Mode-matching Solution Research Articles

On the scattering of cylindrical elastic shell having trifurcation and structural variations at interfaces

The present research focuses on the study of how acoustic radiation modes behave in an elastic shell that has trifurcated junctions and structural variations. To express the vibration of a thin, flexible shell, the Donnell–Mushtari equations are utilized. These modes possess non-orthogonal characteristics and form a system that is linearly dependent. To analyze the scattering phenomenon in an elastic shell with a finite length that is connected to an extended trifurcated inlet and outlet with different ring conditions and step discontinuities, the mode-matching solution is developed. This approach uses continuity conditions for pressure and velocity modes, as well as generalized orthogonality conditions, to transform the differential systems into linear algebraic systems which are numerically solved after truncation. To ensure the accuracy of the algebraic calculations and the convergence of truncated coefficients, an energy flux identity based on the conservation law is developed and verified through continuity conditions. The results of numerical experiments conducted using the truncated solution offer valuable insights for designing effective noise reduction strategies in various industrial and engineering applications.

Scattering analysis of a partitioned membrane-bounded cavity with material contrast.

The acoustic performance of a silencer containing elastic membranes backed by cavities and porous material is investigated. The modeled waveguide configuration contains porous screens as well as the metallic fairings at interfaces between the inlet and outlet and the expansion chamber. The mode-matching solution is developed to analyze the attenuation of the silencer. The governing eigen-sub-systems are Sturm-Liouville and non-Sturm-Liouville types. In the latter case, the exploitation of generalized orthogonality conditions reveals the point-wise convergence of the solution to the governing eigen-systems. The study shows that by tuning the material parameters of the isotropic membranes and altering the bounding wall conditions, the performance of the physical device can be improved. It enables the model configuration to be adopted as a passive or reactive noise control device.

The traveling wave formulation of a splitting chamber containing reactive components

The present work investigates the traveling wave formulation and their scattering characteristics in a waveguide having splitting expansion chamber. The segments of the chamber contain sandwiched elastic components backed by rigid cavities and absorbent linings. The structure is radiated by a plane wave mode incident which scatter on interaction with the expansion chamber. The mode-matching (MM) solution is developed to investigate the attenuation of fluid–structure coupled/uncoupled waves with reference to the properties of elastic components and absorbent material. The eigenmodes of segments comprising elastic components are non-orthogonal, and use of generalized orthogonal characteristics yields the convergent solution. The MM solution is validated altogether through apposite mathematical and physical arguments. Moreover, in low-frequency regime, the MM results are compared with low-frequency approximation, and a good agreement is found. Furthermore, it is observed that the characteristics of elastic components and the properties of absorbent material significantly affect the transmission loss, absorbing power and scattering energies.

A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

A mode-matching solution for the study of cylindrical waveguide bifurcation via closed-form coupling integrals

In this paper, we present a mode-matching technique formulation for the rigorous electromagnetic analysis of a circular waveguide bifurcated into circular and coaxial domains. A generalized scattering matrix is obtained for relating the modal field amplitudes in each of the waveguides, and we were able to obtain closed-form solutions for all the associated coupling integrals. We present a series of validation results showing that the technique introduced in this work can model accurately and efficiently a class of cylindrical waveguide devices.

A Mode-Matching Solution for Scattering from a Large Isosceles Right Triangle Groove in PEC Plane

In this literature, scattering from a large Isosceles Right Triangle Groove (IRTG) in a perfect electric plane (PEC) is studied. An efficient Mode-Matching manner involving physical optics approximation is applied to a large IRTG. Considering two synthetic PEC walls over the groove, the tangential fields inside and outside of the groove are expanded as the sums of infinite harmonics modes. By applying the boundary conditions, the modes are matched on IRTG and a linear system of equations based on series coefficients is constructed. Since matrix elements are computed analytically, finding the inverse an N×N matrix is the most time-consuming operation in this manner. The obtained results are verified by two full numerical techniques MoM and FEM.

Scattering analysis of a partitioned wave-bearing cavity containing different material properties

The scattering analysis of acoustic waves through a wave-bearing cavity containing parallel partitioning is presented. The modeled physical configuration comprises an elastic membrane-bounded expansion chamber separated by horizontal plates. The bounding wall conditions of the splitting plates can be chosen to acoustically rigid, soft or an absorbent lining. The governing boundary value problem is solved by using Mode-Matching technique. The low frequency approximation is used and validated in a low frequency regime via the Mode-Matching solution. It is seen that scattering energies and the transmission loss is significantly affected by the variations of material properties of isotropic membranes and the bounding wall conditions. The considered configuration can be opted for the reactive as well as passive control measure of sound.

