Analysis of Sound Attenuation in Elliptical Chamber Mufflers by Using Green’s Functions

2011 ◽  
Author(s):  
Subhabrata Banerjee ◽  
Anthony M. Jacobi
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Velocity Potential ◽  
Transmission Loss ◽  
Side Wall ◽  
Momentum Equation ◽  
Mathieu Functions ◽  
Expansion Chamber ◽  
Function Solution

The present work aims at finding the transmission loss of an elliptical expansion chamber, the inlet and outlet of which are located at arbitrary locations of the chamber, i.e. the side wall or on the face of the muffler. The analysis is based on the Green’s function solution for an elliptical cavity with homogeneous boundary conditions. Solving field problems with elliptical geometries require the computation of Mathieu and modified Mathieu functions. These are the eigenfunctions of the wave equation in elliptical coordinates and their computations pose a considerable challenge. In our present study, we have tried to develop a formulation for finding the transmission loss using the properties of the Mathieu and the modified Mathieu functions. The Green’s function is found by considering the boundary to be rigid walls with homogeneous boundary conditions. The inlet and outlet are assumed to be uniform velocity piston sources. The velocity potential inside the muffler is found by adding the individual potentials arising from the inlet and outlet pistons. The pressure in the chamber is obtained from the velocity potential through the linear momentum equation. The pressure at the inlet and at the outlet is approximated by the averaging the acoustic pressure over the piston area. The four-pole parameter is derived from the average pressure values and hence the transmission loss is calculated. The results are compared to those available in literature. It is shown that the results obtained from the present work agree well with those reported in literature.

A Green’s Function Solution for Acoustic Attenuation by a Cylindrical Chamber With Concentric Perforated Liners

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
D. Veerababu ◽  
B. Venkatesham
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Velocity Potential ◽  
Transmission Loss ◽  
Analytical Models ◽  
Mean Flow ◽  
Cylindrical Chamber ◽  
Function Solution ◽  
Grazing Flow ◽  
The Mean

Abstract In this study, a Green’s function-based semi-analytical method is presented to predict the transmission loss (TL) of a circular chamber having concentric perforated screens. Initially, the Green’s function is developed for a single-screen configuration as the summation of eigenfunctions of the inner pipe in the absence of the mean flow. The inlet and the outlet ports are modeled as oscillating piston sources. A transfer matrix is formulated from the velocity potential generated by the piston sources. The results obtained from the proposed method are validated with the numerical and analytical models and with the experimental results available in the literature. Later, the method has been extended to the double-screen configuration. The effect of the additional perforated screen on the TL is studied in terms of the surface impedance of the chamber. Along with grazing flow considerations, guidelines are provided to incorporate more concentric perforated screens into the formulation.

Determination of Transmission Loss in Slightly Distorted Circular Mufflers Using a Regular Perturbation Method

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Subhabrata Banerjee ◽  
Anthony M. Jacobi
Keyword(s):  
Helmholtz Equation ◽  
Exact Solutions ◽  
Green’S Function ◽  
Green's Function ◽  
Velocity Potential ◽  
Pressure Field ◽  
Transmission Loss ◽  
General Procedure ◽  
Separation Of Variables ◽  
Fourier Series Expansion

A perturbation-based approach is implemented to study the sound attenuation in distorted cylindrical mufflers with various inlet/outlet orientations. Study of the transmission loss (TL) in mufflers requires solution of the Helmholtz equation. Exact solutions are available only for a limited class of problems where the method of separation of variables can be applied across the cross section of the muffler (e.g., circular, rectangular, elliptic sections). In many practical situations, departures from the regular geometry occur. The present work is aimed at formulating a general procedure for determining the TL in mufflers with small perturbations on the boundary. Distortions in the geometry have been approximated by Fourier series expansion, thereby, allowing for asymmetric perturbations. Using the method of strained parameters, eigensolutions for a distorted muffler are expressed as a series summation of eigensolutions of the unperturbed cylinder having similar dimensions. The resulting eigenvectors, being orthogonal up to the order of truncation, are used to define a Green's function for the Helmholtz equation in the perturbed domain. Assuming that inlet and outlet ports of the muffler are uniform-velocity piston sources, the Green's function is implemented to obtain the velocity potential inside the muffler cavity. The pressure field inside the muffler is obtained from the velocity potential by using conservation of linear momentum. Transmission loss in the muffler is derived from the averaged pressure field. In order to illustrate the method, TL of an elliptical muffler with different inlet/outlet orientations is considered. Comparisons between the perturbation results and the exact solutions show excellent agreement for moderate (0.4∼0.6) eccentricities.

Transmission loss of lined Helmholtz resonator with annular air gap: A Green's function based approach

2021 ◽  
Vol 69 (2) ◽  
pp. 112-121
Author(s):  
D. Veerababu ◽  
B. Venkatesham
Keyword(s):  
Green’S Function ◽  
Transfer Matrix ◽  
Green's Function ◽  
Acoustic Pressure ◽  
Velocity Potential ◽  
Transmission Loss ◽  
Numerical Models ◽  
Cumulative Effect ◽  
Helmholtz Resonator ◽  
Air Gap

The present article discusses a Green's function-based semi-analytical method to predict the transmission loss of a lined Helmholtz resonator with annular air gap. In the analysis, the walls of the chamber are assumed to be acoustically rigid except at the neck portion where it is treated as a piston source moving with uniform velocity. The Green's function is developed as the summation of eigenfunctions of the central duct. The cumulative effect of the lined portion and the annular air gap including the perforated screens is incorporated as the reflection coefficient in the eigenfunctions. By using the Kirchhoff-Helmholtz integral equation, the velocity potential generated by the piston inside the chamber is evaluated. A transfer matrix relating the acoustic pressure and volume velocity across the neck in the main duct is formulated. The effect of the neck length is included as an added inertance to the impedance in the transfer matrix. The results obtained from the proposed method are validated with the developed numerical models and the experimental data available in the literature. A parametric study has been conducted to investigate the effect of porosity of the perforated screens, thickness and flow resistivity of the absorptive material on the transmission loss of the chamber.

A Liouville Type Theorem for Poly-harmonic System with Dirichlet Boundary Conditions in a Half Space

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Zhao Liu ◽  
Wei Dai
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Type Theorem ◽  
Dirichlet Boundary Conditions ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Harmonic System

AbstractIn this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝwherewhereis the Green’s function in ℝ

scholarly journals Green's Function for A Piecewise Continous Potential via Integral Equations Method

2018 ◽  
Vol 24 (2) ◽  
pp. 20-35
Author(s):  
Benali Brahim ◽  
Mohammed Tayeb Meftah ◽  
Rai Vandana
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Integral Equations ◽  
Green's Function ◽  
Hamiltonian Operator ◽  
Potential Part ◽  
Piecewise Continuous ◽  
Discrete Spectra

The aim of this work is to provide Green's function for the Schrodingerequation. The potential part in the Hamiltonian is piecewise continuous operator.It is a zero operator on a disk of radius "a" and a constant V0 outside this disk (intwo dimensions). We have used, to construct the Green's function, the technique ofthe integral equations. We have respected the boundary conditions of the problem.The discrete spectra of the Hamiltonian operator have been also derived.

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