Eigenfunction Expansion Research Articles

Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green’s function, and intermediate asymptotics for a finite thin film with elastic resistance

Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green’s function, and intermediate asymptotics for a finite thin film with elastic resistance

Water wave radiation by an array of bottom-mounted surface-piercing concentric cylinders

This paper develops a wave radiation analysis model in horizontal oscillation mode for concentric cylindrical arrays with arbitrary numbers and arrangements within the framework of linear potential flow theory. The concentric cylinders have solid inner cylinders and porous outer cylinders, and they are both bottom-mounted and surface-piercing. The porous boundary conditions of the outer cylinders are treated using Darcy's law, and the flow field is divided into one outer region and N (number of cylinders) inner regions with the outer cylinders as the boundary. Employing eigenfunction expansion and region-matching methods, expressions for the radiation velocity potential of each region are obtained, and then expressions of the added mass and damping coefficient of each cylinder are obtained. Following a rigorous verification of the results against published literature, this paper proceeds to analyze the impact of various parameters, including oscillation modes, outer cylinder permeability, radius ratio, spacing, and the number of cylinders, on the wave radiation characteristics of tandem-arranged cylindrical arrays. The research reveals that when the oscillation direction of the structure is parallel with its arrangement direction, the hydrodynamic coefficient curves exhibit significant spikes, attributed to the influence of the Dirichlet trapped mode. Outside the peak regions, the hydrodynamic coefficient curves display regular fluctuations, which are attributed to the alternating occurrence of constructive and destructive interference effects within the wave field.

Oblique Wave Scattering by a Pair of Asymmetric Inverse Π-Shaped Breakwater

Abstract The topic of wave scattering by different breakwaters has unfolded over the past few decades, largely driven by the profound impacts of global climate change. In this study, a completely new type of breakwater, inverse Π-shaped breakwater, has been taken into account for serving both purposes from analytical as well as an application point of view. During the formulation, the actual physical problem is transferred into a boundary value problem by employing the small amplitude water wave theory. With the aid of eigenfunction expansion, the problem is reduced to a set of integral equations having square-root singularities at the submerged edges of thin structures. To address such singularities, a multi-term Galerkin approximation technique is used along with Chebyshev polynomials (multiplied with suitable weights) as basis functions. The numerical methodology employed here validates with different previous literatures as special cases. Then, numerical solutions of the reflection and transmission coefficients, hydrodynamic forces are graphically illustrated across various dimensionless structural parameters. In summary, the present paper offers valuable insights into the dynamics of wave scattering by a pair of asymmetric inverse bottom-mounted Π-shaped breakwater, emphasizing the role of different non-dimensional parameters in this system.

Generalised eigenfunction expansion and singularity expansion methods for canonical time-domain wave scattering problems

The generalised eigenfunction expansion method (GEM) and the singularity expansion method (SEM) are applied to solve the canonical problem of wave scattering on an infinite stretched string in the time domain. The GEM, which is shown to be equivalent to d’Alembert’s formula when no scatterer is present, is also derived in the case of a point-mass scatterer coupled to a spring. The discrete GEM, which generalises the discrete Fourier transform, is shown to reduce to matrix multiplication. The SEM, which is derived from the Fourier transform and the residue theorem, is also applied to solve the problem of scattering by the mass–spring system. The GEM and SEM are also used to solve the problem of wave scattering by a mass positioned a fixed distance from an anchor point, which supports more complicated resonant behavior.

Layout Optimization of Wave Energy Park Based on Multi-Objective Optimization Algorithm

Abstract In this paper, both semi-analytical method and numerical simulation is applied to investigate the hydrodynamic behavior of large arrays of point-absorber wave energy converters (WECs). To analyze wave interactions among multiple WECs within an array, a semi-analytical model is developed based on the potential flow theory and the matched eigen-function expansions method. The fluid domain is divided into two kinds of regions: interior regions underneath the cylinders and an exterior region surrounding all the cylinders. The matched eigenfunction expansions method is employed to solve the radiation potential problem in each domain, and the hydrodynamic coefficients and motion response of the cylinders in the array are evaluated. To validate the accuracy of the semi-analytical method, WAMIT is adopted to simulate the wave energy park numerically and compared with the results by the semi-analytical model. The hydrodynamic characteristics and power absorption performance of the WECs within the wave energy park are analyzed. The power performance of a wave energy park is studied as functions of layout geometry, incident wave direction and separation distance between WECs respectively. Finally, MOPSO based on a surrogate model is used to optimize the layout of wave energy array.

