Cylindrical Wave Functions Research Articles

Benchmark solution for vibrations from a moving point source in a tunnel embedded in a half-space

A closed-form semi-analytical solution for the vibrations due to a moving point load in a tunnel embedded in a half-space is given in this paper. The tunnel is modelled as an elastic hollow cylinder and the ground surrounding the tunnel as a linear viscoelastic material. The total wave field in the half-space with a cylindrical hole is represented by outgoing cylindrical waves and down-going plane waves. To apply the boundary conditions on the ground surface and at the tunnel–soil interface, the transformation properties between the plane and cylindrical wave functions are employed. The proposed solution can predict the ground vibration from an underground railway tunnel of circular cross-section with a reasonable computational effort and can serve as a benchmark solution for other computational methods. Numerical results for the ground vibrations on the free surface due to a moving constant load and a moving harmonic load applied at the tunnel invert are presented for different load velocities and excitation frequencies. It is found that Rayleigh waves play an important role in the ground vibrations from a shallow tunnel.

Optical response of cylindrical multilayers in the context of hydrodynamic convection-diffusion model

In this work, we generalize the cascading scattering matrix algorithm for calculating the optical response of concentric multilayered structures comprised of either plasmonic metal or dielectric, within the framework of hydrodynamic convection-diffusion model of electrodynamics. Two additional boundary conditions, namely, the continuity of first order pressure of free electron density and the continuity of normal components of free charge velocity, respectively, are adopted in order to handle the behaviour at interfaces involving metals. Scattering matrices at interfaces can be readily obtained and cascaded to obtain the modal coefficients in each layer by expanding electromagnetic waves in harmonic modes with cylindrical vector wave functions. We have validated the proposed method by analyzing the optical responses of several configurations of nanostructures, including a bi-metallic nanocylinder and a hyperlens. We found that nonlocal effects can be important for small structures, when the characteristic size is comparable to the Fermi wavelength. The proposed method shows its capability and flexibility to solve hybrid metal-dielectric multilayer structures even when the number of layers is large. Although we have discussed our method in the context of the retarded radiation regime, it can be applied in quasi-static scenarios without any difficulties. Furthermore, it may be extended to solve similar problems in other areas of physics, such as acoustics.

Reflection and transmission of Gaussian beam by a chiral slab

Based on the generalized Lorenz–Mie theory framework, the reflection and transmission of an incident Gaussian beam by a chiral slab were investigated, by expanding the incident Gaussian beam, reflected beam, internal beam as well as transmitted beam in terms of cylindrical vector wave functions. The unknown expansion coefficients were determined by virtue of the boundary conditions. For a localized beam model, numerical results of the normalized field intensity distributions are presented, and the propagation characteristics are discussed concisely in this paper.

Acoustic radiation force and spin torque on a viscoelastic cylinder in a quasi-Gaussian cylindrically-focused beam with arbitrary incidence in a non-viscous fluid

The axial and transverse acoustic radiation force components for a viscoelastic phenolic polymer circular cylinder placed arbitrarily in the field of a zeroth-order quasi-Gaussian focused beam in 2D are computed, based on the partial-wave series expansion (PWSE) method and Graf’s addition theorem for the cylindrical wave functions to compute the off-axial beam-shape coefficients. Moreover, the analysis is extended to derive the expression for the acoustic spin radiation torque experienced by the cylinder when the beam is shifted off-axially with respect to the axis of wave propagation. The emergence of the acoustic spin radiation torque experienced by the circular cylinder is a consequence of acoustic attenuation inside its absorptive material. Depending on the direction of the shift, the spin torque is negative or positive, causing the rotation of the cylinder in the polar plane in either the clockwise or the anticlockwise direction, respectively. Moreover, for a fixed position of the cylinder in the off-axial configuration, a spin torque reversal is predicted due to the variation of absorption of both the longitudinal and shear waves. The effect of increasing the surrounding fluid’s density is also investigated, which has a substantial effect on the radiation force and spin torque plots. Numerical results are performed with particular emphasis on the properties of the incident beam, the amount of the shift from the center of the cylinder as well as the properties of the surrounding fluid. The analytical formalism allows evaluating the axial and transverse radiation force components and radiation torque for any 2D beam that is generally an exact solution of the Helmholtz equation. The sought applications are in controlled rotation of a particle and its manipulation using “acoustical sheets” (i.e., finite beams in 2D).

