dissipative part which implies it cannot propagate without attenuation [
42]. In fact, by imposing the boundary conditions given by Eqs. (
2.62) and (
2.63), one obtains the pressurerelease modes of an elliptical duct [
47], while in case of lined ducts, these modes are more popularly referred to as the softwall modes [
48]. The (0, 2)
e and (0, 3)
e radial mode shapes exhibit one and two pressure nodal ellipses, respectively, as their respective eigenfunctions are a solution of the SL problem. The set of figures in the 2nd column, i.e., Fig.
2.12d–f, presents the first three even circumferential mode shapes given by
\(Ce_{1} \left( {\xi ,q_{1,1} } \right)ce_{1} \left( {\eta ,q_{1,1} } \right),\)
\(Ce_{2} \left( {\xi ,q_{2,1} } \right)ce_{2} \left( {\eta ,q_{2,1} } \right),\) and
\(Ce_{3} \left( {\xi ,q_{3,1} } \right)ce_{3} \left( {\eta ,q_{3,1} } \right),\) respectively, while the set of figures in the 3rd column, i.e., Fig.
2.12g–i, presents the first three odd circumferential mode shapes
\(Se_{1} \left( {\xi ,\bar{q}_{1,1} } \right)se_{1} \left( {\eta ,\bar{q}_{1,1} } \right),\)
\(Se_{2} \left( {\xi ,\bar{q}_{2,1} } \right)se_{2} \left( {\eta ,\bar{q}_{2,1} } \right),\) and
\(Se_{3} \left( {\xi ,\bar{q}_{3,1} } \right)se_{3} \left( {\eta ,\bar{q}_{3,1} } \right),\) respectively. The even circumferential modeshapes shown in Fig.
2.12d–f exhibit pressure nodal hyperbolas which divide the elliptical crosssection in regions of alternating phase, and they are found to be similar to their counterpart mode shapes shown in Figs.
2.6a–c Similarly, the odd circumferential mode shapes shown in Fig.
2.12(g–i) resemble their counterpart mode shapes presented in Figs.
2.6d–f, respectively, as may be observed from the pressure nodal hyperbola pattern. An apparent difference between the Dirichlet and Neumann mode shapes is that the former is characterized by zero value of the function field along the elliptical boundaries and a local maximum is formed at the center of the same phase region, while the latter mode shape is characterized by formation of a local maximum near the elliptical boundary.
$$\frac{{D_{2} }}{{D_{1} }} = 0.1,0.2,(0.1), \ldots ,0.8,0.9,1({\text{Circular}}),$$
(2.78)
Table 2.29
Nondimensional resonance frequencies of
radial and
even circumferential/crossmodes of an elliptical membrane clamped at the boundaries for a range of aspectratio
Table 2.30
Nondimensional resonance frequencies of
odd circumferential/crossmodes of an elliptical membrane clamped at the boundaries for a range of aspectratio
$$\begin{aligned} \left. {\left( {k_{0} \frac{{D_{1} }}{2}} \right)} \right_{m,n} & = b_{0} + b_{1} \beta + b_{2} \beta^{2} + b_{3} \beta^{3} + b_{4} \beta^{4} + b_{5} \beta^{5} \\ & \quad + b_{6} \beta^{6} + b_{7} \beta^{7} + b_{8} \beta^{8} + b_{9} \beta^{9,} \\ \end{aligned}$$
(2.79)
$$\beta = \left( {\frac{{D_{2} }}{{D_{1} }}  0.55} \right)/0.30277,$$
(2.80)
Table 2.31
Coefficients of a 9th degree leastsquares polynomial fit for interpolating the nondimensional resonance frequency of the first radial mode and first few even and odd circumferential modes
Coefficients

