Separation of diffracted waves via SVD filter

Diffracted seismic waves may be used to help identify and track geologically heterogeneous bodies or zones.However,the energy of diffracted waves is weaker than that of reflections.Therefore,the extraction of diffracted waves is the basis for the effective utilization of diffracted waves.Based on the difference in travel times between diffracted and reflected waves,we developed a method for separating the diffracted waves via singular value decomposition filters and presented an effective processing flowchart for diffracted wave separation and imaging.The research results show that the horizontally coherent difference between the reflected and diffracted waves can be further improved using normal move-out(NMO) correction.Then,a band-rank or high-rank approximation is used to suppress the reflected waves with better transverse coherence.Following,separation of reflected and diffracted waves is achieved after the filtered data are transformed into the original data domain by inverse NMO.Synthetic and field examples show that our proposed method has the advantages of fewer constraints,fast processing speed and complete extraction of diffracted waves.And the diffracted wave imaging results can effectively improve the identification accuracy of geological heterogeneous bodies or zones.

Angle Compensation and Asymmetry Effect of Light Diffracted by Millimeter Liquid Surface Slosh Wave

The angle compensation method is adopted to detect sloshing waves by laser diffraction, in the case that the wavelength of the sloshing waves is much greater than that of the incident light. The clear diffraction pattern is observed to be of asymmetry, involving orders, position and interval of the diffraction spots that are discovered during the light grazing incidence. It is found that the larger the angle of incidence is, the more obvious the asymmetry is. The higher the negative diffraction orders are, the smaller the intervals between spots are. On the contrary~ in the positive region, the higher the diffraction orders are, the larger the spot intervals are. The positive interval is larger than that of the same negative diffraction order. If the incident angle reaches 1.558 rad in the experiment, all positive diffraction orders completely vanish. Based on the mechanism of phase modulation and with the Fourier transform method, the relations between the incident angle and position, interval spaces, and orders of diffraction spots are derived theoretically. The theoretical calculations are compared with the experimental data, and the comparison shows that the theoretical calculations are in good agreement with the experimental measurement.

Numerical Study of Wave Diffraction Effect Introduced in the SWAN Model

New version of SWAN model includes the wave diffraction effect which is the main improvement compared with the previous versions. Experimental data collected in the wave basin of the University of Delaware were used to test its performance. Wave heights were compared in the four cases （with different wave energies and directional spreading spectra）. The results agreed well with the measurements, especially for the broad directional spectra cases. The effect of wave diffraction was analyzed by switching on/off the corresponding tenn. By introducing the diffraction term, the distributions of wave height and wave direction were smoothed, especially obvious for the narrow spectrum cases. Compared with the calculations without diffraction, the model with diffraction effect gave better results.

Wave Diffraction on Arc-Shaped Floating Perforated Breakwaters

An analytical method is developed to study the sheltering effects on arc-shaped floating perforated breakwaters. In the process of analysis, the tloating breakwater is assumed to be rigid, thin, vertical, and immovable and located in water with constant depth. The fluid domain is divided into two regions by imaginary interface. The velocity potential in each region is expanded by eigenfunction in the context of linear theory. By satisfying continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations can be obtained to determine the unknown coefficients for eigenfunction expansions. The accuracy of the present model was verified by a comparison with existing results for the case of arc-shaped floating breakwater. Numerical results, in the form of contour maps of the non-dimensional wave amplitude around the breakwater and diffracted wave amplitude at typical sections, are presented for a range of wave and breakwater parameters. Results show that the sheltering effects on the arc-shaped floating perforated breakwater are closely related to the incident wavelength, the draft and the porosity of the breakwater.

Diffraction of SH-waves by topographic features in a layered transversely isotropic half-space

Abstract： The scattering of plane SH-waves by topographic features in a layered transversely isotropic （TI） half-space is investigated by using an indirect boundary element method （IBEM）. Firstly, the anti-plane dynamic stiffness matrix of the layered TI half-space is established and the free fields are solved by using the direct stiffness method. Then, Green＇s functions are derived for uniformly distributed loads acting on an inclined line in a layered TI half-space and the scattered fields are constructed with the deduced Green＇s functions. Finally, the free fields are added to the scattered ones to obtain the global dynamic responses. The method is verified by comparing results with the published isotropic ones. Both the steady-state and transient dynamic responses are evaluated and discussed. Numerical results in the frequency domain show that surface motions for the TI media can be significantly different from those for the isotropic case, which are strongly dependent on the anisotropy property, incident angle and incident frequency. Results in the time domain show that the material anisotropy has important effects on the maximum duration and maximum amplitudes of the time histories.