An Approximate Method for Analysis of Solitary Waves in Nonlinear Elastic Materials

Two types of solitary elastic waves are considered: a longitudinal plane displacement wave (longitudinal displacements along the abscissa axis of a Cartesian coordinate system) and a radial cylindrical displacement wave (displacements in the radial direction of a cylindrical coordinate system). The basic innovation is the use of nonlinear wave equations similar in form to describe these waves and the use of the same approximate method to analyze these equations. The distortion of the wave profile described by Whittaker (plane wave) or Macdonald (cylindrical wave) functions is described theoretically

Comparison of inverse techniques for reproducing an extended, partially coherent sound field above a reflecting plane

The relative abilities of two advanced array processing techniques to reconstruct a partially correlated noise source similar to a high-speed jet are compared. M-SONAH is a modification of statistically optimized near-field acoustical holography that includes multiple types of wavefunctions [A. T. Wall et al., J. Acoust. Soc. Am., 137, 963–975 (2015)]. The hybrid method may be described as an advanced cross-beamforming algorithm with regularization [T. Padois et al., AIAA Paper 2013-2212]. A numerical, multiple wavepacket-based source distribution provides a reasonable model of the frequency-dependent directivity and coherence properties of a jet noise sound. An image source was included to account for a reflecting ground plane and the numerically modeled sound field was used an input to both the M-SONAH and hybrid methods. For M-SONAH, the image source was handled using an additional set of cylindrical wavefunctions and for the hybrid method, sources were assumed to be located both above and below the re...

Acoustic backscattering and radiation force on a rigid elliptical cylinder in plane progressive waves

This work proposes a formal analytical theory using the partial-wave series expansion (PWSE) method in cylindrical coordinates, to calculate the acoustic backscattering form function as well as the radiation force-per-length on an infinitely long elliptical (non-circular) cylinder in plane progressive waves. The major (or minor) semi-axis of the ellipse coincides with the direction of the incident waves. The scattering coefficients for the rigid elliptical cylinder are determined by imposing the Neumann boundary condition for an immovable surface and solving a resulting system of linear equations by matrix inversion. The present method, which utilizes standard cylindrical (Bessel and Hankel) wave functions, presents an advantage over the solution for the scattering that is ordinarily expressed in a basis of elliptical Mathieu functions (which are generally non-orthogonal). Furthermore, an integral equation showing the direct connection of the radiation force function with the square of the scattering form function in the far-field from the scatterer (applicable for plane waves only), is noted and discussed. An important application of this integral equation is the adequate evaluation of the radiation force function from a bistatic measurement (i.e., in the polar plane) of the far-field scattering from any 2D object of arbitrary shape. Numerical predictions are evaluated for the acoustic backscattering form function and the radiation force function, which is the radiation force per unit length, per characteristic energy density, and per unit cross-sectional surface of the ellipse, with particular emphasis on the aspect ratio a/b, where a and b are the semi-axes, as well as the dimensionless size parameter kb, without the restriction to a particular range of frequencies. The results are particularly relevant in acoustic levitation, acousto-fluidics and particle dynamics applications.

