Linear Water-wave Problem Research Articles

Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools

We consider the linear water-wave problem in a periodic channel Pi ^h subset {{mathbb {R}}}^3, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough h is proven.

Trapped modes and resonances for thin horizontal cylinders in a two-layer fluid

Exact solutions of the linear water-wave problem describing oblique waves over a submerged horizontal cylinder of small (but otherwise fairly arbitrary) cross-section in a two-layer fluid are constructed in the form of convergent series in powers of the small parameter characterizing the “thinness” of the cylinder. The terms of these series are expressed through the solution of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder. The solutions obtained describe trapped modes corresponding to discrete eigenvalues of the problem (lying close to the cut-off frequency of the continuous spectrum) and resonances lying close to the embedded cut-off. We present certain conditions for the submergence of the cylinder in the upper layer when these resonances convert into previously unobserved embedded trapped modes.

The Dirichlet–Neumann Operator for Oblique Water Waves over a Submerged Thin Cylinder and an Application

The Dirichlet-Neumann operator for the linear water-wave problem describing oblique waves over a submerged horizontal cylinder of small (but otherwise, fairly arbitrary) cross-section in a layer of finite depth is constructed in the form of convergent series in powers of the small parameter characterizing the “thinness” of the cylinder. The terms of these series are expressed through the solution of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder. As an application, discrete eigenvalues of this operator which are frequancies of trapped modes of the problem are obtained.

A fast multipole boundary element method for the three dimensional linear water wave-structure interaction problem with arbitrary bottom topography

We present a numerical method that allows for the efficient solution of general linear water-wave problems with multiple arbitrarily shaped bodies and seabed geometry when the number of unknowns in the problem N is large. The problem domain is divided into an internal region that is modeled using a simple-source boundary integral equation and an outer domain of uniform water depth. The contribution of this work is in introducing an adpative fast-multipole method to the solution procedure, thereby changing the complexity of the problem from O(N3) (or O(N2) with iterative solvers) to O(N) when N is large. The memory requirements scale linearly with N as well. This methodology is needed when the number of bodies is large, when structures with complex shapes are modeled, or when the bottom topography for a problem varies considerably. We use the procedure developed to study the effects of topography over two case examples of floating bodies: 4 truncated vertical cylinders over a bottom protrusion, for which results indicate how variations in topography cause changes to wave loads on the bodies and a configuration of 16 cylinders (with N ~ 150, 000) and the effects of the variable ocean floor are again considered.

Linear Modes for Channels of Constant Cross-Section and Approximate Dirichlet–Neumann Operators

We study normal modes for the linear water wave problem in infinite straight channels of bounded constant cross-section. Our goal is to compare semi-analytic normal mode solutions known in the literature for special triangular cross-sections, namely isosceles triangles of equal angle of \(45^{\circ }\) and \(60^{\circ }\), see Lamb (Hydrodynamics. Cambridge University Press, Cambridge, 1932) , Macdonald (Proc Lond Math Soc 1:101–113, 1893), Greenhill (Am J Math 97–112, 1887), Packham (Q J Mech Appl Math 33:179–187, 1980), and Groves (Q J Mech Appl Math 47:367–404, 1994), to numerical solutions obtained using approximations of the non-local Dirichlet–Neumann operator for linear waves, specifically an ad-hoc approximation proposed in Vargas-Magana and Panayotaros (Wave Motion 65:156–174, 2016), and a first-order truncation of the systematic depth expansion by Craig et al. (Proc R Soc Lond A: Math, Phys Eng Sci 46:839–873, 2005). We consider cases of transverse (i.e. 2-D) modes and longitudinal modes, i.e. 3-D modes with sinusoidal dependence in the longitudinal direction. The triangular geometries considered have slopping beach boundaries that should in principle limit the applicability of the approximate Dirichlet–Neumann operators. We nevertheless see that the approximate operators give remarkably close results for transverse even modes, while for odd transverse modes we have some discrepancies near the boundary. In the case of longitudinal modes, where the theory only yields even modes, the different approximate operators show more discrepancies for the first two longitudinal modes and better agreement for higher modes. The ad-hoc approximation is generally closer to exact modes away from the boundary.

Effect of finite amplitude of bottom corrugations on Fabry-Perot resonance of water waves.

