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This book presents a three-dimensional analysis of acoustic wave propagation in an elliptical waveguide, and applies the equations and concepts to design axially short elliptical end-chamber muffler configurations which are an important component of a complex multi-pass muffler used in a modern-day automotive exhaust system. A general solution of the Helmholtz equation in elliptical cylindrical co-ordinates is presented in terms of the Mathieu and modified Mathieu modal functions. This is followed by the tabulation and analysis, for the first time, of the non-dimensional resonance frequencies of the transverse modes of a rigid-wall elliptical waveguide for a complete range of aspect ratio. The modal shape patterns of the first few circumferential, radial and cross-modes are examined with particular attention to the pressure nodal ellipses and hyperbolae. An analytical formulation is then outlined for characterizing a single-inlet and single-outlet elliptical muffler with the inlet located on the end face and the outlet located either on the end face or side-surface. The ensuing chapter is devoted toward analyzing the Transmission Loss (TL) performance of different short end-chamber mufflers, namely (a) the straight-flow configuration having ports located on the opposite face, (b) the flow-reversal configuration with ports located on the same end face and (c) configuration with inlet port on the end face and outlet on the side surface. Design guidelines are formulated in terms of the optimal location of inlet and outlet ports which suppresses the deteriorating influence of certain higher-order modes, thereby delivering a broadband TL performance. Directions for future work are discussed toward the end.

In summary, this book is a one-stop solution for a practicing automotive engineer designing mufflers, for an applied mathematician studying wave propagation in elliptical geometries, and also as a niche area within noise control engineering.

### Chapter 1. Introduction

Abstract
Exhaust noise of reciprocating internal combustion (IC) engines is certainly one of the biggest pollutants of the present-day urban environment. Fortunately, the use of a well-designed muffler (also known as a silencer) can significantly mitigate this problem by reducing the noise from IC engines [1]. Indeed, all automotive engines are invariably provided with exhaust mufflers, the theory and design practice of which is now over a hundred years! A primary design requirement of an automotive exhaust muffler is obtaining an adequate insertion loss so that the exhaust noise is reduced to the level of the noise from other components of the engine, or as required by the environmental noise pollution limits. Generally speaking, the amount of acoustic attenuation produced by a muffler is proportional to the expansion chamber volume in the low-frequency range which is particularly important because most of the engine noise is limited to the firing frequency and the first few harmonics. The necessity to have a sufficiently large volume is reinforced by the need to have a large expansion ratio, i.e., a sharp impedance mismatch. Table 5.​1 of [2] (reproduced from Bies and Hansen [3]) gives a good estimate of how important the muffler volume is in context with the attenuation produced at different octave band frequency bands; as a rule of thumb, small and large mufflers are approximately characterized by 5 and 15 times the piston displacement capacity of the engine, respectively. Unfortunately, however, the clearing space beneath the automobile body, where the muffler is typically located, is kept small because the stability of a vehicle requires a low center of gravity. Therefore, the space constraint along the vertically downward direction coupled with an additional essential requirement that the muffler shell should not touch the ground (especially on a rough terrain) often leads to the use of silencing chambers having a non-circular shape. Additionally, manufacturing defects or constraints might also force one to use non-axisymmetric chambers. In view of these design constraints, an elliptical and for that matter, a flat-oval chamber is popularly used in modern automobile exhaust systems, see, e.g., Figure 1.1a. Elliptical chamber with a straight-through flow configuration also finds use in the exhaust system of two-wheelers.
Akhilesh Mimani

### Chapter 2. Acoustic Wave Propagation in an Elliptical Cylindrical Waveguide

Abstract
This chapter begins with the three-dimensional (3-D) Helmholtz equation governing acoustic wave propagation in an infinite rigid-wall elliptical cylindrical waveguide carrying a uniform mean flow. The 3-D acoustic pressure field is obtained as modal summation of rigid-wall transverse modes in terms of the angular and radial Mathieu functions and complex exponentials. A well-known 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. Rigid-wall condition is imposed at the elliptical boundaries by setting the derivative of modified Mathieu functions to zero, and root-bracketing in conjunction with the bisection method is used to numerically compute its parametric zeros q. The corresponding non-dimensional resonance frequencies of the transverse (rigid-wall) radial, even and odd circumferential/cross-modes of the elliptical waveguide are tabulated for aspect-ratios 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 aspect-ratio), the resonance frequencies of the even modes are significantly smaller than those of its odd counterpart. With an increase in aspect-ratio, 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 aspect-ratios 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.
Akhilesh Mimani