(0, 1)
e

(1, 1)
e

(1, 1)
o

(2, 1)
e

(2, 1)
o

(3, 1)
e

(3, 1)
o

(4, 1)
e

(4, 1)
o


b
_{0}

3.5119

4.7710

6.2922

6.1178

7.4391

7.5164

8.6476

8.9459

9.9036

b
_{1}

−1.4521

−1.2974

−3.0813

−1.1661

−2.9726

−1.0714

−2.8590

−1.0139

−2.7557

b
_{2}

0.84466

0.82047

1.6823

0.75784

1.6895

0.67326

1.6672

0.57901

1.6147

b
_{3}

−0.46833

−0.47822

−0.92184

−0.47900

−0.92911

−0.47743

−0.94518

−0.4825

−0.9656

b
_{4}

0.43312

0.44103

0.86308

0.44395

0.86276

0.44012

0.86227

0.38452

0.86215

b
_{5}

−0.23705

−0.23904

−0.47527

−0.24689

−0.47422

−0.25547

−0.47438

−0.29668

−0.47725

b
_{6}

−0.20239

−0.20374

−0.40337

−0.20312

−0.40346

−0.2189

−0.40499

−0.19705

−0.4088

b
_{7}

0.11123

0.11242

0.22204

0.11239

0.22186

0.10085

0.22141

0.12699

0.22143

b
_{8}

0.16820

0.16824

0.33619

0.16634

0.33622

0.17245

0.33616

0.17153

0.3374

b
_{9}

−0.09259

−0.092759

−0.18507

−0.093195

−0.18505

−0.08818

−0.18511

−0.09225

−0.18471

Table 2.32
Coefficients of a 9th degree leastsquares polynomial fit for interpolating the nondimensional resonance frequency of the second radial mode and first few crossmodes
Coefficients