Analytical investigation on free vibration of circular double-layer graphene sheets including geometrical defect and surface effects

Effects of a single vacancy defect or a pin hole on free vibration behavior of a double layer graphene sheet are investigated. Using the nonlocal continuum theory as well as the Gurtin–Murdoch theory, the nonlocality and surface effects are considered in equations of motion. Both of in-phase and anti-phase vibration modes are analytically analyzed. Employing the translational addition theorem for cylindrical vector wave functions, the geometrical defect as a circular hole in arbitrary size and location is modeled. The van der Waals interaction between the upper and lower layers is included using the Lennard–Jones pair potential. The computational efficiency and accuracy of results are validated by literature. Effects of boundary conditions, geometrical properties, nonlocality and surface effect parameters on in-phase and anti-phase vibrational modes are investigated. Results reveal that the fundamental natural frequency of an annular double-layer graphene sheet with a free eccentric circular defect is less affected by the size and location of the defect. Moreover, the surface effect parameters have more significant effects on the in-phase vibration modes than the anti-phase ones.

Resonance scattering and radiation force calculations for an elastic cylinder using the translational addition theorem for cylindrical wave functions

The standard Resonance Scattering Theory (RST) of plane waves is extended for the case of any two-dimensional (2D) arbitrarily-shaped monochromatic beam incident upon an elastic cylinder with arbitrary location using an exact methodology based on Graf’s translational addition theorem for the cylindrical wave functions. The analysis is exact as it does not require numerical integration procedures. The formulation is valid for any cylinder of finite size and material that is immersed in a nonviscous fluid. Partial-wave series expansions (PWSEs) for the incident, internal and scattered linear pressure fields are derived, and the analysis is further extended to obtain generalized expressions for the on-axis and off-axis acoustic radiation force components. The wave-fields are expressed using generalized PWSEs involving the beam-shape coefficients (BSCs) and the scattering coefficients of the cylinder. The off-axial BSCs are expressed analytically in terms of an infinite PWSE with emphasis on the translational offset distance d. Numerical computations are considered for a zeroth-order quasi-Gaussian beam chosen as an example to illustrate the analysis. Acoustic resonance scattering directivity diagrams are calculated by subtracting an appropriate background from the expression of the scattered pressure field. In addition, computations for the radiation force exerted on an elastic cylinder centered on the axis of wave propagation of the beam, and shifted off-axially are analyzed and discussed.

Spin-wave bound modes in a circular array of magnetic inclusions embedded into a metallized ferromagnetic thin-film matrix

A general theory of forward volume magnetostatic spin-wave multiple scattering by a finite two-dimensional ensemble of cylindrical magnetic inclusions in a ferromagnetic film (matrix) metallized from both sides is developed. It is predicted that bound spin-wave modes are excited around these inclusions. The set of self-consistent equations for spin-wave multiple scattering inside the matrix with inclusions is obtained by using an invariant addition theorem for the cylindrical wave functions. This set of equations is solved by an iterative process. We derive simple expressions for eigenmodes (bound modes) in a form of Bloch-like spin waves propagating along the inclusion circular array. The theory is illustrated on practically significant examples: (i) bound-mode excitation by propagating spin waves; (ii) helix components of the spin-wave-scattered field around magnetic inclusions; and (iii) a high-quality spin-wave resonator based on a small number of magnetic inclusions.

Scattering of electromagnetic waves from a chiral coated nihility cylinder hosted by isotropic plasma medium

Theoretical analysis of the electromagnetic wave scattering of cylindrical waves from chiral coated nihility cylinder placed in isotropic plasma medium is carried out. The scattering problem is analytically formulated in the frame work of extended classical scattering theory. The cylindrical vector wave functions (CVWFs) are used for the expansion and representation of fields. The appropriate boundary conditions are applied on each interface i.e., Plasma/Chiral and chiral/nihility to get the unknown scattering coefficients. It is concluded that the scattering amplitude can be controlled and tuned by the plasma parameters (plasma density and effective collision frequency) as well as the chirality. Moreover, the present work has practical applications in target protection and microwave controlling devices. Under the special conditions, present work found good agreement with already published literature.