### Chapter 3. Characterization of an Elliptical Chamber Muffler

Abstract
This chapter presents a 3-D semi-analytical approach to characterize a single-inlet and single-outlet (SISO) rigid-wall elliptical (circular) cylindrical chamber muffler having arbitrary port location and evaluate its transmission loss (TL) performance. First, the acoustic pressure field inside an elliptical cylindrical cavity is expressed as modal summation in terms of the angular and radial Mathieu functions and the circular functions. Solving the inhomogeneous Helmholtz equation based on modeling the ports as a point-source, and using the modal expansion, one obtains the 3-D Green’s function acoustic pressure response function. The Green’s function response is then integrated over the end or side ports which are now modeled as rigid oscillating pistons (and divided by their cross-sectional area) to obtain the acoustic pressure response based on the uniform piston-driven model. This yields the impedance [Z] matrix parameters of different elliptical muffler configurations, namely end-inlet and end-outlet as well as end-inlet and side-outlet configuration. Additionally, the analytical formulation enables one to examine the influence of location of the end/side ports on the suppression (or excitation) of a given transverse mode type; this insight will be used for designing short mufflers in the ensuing chapter. Finally, we present an expression for TL in terms of [Z] matrix parameters which is particularly suited to explain and predict the attenuation peak or trough features.
Akhilesh Mimani

### Chapter 4. Double-Tuned Short End-Chamber Mufflers

Abstract
End-chambers are an important component of multi-pass perforated (MPP) tube mufflers which are often used in automotive exhaust systems. Constraints on the physical space available necessitate end-chambers to have a short length which does not allow the higher-order evanescent modes generated at the port-chamber interface to decay sufficiently, signifying the existence of 3-D acoustic field even at low frequencies. This chapter analyzes the transmission loss (TL) performance of such short end-chambers based on the 3-D semi-analytical formulation and presents design guidelines in terms of optimal port location to achieve a broadband attenuation performance. Flow-reversal and straight-through elliptical mufflers are considered; for both configurations, it is shown that end-inlet port offset on the major-axis and end-outlet port offset on the minor-axis, at pressure nodes of $$(2,1)e$$ and $$(0,2)e$$ modes, respectively, yield a broadband TL covering the entire frequency spectrum of interest. This optimal end port location is referred to as double-tuning of short chamber mufflers and is similar to the practise of tuning the extension lengths of inlet and outlet ports of long muffler configurations through appropriate end-corrections. Parametric studies are carried out to investigate the dramatic changes in broadband TL range as the cross-sectional shape changes from an eccentric ellipse to a circle. Furthermore, this chapter also explains the 1-D transverse plane wave model in terms of the transverse mode shapes and discusses the frequency range over which it is valid, and can be used for analyzing and double-tuning highly eccentric elliptical chambers which have an end-centered port and end-offset port located on the major-axis. Finally, short chamber configurations having an end-inlet and a side-outlet are also analyzed, and their TL graph is compared with the double-tuned end-inlet and end-outlet counterpart configuration having the same aspect-ratio. Some guidelines are suggested for the acoustic design of short end-chambers in terms of the recommendation for optimal location of inlet and outlet ports.
Akhilesh Mimani

### Chapter 5. Summary of Contribution and Directions for Future Work

Abstract
This monograph has tabulated for the first time, the parametric zeros and the non-dimensional resonance frequencies of the higher-order transverse modes of a rigid-wall elliptical waveguide for a complete range of aspect-ratio. This is followed by an analysis of the first few mode shapes. Based on the modal summation and uniform piston-driven model, a theoretical formulation is presented for characterizing and evaluating the TL performance of reactive elliptical muffler configurations having an end-inlet and end-/side-outlet [1, 2]. The TL analysis has resulted in formulation of a set of comprehensive guidelines for designing short elliptical and circular mufflers which are used as end-chambers in a modern-day automotive exhaust system. The guidelines suggest optimal port locations which suppress the propagation of certain higher-order transverse modes, thereby yielding a flat but broadband attenuation despite a small expansion volume. More precisely, it was shown that locating the inlet and outlet ports on the major-axis and minor-axis, respectively, at appropriate pressure nodes yields a broadband attenuation for Helmholtz number $$0.5k_{0} D_{1} > 6$$. Similarly, for short chambers of nearly circular cross section, a concentric end-inlet and offset end-outlet centered on the pressure node of the first radial mode yield a broadband attenuation for Helmholtz number as high as $$0.5k_{0} D_{1} \approx 7$$, see Ref. [3].
Akhilesh Mimani