(0, 2)
e

(1, 2)
e

(1, 2)
o

(2, 2)
e

(2, 2)
o

(3, 2)
e

(3, 2)
o


b
_{0}

9.1312

10.249

11.979

11.416

13.083

12.624

14.224

b
_{1}

−4.6619

−4.5561

−6.2391

−4.4318

−6.1401

−4.2945

−6.0252

b
_{2}

2.5248

2.5585

3.3613

2.5850

3.3953

2.6088

3.4253

b
_{3}

−1.3746

−1.3606

−1.8346

−1.3422

−1.8228

−1.3033

−1.8115

b
_{4}

1.2995

1.3021

1.7304

1.3397

1.7359

1.4313

1.7426

b
_{5}

−0.71097

−0.70654

−0.94948

−0.67073

−0.9467

−0.62171

−0.94410

b
_{6}

−0.60458

−0.58434

−0.80615

−0.59082

−0.80378

−0.67581

−0.80515

b
_{7}

0.33410

0.34439

0.44457

0.32129

0.44507

0.24896

0.44198

b
_{8}

0.50566

0.4985

0.67258

0.49323

0.67153

0.5099

0.66964

b
_{9}

−0.27712

−0.28161

−0.37010

−0.28006

−0.37061

−0.26235

−0.37101

×
1.
go back to reference E. Mathieu, Memoire sur le movement vibratoire d’une membrane de forme elliptique. J. Math. Pures Appl. 13, 137 (1868) MATH E. Mathieu, Memoire sur le movement vibratoire d’une membrane de forme elliptique. J. Math. Pures Appl.
13, 137 (1868)
MATH
2.
go back to reference E. Heine, Handbuch der Kugelfunktionen, 2 Vols. (1878/1881) E. Heine,
Handbuch der Kugelfunktionen, 2 Vols. (1878/1881)
3.
go back to reference G. Floquet, Sur les equations differentielles lineaires. Ann. De l’Ecole Normale Superieure 12, 47 (1883) CrossRef G. Floquet, Sur les equations differentielles lineaires. Ann. De l’Ecole Normale Superieure
12, 47 (1883)
CrossRef
4.
go back to reference G.W. Hill, Mean equation of the lunar perigee. Acta. Math. 8, 1 (1886) MathSciNetCrossRef G.W. Hill, Mean equation of the lunar perigee. Acta. Math.
8, 1 (1886)
MathSciNetCrossRef
5.
go back to reference N.W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1947). MATH N.W. McLachlan,
Theory and Application of Mathieu Functions (Oxford University Press, London, 1947).
MATH
6.
go back to reference J. Meixner, F.W. Schäfke, Mathieusche Funktionen Und Sphäroidfunktionen (SpringerVerlag, Berlin, 1954). CrossRef J. Meixner, F.W. Schäfke,
Mathieusche Funktionen Und Sphäroidfunktionen (SpringerVerlag, Berlin, 1954).
CrossRef
7.
go back to reference M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1972). MATH M. Abramowitz, I.A. Stegun,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1972).
MATH
8.
go back to reference E.L. Ince, Researches into the characteristic numbers of the mathieu equation. Proc. Roy. Soc. Edim. 46, 20–29 (1926) MATH E.L. Ince, Researches into the characteristic numbers of the mathieu equation. Proc. Roy. Soc. Edim.
46, 20–29 (1926)
MATH
9.
go back to reference E.T. Kirkpatrick, Tables of values of the modified Mathieu functions. Math. Comput. 14, 118–129 (1970) MathSciNetCrossRef E.T. Kirkpatrick, Tables of values of the modified Mathieu functions. Math. Comput.
14, 118–129 (1970)
MathSciNetCrossRef
10.
go back to reference S. Goldstein, On the asymptotic expansion of the characteristic number of the Mathieu equation. Proc. Roy. Soc. Edim. 49, 203–223 (1929) S. Goldstein, On the asymptotic expansion of the characteristic number of the Mathieu equation. Proc. Roy. Soc. Edim.
49, 203–223 (1929)
11.
go back to reference J. Canosa, Numerical solution of Mathieu’s equation. J. Comput. Phys. 7, 255–272 (1971) CrossRef J. Canosa, Numerical solution of Mathieu’s equation. J. Comput. Phys.
7, 255–272 (1971)
CrossRef
12.
go back to reference D. Frenkel, R. Portugal, Algebraic methods to compute Mathieu functions. J. Phys. Math Gen 34, 3541–3551 (2001) MathSciNetCrossRef D. Frenkel, R. Portugal, Algebraic methods to compute Mathieu functions. J. Phys. Math Gen
34, 3541–3551 (2001)
MathSciNetCrossRef
13.