Recently, the mechanism of Fabry-Perot (F-P) resonance in optics was extended to monochromatic water waves propagating in a domain with two patches of sinusoidal corrugations on an otherwise flat bottom. Assuming small-amplitude surface waves, an asymptotic linear analytical solution (ALAS) was derived by L. A. Couston etal. Phys. Rev. E 92, 043015 (2015).PLEEE81539-375510.1103/PhysRevE.92.043015 When resonance conditions are met, the ALAS predicts large amplification of the incident waves in the resonator area between the two patches of corrugations. Based on the ALAS, the amplitude of these standing waves is expected to increase exponentially with the relative amplitude of bottom corrugations (δ=d/h, where d is the corrugation amplitude and h the still water depth). In the present work, we examine the effects associated with the assumptions made in deriving the ALAS regarding the effect of a finite amplitude of bottom corrugation (i.e., finite value of δ), still in the linear wave framework. F-P resonance is studied by means of highly accurate numerical simulations, considering either the exact linear water wave problem (system A) or an approximate problem with a first-order expansion of the bottom boundary condition (system B). The numerical model is first validated on a Bragg resonance case, through comparisons with the ALAS, experimental measurements, and existing numerical simulations, showing its ability to represent well the so-called wave-number downshift of Bragg resonance (i.e., the slight decrease in the incident wave number where maximum resonance is reached in comparison with the value predicted by the ALAS). We then analyze how this downshift affects the F-P resonance, especially when the corrugations are of finite amplitude, i.e., δ varying from 0.05 to 0.4. The wave-number downshift appears to have a strong effect on the F-P resonance for δ>0.1: very low wave amplification manifests for the wave number predicted by the ALAS. However, when the incident wave number is slightly decreased (by an amount increasing with δ) the F-P resonance case can be recovered, and the maximum amplification values are found to be close to the predictions from the ALAS (e.g., up to a factor of about 27 for δ=0.4). The variations of the reflection coefficient and enhancement factor obtained from systems A and B as a function of the incident wave number are discussed and compared to ALAS predictions. In particular, it is found that the resonance peak is extremely narrow when δ=0.2 and 0.4.

An Iterative Parallel Solver in GPU Applied to Frequency Domain Linear Water Wave Problems by the Boundary Element Method

In this paper the implementation of an iterative solver based on the Generalized Minimum Residual Method (GMRES) with complex-valued coefficients is explored, with application to the Boundary Element Method (BEM). The solver is designed to be implemented in a GPU (Graphic Processing Unit) device, exploiting its massively parallel capabilities. The framework is in the context of linear water wave problems in the frequency domain. The main objective of the proposed solver is the direct acceleration of existing standard BEM codes, by transfering to the GPU the solver task. The CUDA programming language is used, exploiting the particular architecture of the GPU device for complex-valued systems. To explore the performances of the solver, two linear water wave problems have been tested: the frequency-dependent added mass and damping matrices of a 3D floating body, and the Helmholtz equation in a 2D domain. The parallelized GMRES solver is implemented in a NVidia GeForce GTX 970 graphic card, and shows drastic reductions in computing times when compared with its CPU implementation.

A coupled finite and boundary spectral element method for linear water-wave propagation problems

• A coupled finite and boundary spectral element method for linear water-wave propagation problems is proposed. • Boundary spectral element method (BSEM) is a new technique that combines the advantages of the spectral approach and the BEM. • BSEM has been applied to the mild-slope equation with variable bathymetry in one direction. • A convergence study has been made for the BSEM alone and coupled with finite spectral elements. • The proposed formulation has been validated by solving classical water-wave propagation problems. A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p -convergence of spectral methods.

Multiply heaving bodies in the time-domain: Symmetry and complex resonances

The time-domain linear water wave problem for multiple bodies which are constrained to move in heave after being given an arbitrary initial displacement is considered. It is shown that the solution can be calculated in a simple and efficient method using the frequency domain solution. The resonant behaviour of the system is then investigated and an approximation to the solution using the singularity expansion method derived. A method to exploit any symmetry in the multiple body geometry, which makes the calculations more efficient and simplifies the analysis, is presented. Numerical calculations are performed for truncated cylinders, using bespoke code that computes the solution for complex frequencies. A method to approximate the solution for complex frequencies using results for real frequencies is derived and this is used to make calculations for heaving hemispheres using existing general purpose computer code.

Water waves trapped by thin submerged cylinders: Exact solutions

Exact solutions of the linear water-wave problem describing oblique water waves trapped by a submerged horizontal cylinder of small (but otherwise arbitrary) cross-section are constructed in the form of a convergent series in powers of the small parameter characterizing the “thinness” of the cylinder. The terms of this series are expressed through the solutions of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder.

Spectral gaps for the linear surface wave model in periodic channels

Summary We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width � . Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as � → 0: the endpoints are determined within corrections of order � 3/2 . The width of the first bands is shown to be O(� ). Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.

Spectrum of the linear water model for a two‐layer liquid with cuspidal geometries at the interface

We show that the linear water wave problem in a bounded liquid domain may have continuous spectrum, if the interface of a two‐layer liquid touches the basin walls at zero angle. The reason for this phenomenon is the appearance of cuspidal geometries of the liquid phases. We calculate the exact position of the continuous spectrum. We also discuss the physical background of wave propagation processes, which are enabled by the continuous spectrum. Our approach and methods include constructions of a parametrix for the problem operator and singular Weyl sequences.

