Fourier-Bessel Series Expansion Research Articles

Efficient Continuous HRTF Model Using Data Independent Basis Functions: Experimentally Guided Approach

This paper introduces a continuous functional model for head-related transfer functions (HRTFs) in the horizontal auditory scene. The approach uses a separable representation consisting of a Fourier-Bessel series expansion for the spectral components and a conventional Fourier series expansion for the spatial components. Being independent of the data, these two sets of basis functions remain unchanged for all subjects and measurement setups. Hence, the model can transform an individualized HRTF to a subject specific set of coefficients. A continuous functional model is also developed in the time domain. We show the efficient model performance in approximating experimental measurements by using the HRTF measurements from a KEMAR manikin and the synthetic data from the spherical head model. The statistical results are determined from a 50-subject HRTF data set. We also corroborate the predictive capability of the proposed model. The model has near optimal performance, which can be ascertained by comparison with the standard principle component analysis (PCA) and discrete Karhunen-Loeve expansion (KLE) methods at the measurement points and for a given number of parameters.

Amplitude reduction of elastic waves by a row of piles in poroelastic soil

This paper studies the displacements behind a row of elastic piles in a homogeneous unbounded poroelastic soil after excitation by the passage of elastic waves. Biot’s theory is employed to describe soil behavior. Using a Fourier–Bessel series expansion with the aid of a translational addition theorem, closed-form expressions for the scattering coefficients are developed by imposing continuity conditions at the pile–soil interfaces. The influence of certain parameters, such as pile rigidity and soil permeability on the screening performance of the pile barrier is investigated. Particular attention is focused on the influence of the soil permeability on the screening effectiveness of a row of piles in the frequency domain.

Study on dynamic stress intensity factors of disk with a radial edge crack subjected to external impulsive pressure

A dynamic weight function method is presented for dynamic stress intensity factors of circular disk with a radial edge crack under external impulsive pressure. The dynamic stresses in a circular disk are solved under abrupt step external pressure using the eigenfunction method. The solution consists of a quasi-static solution satisfying inhomogeneous boundary conditions and a dynamic solution satisfying homogeneous boundary conditions. By making use of Fourier-Bessel series expansion, the history and distribution of dynamic stresses in the circular disk are derived. Furthermore, the equation for stress intensity factors under uniform pressure is used as the reference case, the weight function equation for the circular disk containing an edge crack is worked out, and the dynamic stress intensity factor equation for the circular disk containing a radial edge crack can be given. The results indicate that the stress intensity factors under sudden step external pressure vary periodically with time, and the ratio of the maximum value of dynamic stress intensity factors to the corresponding static value is about 2.0.

Seismic responses of a hemispherical alluvial valley to SV waves: a three-dimensional analytical approximation

An analytical solution to the three-dimensional scattering and diffraction of plane SV-waves by a saturated hemispherical alluvial valley in elastic half-space is obtained by using Fourier-Bessel series expansion technique. The hemispherical alluvial valley with saturated soil deposits is simulated with Biot's dynamic theory for saturated porous media. The following conclusions based on numerical results can be drawn: (1) there are a significant differences in the seismic response simulation between the previous single-phase models and the present two-phase model; (2) the normalized displacements on the free surface of the alluvial valley depend mainly on the incident wave angles, the dimensionless frequency of the incident SV waves and the porosity of sediments; (3) with the increase of the incident angle, the displacement distributions become more complicated; and the displacements on the free surface of the alluvial valley increase as the porosity of sediments increases.

Free vibration analysis of perforated plate submerged in fluid

An analytical method to estimate the coupled frequencies of the circular plate submerged in fluid is developed using the finite Fourier-Bessel series expansion and Rayleigh-Ritz method. To verify the validity of the analytical method developed, finite element method is used and the frequency comparisons between them are found to be in good agreement. For the perforated plate submerged in fluid, it is almost impossible to develop a finite element model due to the necessity of the fine meshing of the plate and the fluid at the same time. This necessitates the use of solid plate with equivalent material properties. Unfortunately the effective elastic constants suggested by the ASME code are found to be not valid for the modal analysis. Therefore in this study the equivalent material properties of perforated plate are suggested by performing several finite element analyses with respect to the ligament efficiencies.

An analytical solution for three-dimensional diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits

An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion technique. Unlike previous studies, in which the saturated soil deposits were simulated with the single-phase elastic theory, in this paper, they are simulated with Biot's dynamic theory for saturated porous media, and the half space is assumed as a single-phase elastic medium. The effects of the dimensionless frequency, the incidence angle of P-wave and the porosity of soil deposits on the surface displacement magnifications of the hemispherical alluvial valley are investigated. Numerical results show that the existence of a saturated hemispherical alluvial valley has much influence on the surface displacement magnifications. It is more reasonable to simulate soil deposits with Biot's dynamic theory when evaluating the displacement responses of a hemispherical alluvial valley with an incidence of P-waves.