Symmetry, validation, and scattering matrices for time–domain scattering in waveguides

• Channel waveguide problem with scattering regions are presented. • Exploiting symmetry, Mode-matching solutions are converted to scattering and transfer matrices. • Series of identities are presented for scattering matrix. • Time-domain scattering by incident Gaussian wave packets are computed. We outline a method to compute the solution in the frequency–domain for scattering in a waveguide by exploiting symmetry. The method is illustrated by considering a simple scattering example, where soft hard boundary conditions are alternated. We show how the straightforward mode matching or eigenfunction matching solution can be easily converted to scattering and transmission matrices when symmetry is exploited. We then show how the solution for two scatterers can be found explicitly, using symmetry which allows validation of our subsequent solution by scattering matrices. We also give a series of identities which the scattering matrix must satisfy for further numerical validation. Using these frequency–domain solutions we compute the time-domain scattering by incident Gaussian wave–packets.

Nested 2D finite-element function-spaces formulation for the mode-matching problem of arbitrary cross-section waveguide devices

• A novel improved formulation based on hybridized 2D finite-element (FE) and mode-matching methods is proposed. • Numerical modes are expressed in terms of nested FE function spaces thanks to a single inter-cross-section conforming mesh. • FE matrices used to compute numerical modes are reused to directly build the mode-matching solution of all waveguide steps. • Results for real-world waveguide devices are validated through comparison with commercial 3D software and measurements. A large class of microwave, millimeter-wave and terahertz waveguide devices for high-frequency electronic systems are made up of waveguide steps cascaded along the propagation direction, giving rise to diverse modal and numerical analysis techniques to solve Maxwell equations for this problem. In this paper, a novel formulation is proposed to compute the numerical modes in all arbitrary cross-sections and characterize all waveguide steps involved in this kind of structures with a modular and straightforward approach through block-matrix operations. The key idea is expressing the modal fields in terms of 2D nested function spaces (each one for a different cross-section) made up of finite-element basis functions. This leads to finite-element matrices used to compute the modes in all different arbitrary waveguides of the structure from a single inter-cross-section conforming 2D mesh. Moreover, these finite-element matrices and results are used to build directly the mode-matching solution of all steps in the structure. After comparison with analytical results for canonical steps, this flexible and efficient approach is validated with various examples of waveguide devices (two filters, a polarizer, a transformer and a polarization rotator), showing excellent agreement with other numerical methods and measurements.

2.5D Sound Field Reproduction Using Higher Order Loudspeakers

Abstract Using 3-Dimensional (3D) sound sources as secondary sources to 2-Dimensional (2D) sound field reproduction, it is termed 2.5-Dimensional (2.5D) sound field reproduction which is currently drawing broad interest in acoustic signal processing. In this paper we propose a method to reproduce a 2D sound field, using a circular array of 3D High Order (HO) loudspeakers, which provides a mode matching solution based on 3D wave field translation. Using the spherical addition theorem, we first obtain a spherical harmonics representation of a 2D sound field reproduced by an array of HO loudspeakers. Then, the corresponding relationship between the reproduced sound field and the desired sound field is established by spherical/cylindrical harmonic expansions. Finally, the modal weights of HO loudspeakers are designed by using a least squares method. Simulation results show that the proposed method extends the reproduction region and significantly reduces the required minimum number of loudspeakers over the other referenced methods.

Model and analysis of a cylindrical in-line hydraulic suppressor with a solid compressible liner

An in-line hydraulic noise suppressor with a lossy, compressible liner made of foamed polyurethane liner is introduced which is intended to provide an alternative to current in-line silencing devices using compressed nitrogen gas volumes. The liner is engineered to be compressible at elevated pressures, such that it can provide effective noise abatement for practical hydraulic systems. In support of such work, a multimodal model is developed to characterize the device and the liner material. Because the hydraulic system is pressurized after insertion of the liner, the model must address liner compression and the corresponding small gaps introduced in the expansion volume; additionally, both compression and shear wave propagation must be considered in the liner. Several mode matching solutions are investigated, and a pseudoinverse mode matching method is found to provide good convergence characteristics. The multimodal model is validated against a finite element model, and also used in an optimization algorithm to estimate the material properties of a prototype liner using experimental transmission loss data. Experimental results show broadband transmission loss performance at 2.8MPa system pressure; transmission loss decreases with increasing system pressure, and data at 4.1MPa system pressure produces about 4dB less transmission loss than a similarly sized commercial device. The multimodal model with estimated material properties at 2.8MPa achieves a root mean squared error of 1.7dB or less for two different length devices over a frequency range of 50–2000Hz.

Acoustic scattering in flexible waveguide involving step discontinuity.

In this paper, the propagation and scattering of acoustic waves in a flexible wave-guide involving step discontinuity at an interface is considered. The emerging boundary value problem is non-Sturm-Liouville and is solved by employing a hybrid mode-matching technique. The physical scattering process and attenuation of duct modes versus frequency regime and change of height is studied. Moreover, the mode-matching solution is validated through a series of numerical experiments by testifying the power conservation identity and matching interface conditions.