Effect of step bottom and waterway on flexural gravity wave scattering

Flexural gravity wave scattering by two semi-infinite non-identical ice sheets, which are separated by a clean water surface, is investigated in the presence of a step bottom topography. The problem is investigated in the cases of (i) intermediate water depth and (ii) shallow water. The effect of two edge conditions, (i) simply supported edge and (ii) free edge, on wave scattering is also analyzed. Employing linear velocity potential theory, the problem is studied in the frequency domain. The physical phenomenon is modeled as a boundary value problem having the Laplace equation as the governing equation, and it is solved using the eigenfunction expansion method. Considering the bottom topography and upper boundary, the fluid domain is divided into three regions, and in each region, velocity potential is expressed in terms of infinite Fourier series. Velocity and pressure are matched at the intermediate surface of two regions, and a system of algebraic equations with unknown coefficients is obtained. The complete solution of the present problem is recognized by solving the system of equations numerically. The energy relation is derived using Green's theorem. The reflection and transmission coefficients are computed and compared with the energy relation to check the accuracy of the present method. For different parameters (depth ratio, clean waterway, and different ice properties), reflection coefficient and transmission coefficient are computed and presented as a function of angular frequency. The value of the reflection coefficient ranges from zero to unity, whereas at certain frequencies, the transmission coefficient attends value more than unity.

Performance assessment of a Triple Coaxial-Cylinder Wave-Energy Converter (TCWEC)

We propose a novel Triple Coaxial-cylinder Wave-Energy Converter (TCWEC) system, as an evolution of the commonly deployed dual-coaxial cylinder WEC (DCWEC). TCWEC consists of one inner cylinder and two concentric outer cylinders, characterized by two coupled resonant frequencies, improving wave power absorption. A semi-analytical model, based on potential flow theory and matched eigenfunction expansions, is developed to analyze the 3-Degrees of Freedom (DOF) system. The viscous drag coefficients are determined by Computational Fluid Dynamics (CFD) simulations and employed as equivalent linear damping. Analysis of impact of the viscous flow-separation factor fvis on the capture width indicates that the higher energy-absorption capability of TCWEC is not merely attributed to a smaller fvis value, but rather a planned split of the outer cylinder in DCWEC design into two cylinders in TCWEC, so as to produce an effectively broader resonance property. Under identical sea conditions, it is found that, compared to DCWEC, the optimal capture width of TCWEC is increased by as much as 77% in regular waves. In irregular waves, the optimal capture width is increased up to 40% across the entire frequency range. These findings suggest that the energy absorption efficiency of TCWEC is promising and its potential for real-world applications.

Compressible ocean waves generated by sudden seabed rise near a step-type topography

A typical tsunami generation occurs through submarine earthquakes leading to large volume displacement. The corresponding mathematical problem involves modeling surface water waves generated by an arbitrary temporal motion of the ocean floor. The propagation of tsunami wave and the subsequent scattering from a sudden drop in bathymetry away from the ground motion is studied following linearized water wave theory and a weakly compressible ocean, including static oceanic background compression. The Fourier transformation and eigenfunction expansion techniques are employed to find the surface displacement and pressure profiles by leveraging appropriate matching conditions between regions of different depths. A novel energy balance relationship is derived by considering both the pure-gravity and acoustic-gravity modes. The model is validated in the limit that the depth difference approaches zero, showing a vanishing reflection contribution from the depth change. An efficient numerical code is developed that accurately captures the contribution of the cutoff frequencies of acoustic-gravity modes. Apart from the time-domain propagation of tsunami waves away from the origin, standing wave formations are observed within the shallow region, supported by significantly large pressure fluctuations in time. These standing waves or, equivalently, the pressure fluctuations sustain longer for larger ocean depth. The increase in tsunami speed in the deeper region is readily visible in the time-domain simulations. A three-dimensional axisymmetric solution is also developed, and results show a more gradual sloping tsunami wavefront compared to the equivalent two-dimensional solution for shallower depths. Animation movies corresponding to the two- and three-dimensional surface profiles are provided for better visualization.