Evaluation of Diffraction Predictability in Two Phase Averaged Wave Models

Two different methods for incorporating diffraction effect into wave action balance equation based coastal spectral wave models, WABED and SWAN, are discussed and evaluated with respect to their formulations, numerical implementations, and modeling capabilities. Both models were nm to simulate the wave transformation through a gap between two infinitely long breakwaters and that across an elliptical shoal observed in laboratory studies, with the emphasis laid on the diffraction induced by either obstacles or wave amplitude variations. Calculations of WABED were compared with Sommerfeld＇s analytical solutions, experimental observations and SWAN simulations. It is shown that both methods can predict reasonably wave diffraction for the two eases studied herein, and a fairly better performance is provided by WABED for stronger diffraction ease.

Analytical Research on Wave Diffraction on Arc-Shaped Bottom-Mounted Perforated Breakwaters

An analytical method is developed to study wave diffraction on arc-shaped and bottom-mounted perforated breakwaters. The breakwater is assumed to be rigid, thin, vertical, immovable and located in water of constant depth. The fluid domain is divided into two regions by imaginary interface. The velocity potential in each region is expanded by eigenfunctions. By satisfying the continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations can be obtained to determine the unknown coefficients of eigenfunctions. Numerical results, in the form of contour maps of the relative wave amplitude around the breakwater, are presented for a range of wave and breakwater parameters. Results show that the wave diffraction on the arc-shaped and bottom-mounted perforated breakwater is related to the incident wavelength and the porosity of the breakwater. The porosity of the perforated breakwater may have great effect on the diffracted field.

SECOND-ORDER INTERACTION OF IRREGULAR WAVES WITH A TRUNCATED COLUMN

A complete semi-analytical solution is obtained for second-order diffraction of plane bichromatic waves by a fixed truncated circular column.The fluid domain is divided into interior and exterior regions.In the exterior region,the second-order velocity potential is expressed in terms of‘locked-wave’and‘free-wave’ components,both are solved using Fourier and eigenfunction expansions.The re- sulting‘locked wave’potential is expressed by one-dimensional Green's integrals with oscillating integrands.In order to increase computational efficiency,the far-field part of the integrals are carried out analytically.Solutions in both regions are matched on the interface by the potential and its normal derivative continuity conditions.Based on the present approach,the sum-and difference-frequency potentials are efficiently evaluated and are used to generate the quadratic transfer functions which correlates the incident wave spectrum with second-order forcing spectrum on the column.The sum-frequency QTFs for a TLP column are present,which are compared for some frequency pairs with those from a fully numerical procedure.Satisfactory agreement has been obtained.QTF spectra for a case study TLP column,generated using the semi-analytical solution are presented.Also given are the results for nonlinear wave field around the column.

A composite numerical model for wave diffraction in a harbor with varying water depth

A composite numerical model is presented for computing the wave field in a harbor. The mild slope equation is discretized by a finite element method in the domain concerned. Out of the computational domain, the water depth is assumed to be constant. The boundary element method is applied to the outer boundary for dealing with the infinite boundary condition. Because the model satisfies strictly the infinite boundary condition, more accurate results can be obtained. The model is firstly applied to compute the wave diffraction in a narrow rectangular bay and the wave diffraction from a porous cylinder. The numerical results are compared with the analytical solution, experimental data and other numerical results. Good agreements are obtained. Then the model is applied to computing the wave diffraction in a square harbor with varying water depth. The effects of the water depth in the harbor and the incoming wave direction on the wave height distribution are discussed.

Hydroelastic Response of A Circular Plate in Waves on A Two-Layer Fluid of Finite Depth

The hydroelastic response of a circular, very large floating structure(VLFS), idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the platecovered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presented to investigate the effects of different physical quantities, such as the thickness of the plate, Young’s modulus, the ratios of the densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid. Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.