go back to reference F.A. Alhargan, A complete method for the computations of Mathieu characteristic numbers of integer orders. SIAM Rev. 38, 239–255 (1996) MathSciNetCrossRef F.A. Alhargan, A complete method for the computations of Mathieu characteristic numbers of integer orders. SIAM Rev.
38, 239–255 (1996)
MathSciNetCrossRef
14.
go back to reference F.A. Alhargan, Algorithms for the computation of all Mathieu functions of integer orders. ACM Trans. Math. Software 26, 390–407 (2000) CrossRef F.A. Alhargan, Algorithms for the computation of all Mathieu functions of integer orders. ACM Trans. Math. Software
26, 390–407 (2000)
CrossRef
15.
go back to reference F.A. Alhargan, Algorithm 804: subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Softw. 26, 408–414 (2000) CrossRef F.A. Alhargan, Algorithm 804: subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Softw.
26, 408–414 (2000)
CrossRef
16.
go back to reference D. Clemm, Algorithm 352 Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12, 399–408 (1969) CrossRef D. Clemm, Algorithm 352 Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM
12, 399–408 (1969)
CrossRef
17.
go back to reference S.R. Rengarajan, J.E. Lewis, Mathieu functions of integral order and real arguments. IEEE Trans. Microw. Theory and Tech. MTT 28, 276–277 (1980) S.R. Rengarajan, J.E. Lewis, Mathieu functions of integral order and real arguments. IEEE Trans. Microw. Theory and Tech. MTT
28, 276–277 (1980)
18.
go back to reference N. Toyama, K. Shogen, Computation of the value of the even and odd Mathieu functions of order n for a given parameter s and an argument x. IEEE T. Antenn. Propag. AP 32, 537–539 (1994) N. Toyama, K. Shogen, Computation of the value of the even and odd Mathieu functions of order
n for a given parameter
s and an argument
x. IEEE T. Antenn. Propag. AP
32, 537–539 (1994)
19.
go back to reference W. Leeb, Algorithm 537 Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Soft. 5, 112–117 (1979) CrossRef W. Leeb, Algorithm 537 Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Soft.
5, 112–117 (1979)
CrossRef
20.
go back to reference R.B. Shirts, Algorithm 721 MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order. ACM Trans. Math. Soft. 19, 391–406 (1993) CrossRef R.B. Shirts, Algorithm 721 MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order. ACM Trans. Math. Soft.
19, 391–406 (1993)
CrossRef
21.
go back to reference J.J. Stamnes, B. Spjelkavik, New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders. Pure Appl. Opt. 4, 251–262 (1995) CrossRef J.J. Stamnes, B. Spjelkavik, New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders. Pure Appl. Opt.
4, 251–262 (1995)
CrossRef
22.
go back to reference S. Zhang, J. Jin, Computation of Special Functions (Wiley, New York, 1996). S. Zhang, J. Jin,
Computation of Special Functions (Wiley, New York, 1996).
23.
go back to reference C. Julio, GutiérrezVega, Formal Analysis of the Propagation of Invariant Optical Fields in Elliptic Coordinates, Ph. D. Thesis, INAOE, México (20000 C. Julio,
GutiérrezVega, Formal Analysis of the Propagation of Invariant Optical Fields in Elliptic Coordinates, Ph. D. Thesis, INAOE, México (20000
24.
go back to reference E. Cojocaru, Mathieu Functions Computational Toolbox Implemented in MATLAB. https://arxiv.org/abs/0811.1970, https://www.mathworks.com/matlabcentral/fileexchange/22081mathieufunctionstoolboxv10 E. Cojocaru,
Mathieu Functions Computational Toolbox Implemented in MATLAB.
https://arxiv.org/abs/0811.1970,
https://www.mathworks.com/matlabcentral/fileexchange/22081mathieufunctionstoolboxv10
25.