Multiple eigenfunction expansion for one dimensional Schrӧdinger equation with piecewise-constant coefficient

In this work, direct scattering problems on the real axis for the onedimensional Schr¨odinger equation with a piecewise constant coefficient is studied.

A frequency domain analysis method for the dynamic response of a submerged floating tunnel under wave action

To investigate the dynamic response of a submerged floating tunnel (SFT) under the action of waves, a modal superposition method in the frequency domain was established, and the amplitudes of the modals were determined using the generalized wave loads and hydrodynamic coefficients of the structural modals. In this model, the SFT was simplified as a constant-cross-section Euler–Bernoulli beam with multiple anchor cables, and its structural modals were resolved using the finite element method (FEM). The generalized wave loads and generalized hydrodynamic coefficients were obtained using the Fourier expansion strategy of the modal function, by which the hydrodynamic problem can be converted into 2D problems with the modified Helmholtz controlling equation and solved using the matched higher-order boundary element method (HOBEM) and eigenfunction expansion. The effects of the diameter of the anchor cables and length of the SFT on the stiffness of the structure and the dynamic response of an SFT with anchor cables under wave action were investigated. Furthermore, the effects of different added mass calculation methods on the structural response were studied. In addition, the wave exciting forces were calculated using potential flow theory or linearized Morison equations according to the wave scale. Finally, the effects of the incident wave frequencies and incident angles were investigated in detail to reveal the combined resonance mechanism.

Application of Lagrange Multiplier for Solving Non – Homogenous Differential Equations

The Lagrange Multiplier method has been applied in solving the regular Sturm Liouville (RSL) equation under a boundary condition of the first kind (Dirichlet boundary condition). This method is a very powerful tool for solving the RSL equation and it involves the solution of the RSL equation with the expansion of eigenfunctions into trigonometric series. The efficiency of this approach is emphasized by solving two examples of regular Sturm Liouville problem under homogenous Dirichlet boundary conditions. The methodology is effectively demonstrated, and the results show a high degree of accuracy of the solution in comparison with the exact solution and reasonably fast convergence.

Wave excited motion of a body floating on water confined between a semi-infinite ice sheet and a side wall

In this paper, we investigate the diffraction and radiation coupling problem of a floating body confined between a fixed wall and a semi-infinite ice sheet under the action of incident waves. Based on the linear water wave theory and Kirchhoff Love elastic thin plate assumption, the ice sheet is considered as an elastic beam, and the boundary conditions of fluid boundaries covered by the ice sheet are obtained. Dividing flow domains with different upper surfaces, the eigenfunction expansion method was used to obtain the velocity potential expansion equations for each sub-domain. Then, through the matching conditions at the interface of sub-domains, a system of equations was constructed to solve the unknown coefficients of the expansion equations. The influence of the existence of the ice sheet on the diffraction and radiation motion of the floating body is analyzed. The effects of changes in the draft and open water width of the floating body on the wave excitation force and hydrodynamic coefficient of the floating body are discussed. The influence of the reflection effect of the ice sheet on the added mass and damping coefficient of the floating body has been discovered, especially the changes and oscillation rules of their extreme points. This research achievement provides theoretical support for the safety design of floating structures in frozen harbors.

