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.

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.

On seismic deghosting using Green’s theorem

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. V317-V325 ◽  
Author(s):  
Lasse Amundsen ◽  
Arne Reitan ◽  
Arthur B. Weglein ◽  
Bjørn Ursin
Keyword(s):  
Green’S Function ◽  
Series Expansion ◽  
Green's Function ◽  
Pressure Field ◽  
Normal Derivative ◽  
Fourier Series Expansion ◽  
Receiver Side ◽  
Green's Theorem ◽  
Green’S Theorem ◽  
Receiver Plane

We have examined theoretically how receiver-side deghosting of pressure measurements can be derived from the Green’s theorem method. We split the Green’s function that obeys Dirichlet boundary conditions on the sea surface and at the receiver plane into two contributions: the first emitting energy downward only from its source location and the other emitting energy only upward. Using the normal derivative of the source-side downgoing Green’s function in the Green’s theorem evaluation over the receiver plane, the upgoing part of the pressure field is predicted. This is the receiver-side deghosted field. By inserting the source-side upgoing normal derivative Green’s function in Green’s theorem, its evaluation over the receiver plane predicts the downgoing part of the pressure field. For a plane horizontal receiver surface, the required Green’s function can be derived using the image series expansion method. To display the fundamental frequencies of this Green’s function, we have applied a Fourier series expansion of the Green’s function. Our theory gives a new understanding of and generalizes and simplifies previously published theories on Green’s theorem-based receiver-side deghosting of pressure wavefields.

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.

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.

Integral Solution of Diffusion Equation: Part 2—Boundary Conditions of Second and Third Kinds

1987 ◽  
Vol 109 (3) ◽  
pp. 557-562 ◽  
Author(s):  
A. Haji-Sheikh ◽  
R. Lakshminarayanan
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Green’S Function ◽  
Green's Function ◽  
Buried Pipe ◽  
Square Enclosure ◽  
Complex Boundary ◽  
The Galerkin Method ◽  
Numerical Complexity

An analytical solution of the diffusion equation using the Galerkin method to calculate the eigenvalues is currently available for boundary conditions of the first kind. This paper includes algebraic techniques to accommodate boundary conditions of the second and third kinds. Several case studies are presented to illustrate the utility and accuracy of the procedure. Selected examples either have no exact solutions or their exact solutions have not been cited because of the mathematical or numerical complexity. The illustrations include transient conduction in hemielliptical solids with either external convective surfaces or convective bases, and buried pipe in a square enclosure. Whenever possible, symbolic programming is used to carry out the differentiations and integrations. In some cases, however, the integrations must be strictly numerical. It is also demonstrated that a Green’s function can be defined to accommodate many geometries with nonorthogonal boundaries subject to more complex boundary conditions for which an exact Green’s function does not exist.

Sign in / Sign up

Export Citation Format

Share Document