Scattering of plane P waves by circular-arc alluvial valleys with saturated soil deposits

This paper presents an analytical solution for two-dimensional scattering and diffraction of plane P waves by circular-arc alluvial valleys with shallow saturated soil deposits. The solution is based on Biot's dynamic theory for saturated porous media, and derived by employing Fourier–Bessel series expansion technique. In this analysis, soil deposits in the circular-arc valley are modeled as saturated porous media based on Biot's dynamic theory, and the circular-arc valley is assumed to be imbedded in an infinite half-space, filled with elastic single-phase media. Numerical results from this solution show that the amplitudes of displacement at the surface of an alluvial valley are mainly relative to the angle of incidence, the dimensionless frequency of incident P wave, the degree of saturation and porosity of soil deposits, and the stiffness and Poisson's ratio of the solid skeleton of the soil deposits. Furthermore, the proposed solution is compared with the previous solution, in which the soil deposit was modeled as an elastic single-phase solid.

An Analytic Solution for the Three-Dimensional Diffraction of Plane SV-Waves by Hemispherical Canyon in a Fluid-Saturated Porous Medium Half Space

On the basis of the Biot dynamic theory, an analytic solution of three-dimensional scattering of incident plane SV waves by hemispherical canyons in a fluid saturated porous media half space is derived. The Fourier-Bessel series expansion technique is used to get the analytic solution for the first time. The results of surface displacement amplitudes are presented, the effects of frequency and angle of incident waves on the surface displacement amplitudes are discussed. These results are compared with those from the same situation but with an elastic one-phase medium. We conclude that (1) there is a significant difference between the previous single-phase model and the present two-phase model; (2) the normalized displacements on the free surface of alluvial valley are mainly dependent on the incident wave angles, the dimensionless frequency of the incident SV-waves; (3) with the increases of the incident angle, frequency, the displacements become more complicated; and the amplification of the displacement on the free surface of alluvial valley is more prominent than that modeled by single-phase.

Scattering of plane SV waves by cylindrical canyons in saturated porous medium

On the basis of Biot dynamic theory, an analytic solution of two-dimensional scattering and diffraction of plane SV waves by circular cylindrical canyons in a half space of saturated porous media is presented in this paper for the first time. The solution is obtained by employing the Fourier–Bessel series expansion technique. Parametric studies had been carried out, which includes: the angle of incidence, the frequency of the incident SV wave, the porosity of saturated porous medium and the stiffness and Poisson's ratio of the solid-skeleton. All the outcomes are useful for the seismic analysis of the surface topography conditions.

Natural vibration characteristics of a clamped circular plate in contact with fluid

This study deals with the free vibration of a circular plate in contact with a fluid; submerged in fluid, beneath fluid or on fluid. An analytical method based on the finite Fourier-Bessel series expansion and Rayleigh-Ritz method is suggested. The proposed method is verified by the finite element analysis using commercial program with a good accuracy. The normalized natural frequencies are obtained in order to estimate the relative added mass effect of fluid on each vibration mode of the plate. Also, the location of plate coupled with fluid and the cases of free and bounded fluid surface are studied for the effect on the vibration characteristics.

An Analytical Solution for the Diffraction of Plane P-Waves by Three-Dimensional Hemispherical Canyon in a Fluid-Saturated Porous Media Half Space

On the basis of the Biot dynamic theory, an analytic solution of three-dimensional scattering and diffraction by hemispherical canyon “ in a fluid saturated porous media half space under plane P-waves incidence is presented. The Fourier-Bessel series expansion technique is used to get the analytic solution for the first time. The results of surface displacement amplitudes are presented, and the effects of frequency and angle of incident waves on the surface displacement amplitudes are discussed. These results are compared with those from the same situation but with an elastic one-phase media and two-dimensional circular cylindrical canyons in two-phase media, respectively.

Hydroelastic vibration of a circular plate submerged in a bounded compressible fluid

An analytical method for the linear free vibration of a circular plate submerged in a fluid was developed by the Rayleigh–Ritz method based on the Fourier–Bessel series expansion. It is assumed that the plate is clamped at an offcenter location of a rigid cylindrical container and the fluid bisected by the plate in the container is non-viscous and compressible. Since a combination of the dry modal functions of the circular plate can approximate the wet vibration modes, the functions were used to form a set of linearly independent functions of the Rayleigh–Ritz method. It was found that the theoretical results could predict well the fluid-coupled natural frequencies with excellent accuracy compared with the finite element analysis results. The effects of the fluid compressibility and the offcenter distance on the natural frequencies were also observed.