go back to reference B.A. Troesch, H.R. Troesch, Eigenfrequencies of an elliptic membrane. Math. Comp. 27, 755–765 (1973) MathSciNetCrossRef B.A. Troesch, H.R. Troesch, Eigenfrequencies of an elliptic membrane. Math. Comp.
27, 755–765 (1973)
MathSciNetCrossRef
26.
go back to reference G. Chen, P.J. Morris, J. Zhou, Visualization of special eigenmode shapes of a vibrating elliptical membrane. SIAM Rev. 36, 453–469 (1994) MathSciNetCrossRef G. Chen, P.J. Morris, J. Zhou, Visualization of special eigenmode shapes of a vibrating elliptical membrane. SIAM Rev.
36, 453–469 (1994)
MathSciNetCrossRef
27.
go back to reference J.B. Keller, S.I. Rubinow, Asymptotic solution of eigenvalue problems. Ann. Phys. 9, 24–75 (1960) CrossRef J.B. Keller, S.I. Rubinow, Asymptotic solution of eigenvalue problems. Ann. Phys.
9, 24–75 (1960)
CrossRef
28.
go back to reference J.C. GutiérrezVega, R.M. RodríguezDagnino, M.A. MenesesNava, S. ChávezCerda, Mathieu functions, a visual approach. Am. J. Phys. 71, 233–242 (2003) CrossRef J.C. GutiérrezVega, R.M. RodríguezDagnino, M.A. MenesesNava, S. ChávezCerda, Mathieu functions, a visual approach. Am. J. Phys.
71, 233–242 (2003)
CrossRef
29.
go back to reference S. Ancey, A. Folacci, P. Gabrielli, Whisperinggallery modes and resonances of an elliptic cavity. J. Phys. A Math. Gen. 34, 1341–1359 (2001) MathSciNetCrossRef S. Ancey, A. Folacci, P. Gabrielli, Whisperinggallery modes and resonances of an elliptic cavity. J. Phys. A Math. Gen.
34, 1341–1359 (2001)
MathSciNetCrossRef
30.
go back to reference L.D. Akulenko, S.V. Nesterov, Free vibrations of a homogeneous elliptic membrane. Mech. Solids 35, 153–162 (2000) L.D. Akulenko, S.V. Nesterov, Free vibrations of a homogeneous elliptic membrane. Mech. Solids
35, 153–162 (2000)
31.
go back to reference H.B. Wilson, R.W. Scharstein, Computing elliptic membrane high frequencies by Mathieu and Galerkin methods. J. Eng. Math. 57, 41–55 (2007) MathSciNetCrossRef H.B. Wilson, R.W. Scharstein, Computing elliptic membrane high frequencies by Mathieu and Galerkin methods. J. Eng. Math.
57, 41–55 (2007)
MathSciNetCrossRef
32.
go back to reference L.J. Chu, Electromagnetic waves in elliptic hollow metal pipes of metal. J. Appl. Phys. 9, 583–591 (1938) CrossRef L.J. Chu, Electromagnetic waves in elliptic hollow metal pipes of metal. J. Appl. Phys.
9, 583–591 (1938)
CrossRef
33.
go back to reference S.D. Daymond, The principal frequencies of vibrating systems with elliptic boundaries. J . Mech. Appl. Math. VIII, 361–372 (1955) S.D. Daymond, The principal frequencies of vibrating systems with elliptic boundaries. J
. Mech. Appl. Math.
VIII, 361–372 (1955)
34.
go back to reference J.G. Kretzschmar: Wave propagation in hollow conducting elliptical waveguides. IEEE Trans. Microw. Theory MTT 18 (1970) J.G. Kretzschmar: Wave propagation in hollow conducting elliptical waveguides. IEEE Trans. Microw. Theory MTT
18 (1970)
35.
go back to reference M.V. Lowson, S. Baskaran, Propagation of sound in elliptic ducts. J. Sound Vib. 38, 185–194 (1975) CrossRef M.V. Lowson, S. Baskaran, Propagation of sound in elliptic ducts. J. Sound Vib.
38, 185–194 (1975)
CrossRef
36.
go back to reference F.D. Denia, J. Albelda, F.J. Fuenmayor, A.J. Torregrosa, Acoustic behaviour of elliptical chamber mufflers. J. Sound Vib. 241, 401–421 (2001) F.D. Denia, J. Albelda, F.J. Fuenmayor, A.J. Torregrosa, Acoustic behaviour of elliptical chamber mufflers. J. Sound Vib.
241, 401–421 (2001)
37.
go back to reference A. Mimani, M.L. Munjal, 3D acoustic analysis of elliptical chamber mufflers having an end inlet and a side outlet: an impedance matrix approach. Wave Motion 49, 271–295 (2012) MathSciNetCrossRef A. Mimani, M.L. Munjal, 3D acoustic analysis of elliptical chamber mufflers having an end inlet and a side outlet: an impedance matrix approach. Wave Motion