Shock wave mitigation using zig-zag structures and cylindrical obstructions

The present study focuses on the mitigation of shock wave using novel geometric passages in the flow field.The strategy is to produce multiple shock reflections and diffractions in the passage with minimum flow obstruction,which in turn is expected to reduce the shock wave strength at the target location.In the present study the interaction of a plane shock front(generated from a shock tube)with various geometric designs such as,1)zig-zag geometric passage,2)staggered cylindrical obstructions and 3)zigzag passage with cylindrical obstructions have been investigated using computational technique.It is seen from the numerical simulation that,among the various designs,the maximum shock attenuation is produced by the zig-zag passage with cylindrical obstructions which is then followed by zig-zag passage and staggered cylindrical obstructions.A comprehensive investigation on the shock wave reflection and diffraction phenomena happening in the proposed complex passages have also been carried out.In the new zig-zag design,the initial shock wave undergoes shock wave reflection and diffraction process which swaps alternatively as the shock front moves from one turn to the other turn.This cyclic shock reflection and diffraction process helps in diffusing the shock wave energy with practically no obstruction to the flow field.It is found that by combining the shock attenuation ability of zig-zag passage(using shock reflection and diffraction)with the shock attenuation ability of cylindrical blocks(by flow obstruction),a drastic attenuation in shock strength can be achieved with moderate level of flow blocking.

A Fully Nonlinear HOBEM with the Domain Decomposition Method for Simulation of Wave Propagation and Diffraction

A higher-order boundary element method(HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with continuity conditions enforced on the interfaces between the adjacent sub-domains is implemented for reducing the computational cost. By adjusting the algorithm of iterative procedure on the interfaces, four types of coupling strategies are established, that is, Dirchlet/Dirchlet-Neumman/Neumman(D/D-N/N), Dirchlet-Neumman(D-N),Neumman-Dirchlet(N-D) and Mixed Dirchlet-Neumman/Neumman-Dirchlet(Mixed D-N/N-D). Numerical simulations indicate that the domain decomposition methods can provide accurate results compared with that of the single domain method. According to the comparisons of computational efficiency, the D/D-N/N coupling strategy is recommended for the wave propagation problem. As for the wave-body interaction problem, the Mixed D-N/N-D coupling strategy can obtain the highest computational efficiency.

A SECOND ORDER STATIONARY SOLUTION OF THE THREE DIMENSIONAL WAVE DIFFRACTION POTENTIAL

A theory on the second order wave diffraction by a three dimensional body fixed in a regular sea has been developed in the present paper. By regarding the sinusoidal disturb potential as a stationary solu- tion of an initial value problem, and using Laplace transformation method and Tauberian theorem, the boundary value problems of stationary solution of the first and second order diffraction potential have been de- rived in this paper. Furthermore, the explicit solution of the second order stationary diffraction potential has been obtained with the method of the double Fourier transformation. It is found that the asymptotic behaviour of the second order stationary solution at far field is dependent on two wave systems, the first is 'free wave', travelling independently of the first order wave system, the other is 'phase locked waves', which accompany the first order waves. At the same time, the radiation conditions of the second order diffraction problems are derived. We also find that one can still pursue a steady state formulation with the inclusion of Rayleigh damping. Finally, as an example, the second order wave forces upon a fixed vertical cir- cular cylinder have been calculated, and the numerical results agree well with the experimental data.

Wave Radiation and Diffraction by A Two-Dimensional Floating Body with An Opening Near A Side Wall

The radiation and diffraction problem of a two-dimensional rectangular body with an opening floating on a semi- infinite fluid domain of finite water depth is analysed based on the linearized velocity potential theory through an analytical solution procedure. The expressions for potentials are obtained by the method of variation separation, in which the unknown coefficients are determined by the boundary condition and matching requirement on the interface. The effects of the position of the hole and the gap between the body and side wall on hydrodynamic characteristics are investigated. Some resonance is observed like piston motion in a moon pool and sloshing in a closed tank because of the existence of restricted fluid domains.

AN ANALYSIS FOR DIFFRACTION OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN THICK PLATES WITH A CAVITY

By using the complex variable method and conformal mapping, the diffraction of flexural waves and dynamic stress concentrations in thick plates with a cavity have been studied. A general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of an arbitrary cavity is obtained. By employing the orthogonal function expansion technique, the dynamic stress problem can be reduced to the solution of an infinite algebraic equation series. As an example, the numerical results for the dynamic stress concentration factor in thick plates with a circular, elliptic cavity are graphically presented. The numerical results are discussed.