Linear water-wave problem in a pond with a shallow beach

We consider the linear water-wave problem in a bounded water-basin with a shallow beach. In spite of the boundedness of the domain, the spectrum of the problem may have a continuous component, if the beach of the basin has a cuspidal form. Following the approach and methods of an earlier paper by S.A.Nazarov and the second named author, we improve the results of the citation by determining the spectrum in an open borderline case under weaker geometric assumptions.

Localization Estimates for Eigenfrequencies of Waves Trapped by a Freely Floating Body in a Channel

The linear water-wave problem of F. John is studied in the case of a freely floating body in an unbounded cylindrical channel. We obtain a sufficient condition for a trapped mode eigenfrequency as well as a localization estimate for its position in the spectrum: the eigenfrequency is compared with an eigenfrequency of a reference problem involving an integro-differential boundary condition on the wetted surface of the body. The localization estimate is derived by reducing the original John problem to a standard spectral problem for a self-adjoint operator in a complicated Hilbert space, and by an application of basic spectral measure theory. Applications of the sufficient condition to concrete cases are given.

On the Linear Water Wave Problem in the Presence of a Critically Submerged Body

We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a critically submerged body (i.e., the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyze its properties. Under a geometric condition introduced by V. Maz'ya in 1978, we prove an important property of the scattering matrix, which may be interpreted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g., $H^2_{loc}\cap L^\infty$) without use of the radiation conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The fact that the existence and uniqueness result does not rely on either the radiation conditions or the limiting absorption principle is the first result of this type known to us in the theory of linear wave problems in unbounded domains.

A body traps as many water-wave modes in a symmetric channel as it wishes

It is proved that, under a certain geometrical assumption in the linear water-wave problem, a body approaching the water surface of a symmetric three-dimensional channel gets any number of trapped modes with frequencies in the interval (0, δ) of the continuous spectrum; here δ can be any given positive number.

A General Spectral Approach to the Time-Domain Evolution of Linear Water Waves Impacting on a Vertical Elastic Plate

We present a solution in the time-domain to the two-dimensional linear water-wave problem, in which a semi-infinite fluid region is bounded on one side by a vertical elastic plate. The problem is solved using a generalized eigenfunction expansion from the solutions for single frequencies, and we begin with a novel solution of the single-frequency problem. By formulating the problem using the acceleration potential, we find an inner-product space, in which the evolution operator with continuous spectrum is self-adjoint. This inner-product space is required for the generalized eigenfunction solution, which allows us to prescribe arbitrary initial water-surface and plate displacements and velocities. Furthermore, using the generalized eigenfunction expansion, the solution is approximated by deforming the contour of integration and using the contributions from the singularities of the analytic continuation. Numerical experiments show that the long-time behavior in certain situations can be closely approximated by this method.

The effect of surface tension on localized free-surface oscillations about surface-piercing bodies

It is well known that the inclusion of surface tension in the linear water-wave problem introduces an additional term in the free-surface boundary condition. Furthermore, if the fluid contains one or more partially immersed bodies, edge conditions describing the motion of the fluid at each contact line need to be applied. In this paper, an inverse procedure is used to construct examples of two-dimensional, surface-piercing trapping structures for non-zero values of surface tension. The problem is considered with two separate edge conditions that are appropriate for investigations involving trapped modes. The first edge condition fixes each contact line and the procedure used forces the bodies to be horizontal at the contact points; it is shown that results can be found for all values of surface tension. The second condition forces the free-surface slope to be zero at the contact points, and results are obtained for a restricted range of surface tension values.

The effect of surface tension on trapped modes in water-wave problems

In this paper the effect of surface tension is considered on two two-dimensional water-wave problems involving pairs of immersed bodies. Both models, having fluid of infinite depth, support localized oscillations, or trapped modes, when capillary effects are excluded. The first pair of bodies is surface-piercing whereas the second pair is fully submerged. In the former case it is shown that the qualitative nature of the streamline shape is unaffected by the addition of surface tension in the free surface condition, no matter how large this parameter becomes. The main objective of this paper, however, is to study the submerged body problem. For this case it is found, by contrast, that there exists a critical value of the surface tension above which it is no longer possible to produce a completely submerged pair of bodies which support trapped modes. This critical value varies as a function of the separation of the two bodies. It can be inferred from this that surface tension does not always play a qualitatively irrelevant role in the linear water-wave problem.

Water-wave problems, their mathematical solution and physical interpretation

A typical linear water-wave problem is a boundary-value problem involving partial differential equations and boundary conditions. Such a problem is usually solved by applying a sequence of mathematical arguments, and it would be helpful if some or all of the successive steps in this sequence could be given a physical interpretation. In the author’s experience this is generally not possible. Illustrative examples are presented.