Wavefield Analysis Over Large Areas Using Distributed Higher Order Microphones

Successful recording of large spatial soundfields is a prevailing challenge in acoustic signal processing due to the enormous numbers of microphones required. This paper presents the design and analysis of an array of higher order microphones that uses 2D wavefield translation to provide a mode matching solution to the height invariant recording problem. It is shown that the use of Mth order microphones significantly reduces the number of microphone units by a factor of 1/(2M + 1) at the expense of increased complexity at each microphone unit. Robustness of the proposed array is also analyzed based on the condition number of the translation matrix while discussing array configurations that result in low condition numbers. The white-noise gain (WNG) of the array is then derived to verify that improved WNG can be achieved when the translation matrix is well conditioned. Furthermore, the array's performance is studied for interior soundfield recording as well as exterior soundfield recording using appropriate simulation examples.

Analytic mode-matching for acoustic scattering in three dimensional waveguides with flexible walls: Application to a triangular duct

An analytic mode-matching method suitable for the solution of problems involving scattering in three-dimensional waveguides with flexible walls is presented. Prerequisite to the development of such methods is knowledge of closed form analytic expressions for the natural fluid–structure coupled waveforms that propagate in each duct section and the corresponding orthogonality relations. In this article recent theory [J.B. Lawrie, Orthogonality relations for fluid–structural waves in a 3-D rectangular duct with flexible walls, Proc. R. Soc. A. 465 (2009) 2347–2367] is extended to construct the non-separable eigenfunctions for acoustic propagation in a three-dimensional rectangular duct with four flexible walls. For the special case in which the duct cross-section is square, the symmetrical nature of the eigenfunctions enables the eigenmodes for a right-angled, isosceles triangular duct with flexible hypotenuse to be deduced. The partial orthogonality relation together with other important properties of the triangular modes are discussed. A mode-matching solution to the scattering of a fluid–structure coupled wave at the junction of two identical semi-infinite ducts of triangular cross-section is demonstrated for two different sets of “junction” conditions.

Sound-field reproduction systems using fixed-directivity loudspeakers

Sound reproduction systems using open arrays of loudspeakers in rooms suffer from degradations due to room reflections. These reflections can be reduced using pre-compensation of the loudspeaker signals, but this requires calibration of the array in the room, and is processor-intensive. This paper examines 3D sound reproduction systems using spherical arrays of fixed-directivity loudspeakers which reduce the sound field radiated outside the array. A generalized form of the simple source formulation and a mode-matching solution are derived for the required loudspeaker weights. The exterior field is derived and expressions for the exterior power and direct to reverberant ratio are derived. The theoretical results and simulations confirm that minimum interference occurs for loudspeakers which have hyper-cardioid polar responses.

Evaluation of the input impedance of a top-loaded monopole in a parallel-plate waveguide by the MoM/Green's function method

An accurate modeling of a top-hat monopole transition in a parallel-plate waveguide is performed using the method of moments/Green's function method. The selection of the appropriate source-field relationship to override divergence series is discussed in detail. Numerical results are given for the input impedance of a top-loaded coaxial transition. A mode-matching solution is used as a reference to validate the results.

An improved mode-matching method for large cavities

The calculation of electromagnetic fields for two-dimensional (2-D) large open cavities filled with a layered material is studied. By defining two extension operators, the exact solution and the mode-matching solution may be formulated under the same framework, which makes it possible to analyze the difference. Using an asymptotic technique, we present a new (improved) mode-matching method. In particular, the explicit solution of the model problem is given. Numerical examples are presented.

On the convergence of MoM and mode matching solutions for infinite array and waveguide problems

It is shown that the method of moment (MoM) fails to analyze an infinite array of strip discontinuities if the number of basis functions and the number of expansion modes are set equal, however large they individually may be. Using spectral analysis and linear vector space theory it is found that for a given number of basis functions, there exists a minimum number of expansion modes required for a reasonable accuracy. Subsequently, for a given number of expansion modes, there is an upper limit for the number of basis functions. The mode matching (MM) analysis is viewed as a special case of the MoM analysis and the accuracy of the analysis is discussed with respect to the number of modes in the waveguides.

Three‐dimensional induction logging problems, Part 2: A finite‐difference solution

A 3‐D finite‐difference solution is implemented for simulating induction log responses in the quasi‐static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite‐difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl‐free and divergence‐free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.

Efficient computation of propagation factors in ferrite-loaded rectangular waveguides

The mode matching solution of ferrite-loaded waveguide devices requires accurate knowledge of the pertinent modes. Computation of the propagating factor of such modes is essential but non-trivial. Here an efficient technique for computing these factors for an arbitrary number of modes is presented.