UNSTEADY INTERACTION OF SHOCK WAVE DIFFRACTING AROUND A CIRCULAR CYLINDER IN AIR

The reflection and diffraction of a planar shock wave around a circular cylinder are a typical problem of the complex nonlinear shock wave phenomena in literature.It has long been studied experimentally,analytically as well as numerically.Takayama in 1987 obtained clear experimental pictures ofisopycnics in shock tube under the condi- tion that the impinging shock wave propagates as far as 3 diameters away from the cylinder.To know more complete- ly the whole unsteady process,it is desirable to get experimental results in a region which is more than 10 diameters away from the cylinder.This is what has been done in this paper by using the pulsed laser holographic interferometry for several shock Mach numbers of the impinging shock. Results for several moments are shown,giving more know- ledge about the whole unsteady flow field.This is useful for a reliable and complete understanding of the changing force acting on the cylinder,and provides interesting data to check the performance of many recently developed high resolution numerical methods for unsteady shock wave calculation.

Theoretical solution for wave diffraction by wedge or corner with arbitrary reflection characteristics

An exact analytic solution for wave diffraction by wedge or corner with arbitrary angle (rπ) and reflection coefficients (R0 and Rr) is presented in this paper. It is expressed in two forms-series and integral representations, corresponding recurrence relation and asymptotic expressions are also derived. The solution is simplified for some special cases of rπ. For Rr= R0,r= 1/N and Rr≠R0,r = 1/2N, the solution can be reduced to linear superpositions of incident and partially reflected waves, hence a nonlinear solution of forth order for this problem can be obtained by using the author's theory of nonlinear interaction among gravity surface waves. The given solution is related to inhomogeneous Robin boundary conditions, which include the Neumann boundary conditions usually accepted in wave diffraction theory.

Interaction of Oblique Waves and A Rectangular Structure with An Opening Near A Vertical Wall

With the method of separation of variables and the eigenfunction expansion employed, an analytical solution is presented for the radiation and diffraction of a rectangular structure with an opening near a vertical wall in oblique seas, in which the unknown coefficients are determined by the boundary conditions and matching requirement on the interface. The effects of the width of the opening and the angle of incidence on the hydrodynamic characteristics of a rectangular structure with an opening near a vertical wall are mainly studied. The comparisons of the calculation results with wall-present and with wall-absent are also made. The results indicate that the variation trends of the heave added mass and excitation force with wall-present are almost the same as those with wall-absent, and that the peak values in the former case are obviously larger than those in the latter due to the reflection of the vertical wall.

Wave Defending Effects of V-Type Bottom-Mounted Breakwaters

An analytical method is developed for the study of the wave defending effects of the V-type bottom-mounted breakwater. The breakwater is assumed to be rigid, thin, impermeable and vertically located in water of constant depth. The fluid domain is divided into three sub-regions by an imaginary interface. The velocity potential in each region is expanded by eigenfunctions. By satisfying the corresponding boundary conditions and matching conditions in and between sub-regions, a set of linear algebraic equations can be obtained to determine the unknown coefficients for the eigenfunction expansions for each sub-region. The accuracy of the present model is verified by a comparison with existing results for the case of an isolated breakwater. Numerical results, in the form of contour maps of the relative wave amplitude around the breakwater, are presented for a range of wave and breakwater parameters. The results show that the V-type bottom-mounted breakwater is generally effective in defending against waves. In general, the wave height in the protected area is about 20～50 percent of the incident wave height.

Wave Action on Structures of Combined Cylinders

The scattering of water wave by an array of combined cylinders with solid interior columns and partially perforated exterior columns at the upper section of an outer wall is solved with a linear model for the pressure difference between the two sides of a porous wall and the fluid velocity inside it. A semi-analytic linear solution is established by the eigenfunction expansion method for wave diffraction from the array of combined cylinders. Both numerical experiment and physical experiment are carried out to verity the theory. Also presented are some numerical examples to show the situation of wave force on the cylinders influenced by the effeet factor of porosity G, relative wave number ka and the ratio of water depth inside and outside the cylinders.