Improvements for scattering from a large-sized chiral cylinder at an oblique incidence

Light scattering from a chiral cylinder is reconsidered based on Lorenz–Mie regime. The incident, internal, and scattered fields are expressed in terms of cylindrical vector wave functions. The scattering coefficients at an oblique incidence are obtained according to the boundary conditions and expressed in forms of ratios and logarithmic derivatives of cylindrical Bessel functions, which can be used for large-sized chiral cylinder scattering. Effects of the cylinder parameters such as chirality, loss and incident angle on scattering properties are numerically analyzed. Numerical results show that rainbow phenomenon similar to that of a chiral sphere also occurs for a large-sized chiral cylinder.

From Bessel beam to complex-source-point cylindrical wave-function

This investigation shows that a scalar Bessel beam can be transformed into the non-paraxial complex-source-point cylindrical wave (CSPCW). High-order CSPCW solutions, termed here high-order quasi-Gaussian cylindrical beams, which exactly satisfy the Helmholtz equation, are derived analytically. Moreover, partial-derivatives of the high-order CSPCW solutions satisfy the Helmholtz equation. In addition, the CSPCW solutions satisfy the nonrelativistic Schrödinger equation within standard quantum mechanics, thus, the results can be used in the description of elementary particle/matter motion and related applications in quantum scattering theory. Furthermore, the analysis is extended to the case of vector beams in which the components of the electromagnetic (EM) field are obtained based on different polarizations of the magnetic and electric vector potentials, which exactly satisfy Maxwell’s vectorial equations and Lorenz’ gauge condition. An attractive feature of the high-order solutions is the rigorous description of strongly focused (or strongly divergent) cylindrical wave-fields without any approximations, nor the need for numerical methods. Possible applications are in beam-forming design using high-aperture or collimated cylindrical laser/electron quasi-Gaussian beams in imaging microscopy, particle manipulation, optical tweezers, and the study of the scattering, and radiation forces on objects.

Multisource statistically optimized near-field acoustical holography.

This paper presents a reduced-order approach to near-field acoustical holography (NAH) that allows the user to account for sound fields generated by multiple spatially separated sources. In this method, an equivalent wave model (EWM) of a given field is formulated to include combinations of planar, cylindrical, spherical, or other elementary wave functions in contrast to an EWM restricted to a single separable coordinate system. This can alleviate the need for higher-order functions, reduce the number of measurements, and decrease error. The statistically optimized near-field acoustical holography (SONAH) algorithm is utilized to perform the NAH projection after the formulation of the multisource EWM. The combined process is called multisource statistically optimized near-field acoustical holography (M-SONAH). This method is used to reconstruct simulated sound fields generated by combinations of a vibrating piston in a sphere and linear arrays of monopole sources. It is shown that M-SONAH can reconstruct near-field pressures in multisource environments with lower errors and fewer measurements than a strictly plane or cylindrical-wave formulation using the same simulated measurement.

Transmission of Gaussian beam through a coated chiral cylinder

Based on the generalized Lorenz–Mie theory (GLMT) framework, an exact analytical solution to the transmission of a Gaussian beam through a chiral cylinder with dielectric coating is formulated, by expanding the incident Gaussian beam, internal beam as well as scattered beam in terms of cylindrical vector wave functions (CVWFs). The unknown expansion coefficients are determined by a system of equations derived from the boundary conditions. For a localized beam model, numerical results of the normalized field intensity distributions are presented, and the propagation characteristics are discussed concisely.

Dynamic Interaction Between Two Fluid-Filled Circular Pipelines in Saturated Poroelastic Medium Subjected to Harmonic Waves

A semi-analytical method is developed to investigate the dynamic interaction of two fluid-filled circular pipelines in a porous elastic fluid-saturated medium subjected to harmonic plane waves. The harmonic equations based on Biot's theory are reduced by Helmholtz decomposition theorem. The potentials in the fluid-saturated medium, in the linings, and inside the pipelines are expressed by wave function expansion method. The addition theorem for cylindrical wave functions is employed to obtain the closed-form solution in the form of infinite series. The hoop stress amplitudes around the pipelines are evaluated and discussed for the representative values of parameters characterizing the model. The effects of the proximity of two pipelines, the geometrical and material properties of linings, and the incident wave frequency on the dynamic stress around the pipelines are examined.