Scattering of plane SH waves by a circulararc hill with a circular tunnel

An analytical solution for scattering of incident SH waves by a circular-arc hill with a concentric circular tunnel was derived by Fourier-Bessel series expansion and auxiliary functions technique. The solution is reduced to solving a set of infinite linear algebraic equations finally. The accuracies of the numerical results are checked by the residual errors of boundary conditions with the truncation order increasing. The numerical results show that the existence and dimension of the tunnel have great effect on motion of the ground surface nearby and dynamic stress concentration of the tunnel.

Scattering of Plane SV Waves by Circular‐Arc Alluvial Valleys with Saturated Soil Deposits

AbstractOn the basis of Biot dynamic theory for saturated porous media, an analytic solution of twodimensional scattering and diffraction of plane SV waves by circular‐arc alluvial valleys with saturated soil deposits is presented using the Fourier‐Bessel series expansion technique with the deposits simulated by Biot dynamic theory for saturated porous media, and the half space assumed to be elastic single‐phase media. For incidence beyond critical angle, the surface waves generated are expanded in terms of finite Fourier series. The method is applicable to the case of a wide range of frequencies, and thus makes it preferable to the presently available numerical methods. The surface displacement amplitudes presented by numerical results show that the results depend on the angle of incidence, the dimensionless frequency of the incident SV wave.

A series solution for surface motion amplification due to underground group cavities: Incident P waves

A series solution for surface motion amplification due to underground group cavities for incident plane P waves is derived by Fourier-Bessel series expansion method. It is shown that underground group cavities significantly amplify the surface ground motion nearby. It is suggested that the effect of subways on ground motion should be considered when the subways are planned and designed.

A series solution for surface motion amplification due to underground twin tunnels: incident SV waves

A series solution of displacement response of the ground surface in the presence of underground twin tunnels subjected to excitation of incident plane SV waves is derived by using Fourier-Bessel series expansion method. The numerical parametric study shows that underground twin tunnels significantly amplify the nearby surface ground motion. It is suggested that the effect of subways on ground motion should be considered when the subways are planned and designed.

Fluid bounding effect on natural frequencies of fluid-coupled circular plates

This study deals with the free vibration of two identical circular plates coupled with a bounded or unbounded fluid. An analytical method based on the finite Fourier-Bessel series expansion and Rayleigh-Ritz method is suggested. The proposed method is verified by the finite element analysis using commercial program with a good accuracy. The case of bounded or unbounded fluid is studied for the effect on the vibration characteristics of two circular plates. Also, the effect of gap between the plates on the fluid-coupled natural frequencies is investigated.

An Analytical Solution for the Diffraction of Plane P Waves by Circular Cylindrical Canyons in a Fluid‐Saturated Porous Media Half Space

AbstractOn the basis of Biot dynamic theory, an analytic solution of two‐dimensional scattering and diffraction of plane P waves by circular cylindrical canyons in a fluid‐saturated porous media half space is presented by Fourier‐Bessel series expansion technique for the first time in this paper. The surface displacement amplitudes are presented, and the effects of frequency and angle of incident waves, and the ratio of canyon depth to its width on the surface displacement amplitudes are discussed and compared with those in which the half space are regarded as ordinary elastic one‐phase solid.

Free vibration of two identical circular plates coupled with bounded fluid

This paper examines the hydroelastic vibration of two identical circular plates coupled with a bounded fluid. An analytical method based on the finite Fourier–Bessel series expansion and the Rayleigh–Ritz method is suggested. In the theory, it is assumed that a rigid cylindrical container is filled with the ideal fluid and the two plates are clamped along the container edges. The proposed method is verified by finite element analysis using commercial software with excellent accuracy. Two transverse vibration modes, in-phase and out-of-phase, are observed alternately in the fluid-coupled system when the number of nodal circles increases for the fixed nodal diameter. It is found that the normalized natural frequency of the system monotonically increases with an increase in the number of nodal diameters and circles by virtue of a decrease in relative hydrodynamic mass.

The Gibbs' phenomenon for Fourier-Bessel series

The paper investigates the Gibbs' phenomenon at a jump discontinuity for Fourier-Bessel series expansions. The unexpected thing is that the Gibbs' constant for Fourier-Bessel series appears to be the same as that for Fourier series expansions. In order to compute the coefficients for Fourier-Bessel functions efficiently, several integral formulas are derived and the Struve functions and their asymptotic expansions discussed, all of which significantly ease the computations. Three numerical examples are investigated. Findings suggest further investigations suitable for undergraduate research projects or small student group investigations.