Hydrodynamic analysis of an oscillating water column array in presence of variable bathymetry

In this study, hydrodynamic analysis of an oscillating water column (OWC) array in the presence of the variable bathymetry (seafloor depth) is performed by using a semi-analytical potential-flow solver. Within the framework of linear theory, the boundary-value problem associated with wave diffraction and radiation by an OWC array with variable bathymetry is formulated and the semi-analytical hydrodynamic model is developed using the matched eigenfunction expansion method. The fluid domain above variable bathymetry is mathematically discretized into multiple subdomains using the boundary approximation method. The Haskind relation and the energy flux conservation law are used to verify the semi-analytical model. The hydrodynamic performance of the OWC array in the presence of the ripple and coral reef bathymetries are investigated in further. Pronounced oscillations are observed for the curve of reflection coefficient Cr and hydrodynamic efficiency η for cases with ripple and coral reef bathymetries. Bragg resonance is identified as the triggering of the strong oscillations in Cr and η for case of ripple bathymetry. The primary frequency of Bragg resonance and the peaked value of Cr converge as the ripple number increases (i.e., Ns > 3). Particularly, for cases with coral reef bathymetry, the wave resonance modal in open-ended basins results in a significant wave amplification at the weather side of the OWC array.

Viscous effects on the hydrodynamic performance of a two-body wave energy converter with a damping plate

In this study, the hydrodynamic forces and power absorption performance of an autonomous underwater vehicle (AUV)-based two-body wave energy converter (2BWEC) are investigated. A theoretical model is developed within the framework of linear potential flow to solve for added mass, radiation damping, and wave excitation force using the matched eigenfunction expansion method (MEEM). A computational fluid dynamics (CFD) model is employed to account for vortex-shedding effects of the floater and inner cylinder with a damping plate under various excitation conditions. Empirical formulas for supplementary added mass and drag coefficients caused by flow separation are proposed based on curve-fitting the differences between CFD results and MEEM calculations. These formulas are integrated into motion equations to enhance accuracy in evaluating the power absorption of the 2BWEC. It has been found that in the context of viscous flow, both the added mass and damping coefficients are increased, particularly for the inner cylinder with a damping plate. In addition, the viscous hydrodynamic coefficients exhibit strong dependence on the Keulegan–Carpenter number, while showing insensitivity to changes in the frequency parameter β. The supplementary (viscous) added mass provides additional inertia for the AUV with a limited mass itself, which is advantageous for the power absorption of the AUV-based 2BWEC. Conversely, the presence of viscous damping from the damping plate impedes wave energy capture.

Scattering of elastic waves by a cluster of cylindrical inclusions: multiple scattering analysis and equivalent scatterer modeling

Abstract The two-dimensional scattering of elastic waves by a cluster of aligned circular cylindrical elastic inclusions embedded in a circular cylindrical domain of an infinitely extended elastic medium is analysed numerically. The analysis is based on a semi-analytical method where the scattered waves are expressed as the eigenfunction expansions of the time-harmonic displacement potentials and the expansion coefficients are determined by a numerical collocation technique. The wave fields within and around the cluster are demonstrated for different spatial arrangements of inclusions in the cluster. It is also examined if the scattering characteristics of the cluster can be simulated by a single equivalent scatterer which has the effective homogenized properties of the matrix/inclusion composite. As a result, it is shown that for sufficiently small inclusion radius-wavelength ratios, the scattering characteristics of the cluster are relatively insensitive to the spatial inclusion arrangements therein, and can be reasonably simulated by those of the equivalent scatterer.

Wave Pressure and Wave Height Distribution around Seawall Structure Constructed by an Array of TSP Circular Piles

An analytic solution for the interaction between an array of circular piles made by joining trapezoid steel pipes (TSP) and waves was obtained using an eigenfunction expansion method. First, an analytic model for the wave scattering of multiple piles fixed at arbitrary positions was derived, and then a simplified model was obtained assuming that an infinite array of identical piles were deployed perpendicular to the propagating direction of incident waves. A regular wave experiment was conducted using an experimental model with a scale ratio of 1/100 in a two-dimensional wave tank to verify the analytic solutions. The analytic results and experimental results were qualitatively consistent with each other. Using a developed analytic model, we examined the wave force on the multiple piles and the wave deformation in front of the arrayed piles. The period for the installation is greatly reduced as the TSP pile can be prefabricated in a factory. In particular, it is possible to install at the soft seabed. A seawall structure using arrayed TSP piles will be an ideal complement for a concrete seawall in future.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

Unified theory of alpha, mu, and tau rhythms via eigenmodes of brain activity.