49, 271–295 (2012)
MathSciNetCrossRef
38.
go back to reference A. Mimani, M.L. Munjal, Acoustic endcorrection in a flowreversal end chamber muffler: a semianalytical approach. J. Comput. Acoust. 24, 1650004 (2016) MathSciNetCrossRef A. Mimani, M.L. Munjal, Acoustic endcorrection in a flowreversal end chamber muffler: a semianalytical approach. J. Comput. Acoust.
24, 1650004 (2016)
MathSciNetCrossRef
39.
go back to reference K. Hong, J. Kim, Natural mode analysis of hollow and annular elliptical cylindrical cavities. J. Sound Vib. 183, 327–351 (1995) CrossRef K. Hong, J. Kim, Natural mode analysis of hollow and annular elliptical cylindrical cavities. J. Sound Vib.
183, 327–351 (1995)
CrossRef
40.
go back to reference W.M. Lee., Acoustic eigenproblems of elliptical cylindrical cavities with multiple elliptical cylinders by using the collocation multipole method. Int. J. Mech. Sci. 78, 203–214 (2014) W.M. Lee., Acoustic eigenproblems of elliptical cylindrical cavities with multiple elliptical cylinders by using the collocation multipole method. Int. J. Mech. Sci.
78, 203–214 (2014)
41.
go back to reference M.L. Munjal, Acoustics of Ducts and Mufflers , 2nd edn. (Wiley, Chichester, UK, 2014). M.L. Munjal,
Acoustics of Ducts and Mufflers , 2nd edn. (Wiley, Chichester, UK, 2014).
42.
go back to reference D.T. Blackstock, Fundamentals of Physical Acoustics, Chap. 3 (John Wiley, New York, 2000) D.T. Blackstock,
Fundamentals of Physical Acoustics, Chap. 3 (John Wiley, New York, 2000)
43.
go back to reference J.M.G.S. Oliveira, P.J.S. Gil, Sound propagation in acoustically lined elliptical ducts. J. Sound Vib. 333, 3743–3758 (2014) CrossRef J.M.G.S. Oliveira, P.J.S. Gil, Sound propagation in acoustically lined elliptical ducts. J. Sound Vib.
333, 3743–3758 (2014)
CrossRef
44.
go back to reference M. Willatzen, L.C.L. Yan Voon, Flowacoustic properties of ellipticalcylinder waveguides and enclosures. J. Phys. Conf. Ser. 52, 1–13 (2006) M. Willatzen, L.C.L. Yan Voon, Flowacoustic properties of ellipticalcylinder waveguides and enclosures. J. Phys. Conf. Ser.
52, 1–13 (2006)
45.
go back to reference G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists (Academic Press Elsevier, London, 2005), pp. 869–879 MATH G.B. Arfken, H.J. Weber,
Mathematical Methods for Physicists (Academic Press Elsevier, London, 2005), pp. 869–879
MATH
46.
go back to reference E. Kreyszig, Advanced Engineering Mathematics, 10th edn. (John Wiley & Sons, New Jersey, USA, 2011). MATH E. Kreyszig,
Advanced Engineering Mathematics, 10th edn. (John Wiley & Sons, New Jersey, USA, 2011).
MATH
47.
go back to reference A. Sarkar, V.R. Sonti, Wave equations and solutions of invacuo and fluidfilled elliptical cylindrical shells. Int. J. Acoust. Vib. 14, 35–45 (2009) A. Sarkar, V.R. Sonti, Wave equations and solutions of invacuo and fluidfilled elliptical cylindrical shells. Int. J. Acoust. Vib.
14, 35–45 (2009)
48.
go back to reference S.W. Rienstra, B.J. Tester, An analytic Green’s function for a lined circular duct containing uniform mean flow. J. Sound Vib. 317, 994–1016 (2008) CrossRef S.W. Rienstra, B.J. Tester, An analytic Green’s function for a lined circular duct containing uniform mean flow. J. Sound Vib.
317, 994–1016 (2008)
CrossRef
 Title
 Acoustic Wave Propagation in an Elliptical Cylindrical Waveguide
 DOI
 https://doi.org/10.1007/9789811048289_2
 Author:

PhD Akhilesh Mimani
 Publisher
 Springer Singapore
 Sequence number
 2
 Chapter number
 Chapter 2