Pseudo-Gaussian cylindrical acoustical beam – Axial scattering and radiation force on an elastic cylinder

Making use of the addition theorem for the cylindrical wave functions and the complex-source-point method in cylindrical coordinates, an exact solution to the Helmholtz equation is derived, which corresponds to a tightly focused (or collimated) cylindrical quasi-Gaussian beam with arbitrary waist. The solution is termed “quasi-Gaussian” to make a distinction from the standard Gaussian beam solution obtained in the paraxial approximation. The advantage of introducing this new solution is the efficient and fast computational modeling of tightly focused or quasi-collimated cylindrical wave-fronts depending on the dimensionless waist parameter kw0, where k is the wavenumber of the acoustical radiation. Moreover, a closed-form partial-wave series expansion is obtained for the incident field, which has the property that the axial scattering (i.e. along the direction of wave propagation) and the axial acoustic radiation force (which is a time-averaged quantity) on a cylinder, can be calculated without any approximations in the limit of linear acoustical waves in a nonviscous fluid. Examples are found where the extinction in the radiation force function plot is found to be correlated with conditions giving reduction of the backscattering from an elastic cylinder. Those results are useful in beam-forming design, particle manipulation in acoustic tweezers operating with focused cylindrical beams, and the prediction of the scattering and radiation forces on a cylindrical particle or liquid bridges.

Cylindrical acoustical holography applied to full-scale jet noise.

Near-field acoustical holography methods are used to predict sound radiation from an engine installed on a high-performance military fighter aircraft. Cylindrical holography techniques are an efficient approach to measure the large and complex sound fields produced by full-scale jets. It is shown that a ground-based, one-dimensional array of microphones can be used in conjunction with a cylindrical wave function field representation to provide a holographic reconstruction of the radiated sound field at low frequencies. In the current work, partial field decomposition methods and numerical extrapolation of data beyond the boundaries of the hologram aperture are required prior to holographic projection. Predicted jet noise source distributions and directionality are shown for four frequencies between 63 and 250 Hz. It is shown that the source distribution narrows and moves upstream, and that radiation directionality shifts toward the forward direction, with increasing frequency. A double-lobe feature of full-scale jet radiation is also demonstrated.

Transmission of a Gaussian beam through a gyrotropic cylinder.

An exact analytical solution to the transmission of a Gaussian beam through a gyrotropic cylinder is formulated in terms of a cylindrical vector wave function expansion form. By applying the continuous boundary conditions of electromagnetic fields, the unknown expansion coefficients of the scattered and internal fields are determined. For a localized beam model, numerical results are presented for the normalized near-surface and internal field intensity distributions, and the propagation characteristics are discussed concisely.

Single and Multiple Scattering of Inplane Waves by Nanofibers with Consideration of Interface Effects

The single scattering by an individual nanofiber and the multiple scattering by unidirectionally arranged nanofibers embedded randomly in a host material for the propagation of incident plane P and SV waves perpendicularly to the axis of fibers are studied. By using the classical expansion method for wave functions, the incident, scattered, and refracted waves are expanded in series of cylindrical wave functions. The unknown expansion coefficients can be determined from interface conditions between the fiber and the host. The surface elasticity theory is used to consider the surface effects of nanofibers and to derive nontraditional interface conditions. The scattering cross section and the far-field scattering amplitude, which represent important information on the scattered wave, are discussed first. Then, the phase speed and the attenuation of the effective wave with multiple scattering effects are estimated based on the effective field approach and the configuration average. The influence of the interface effects of nanofibers on the scattering cross section, the far-field scattering amplitude and the effective propagation constants of elastic waves are discussed in detail on the basis of the numerical results obtained.