A compact description of the frequency structure and topography of human alpha-band rhythms is obtained by use of the first four brain activity eigenmodes previously derived from corticothalamic neural field theory. Just two eigenmodes that overlap in frequency are found to reproduce the observed topography of the classical alpha rhythm for subjects with a single, occipitally concentrated alpha peak in their electroencephalograms. Alpha frequency splitting and relative amplitudes of double alpha peaks are explored analytically and numerically within this four-mode framework using eigenfunction expansion and perturbation methods. These effects are found to result primarily from the different eigenvalues and corticothalamic gains corresponding to the eigenmodes. Three modes with two non-overlapping frequencies suffice to reproduce the observed topography for subjects with a double alpha peak, where the appearance of a distinct second alpha peak requires an increase of the corticothalamic gain of higher eigenmodes relative to the first. Conversely, alpha blocking is inferred to be linked to a relatively small attention-dependent reduction of the gain of the relevant eigenmodes, whose effect is enhanced by the near-critical state of the brain and whose sign is consistent with inferences from neural field theory. The topographies and blocking of the mu and tau rhythms within the alpha-band are explained analogously via eigenmodes. Moreover, the observation of three rhythms in the alpha band is due to there being exactly three members of the first family of spatially nonuniform modes. These results thus provide a simple, unified description of alpha band rhythms and enable experimental observations of spectral structure and topography to be linked directly to theory and underlying physiology.

Scattering of Oblique Water Waves by Two Vertical Barriers Over Trench-Type Bottom

The problem of scattering of oblique water waves by two surface-piercing vertical thin rigid barriers over a trench-like bottom is investigated within the framework of linear water wave theory. The eigenfunction expansion technique is used to obtain a system of algebraic equations, which is further solved using the algebraic Least squares method. The reflection and transmission coefficients are computed and contrasted with the prior results. Physical characteristics such as surface elevation and horizontal force on both barriers are also determined and depicted graphically against various parameters to see their behavior. It is possible to strategically construct barriers and trench-like bottoms to regulate how waves reflect and propagate. This might mitigate the wave impact on ships, improving the safety, effectiveness and longevity of ships navigating through the channels. The highlights of the given problem include comparing the impacts of different depth ratios and barrier placements, followed by a study of the modulation parameter to evaluate standing wave patterns in the trench region. The specific circumstances for the formation of the standing wave are identified. Hence, understanding this study is critical for safe navigation because standing waves may produce hazardous conditions for boats, making navigation difficult and causing substantial damage.

Active polar ring polymer in shear flow-An analytical study.

We theoretically study the conformational and dynamical properties of semiflexible active polar ring polymers under linear shear flow. A ring is described as a continuous semiflexible Gaussian polymer with a tangential active force of a constant density along its contour. The linear but non-Hermitian equation of motion is solved using an eigenfunction expansion, which yields activity-independent, but shear-rate-dependent, relaxation times and activity-dependent frequencies. As a consequence, the ring's stationary-state properties are independent of activity, and its conformations and rheological properties are equal to those of a passive ring under shear. The presence of characteristic time scales by relaxation and the activity-dependent frequencies give rise to a particular dynamical behavior. A tank-treading-like motion emerges for long relaxation times and high activities, specifically for stiff rings. In the case of very flexible polymers, the relaxation behavior dominates over activity contributions suppressing tank-treading. Shear strongly affects the crossover from a tank-treading to a relaxation-dominated dynamics, and the ring polymer exhibits tumbling motion at high shear rates. This is reflected in the tumbling frequency, which displays two shear-rate dependent regimes, with an activity-dependent plateau at low shear rates followed by a power-law regime with increasing tumbling frequency for high shear rates.