Search

The characteristics of wave spectra in the swell-dominated seas in the Chancay Bay, Peru, and off the coast of Hambantota, Sri Lanka, were studied based on in situ measurements. According to the characteristics of the spectral shapes, the measured wave spectra were divided into single- and double-peaked spectra. For the single-peaked measured spectra, many widely used wave spectrum models were adopted to fit the measured wave spectra. By comparing the fitting results with the measured wave spectra, these wave spectra were insufficient for describing the peak enhancement or the high-frequency tail shape of the measured spectra. The spectral shape parameters were found to play an important role in the expression of the spectrum models. The peak enhancement factor γ in the Joint North Sea Wave Project (JONSWAP) spectra could adjust the peak height of the spectrum and the shape parameter m of the spectrum high-frequency tail in the Wallops spectra made the wave spectrum more flexible. Therefore, an improved single-peaked wave spectrum was proposed by combining the JONSWAP and Wallops spectra. It performed better than other spectrum models in describing the swell and has advantages in both the wave characteristic parameters and wave spectral shapes. The spectrum peak width parameter σ was relatively consistent in the two measurement stations and both around 0.24. Moreover, the proposed model overcame the limitations of the excessively high estimated slope in the high-frequency range of the Wallops spectrum; the shape parameter m of the spectrum high-frequency tail is primarily distributed in the range (1, 5). For the double-peaked measured spectra, the Ochi–Hubble spectrum was found to work better than the Torsethaugen spectrum.

In order to meet the needs of marine tourism development, this paper analyzes the water resonance in the inner domain of a fixed floating tourist platform based on a 3D non-hydrostatic model. The tourist platform has a complex structure that is prone to the resonance of the water in the inner domain under wave actions. The water’s motion response in the platform’s inner domain under wave action is simulated numerically using numerical simulation methods. The non-hydrostatic model used in this paper is based on a semi-implicit method to solve the incompressible Navier–Stokes equations (NSE) and combines the immersed boundary method with the global continuity equations in the pressure zone (the flow zone under the structure) to approach the problem. In this paper, firstly, the resonant frequencies and resonant amplitudes of the water inside the box-shaped ship are calculated and compared with the experimental data to demonstrate the non-hydrostatic model’s accuracy in calculating the resonance problem of the water in the inner domain of a fixed floating platform. Secondly, the water’s resonant frequencies and resonant modes in the inner domain are calculated by numerical modeling of the floating tourist platform, including rare high-order resonance modes. We found that there are six resonance modes with complex shapes in the water of the inner domain of the tourist platform. Furthermore, the effects of the wall thickness, chord distance and draft of the annular tourist platform on the resonant amplitude values and modes of the water domain within the platform are analyzed in depth.

In this paper, a 2-stage cascaded deep learning framework, Port Wave Prediction Network (PWPNet), is proposed for real-time prediction of significant wave height (SWH) distribution in a port. The PWP-out model of the first stage, predicting port-entrance wave parameters, utilizes three branches, the first branch using a Long Short Term Memory (LSTM) module to learn the temporal dependencies of time sequences of port-entrance wave parameters, the second branch using Wave and Wind field Feature Extraction (WWFE) modules, composed of a residual network with spatial and channel attention, to capture spatiotemporal characteristics of outside-port 2D wave and wind field data, the third branch using multi-scale time encoding to capture the periodic characteristics of waves and wind. The PWP-in model of the second stage, estimating the in-port SWH distribution, uses port-entrance wave parameters based on a customized Artificial Neural Network (ANN) and takes PWP-out’s output as its input. A comparison of the performance of PWP-out and mainstream machine learning models including LSTM, GRU, BPNN, SVR, ELM, and RF at Hambantota Port shows that PWP-out outperforms all other models regarding medium-term (25–48 h), med–long-term (49–72 h), and long-term (73–96 h) predictions, and ablation experiments proved the effectiveness of the three branches. Furthermore, the performance comparison of our PWPNet and other 2-stage models of LSTM, GRU, BPNN, SVR, ELM, and RF cascaded with PWP-in shows that PWPNet outperforms those cascaded models for medium-term to long-term predictions of SWH distribution in a port.

Numerical simulations were performed to study the long-distance evolution of irregular waves in deep water. It was observed that some solitons, which are the theoretical solutions of the nonlinear Schrödinger equation, emerged spontaneously as irregular wave trains propagated in deep water. The solitons propagated approximately at a speed of the linear group velocity. All the solitons had a relatively large amplitude and one detected soliton’s height was two times larger than the significant wave height of the wave train, therefore satisfying the rogue wave definition. The numerical results showed that solitons can persist for a long distance, reaching about 65 times the peak wavelength. By analyzing the spatial variations of these solitons in both time and spectral domains, it is found that the third-and higher-order resonant interactions and dispersion effects played significant roles in the formation of solitons.

Focused wave groups are traveling waves characterized by extremely-large transient wave amplitudes and very short durations. These waves usually cause serious damage to marine/offshore structures and coastal infrastructures, and can even result in human casualties (Nikolkina and Didenkulova, 2011). The studies on natural disasters related to the focused wave groups near the coastal zone have been mostly confined to wave evolution over beaches, wave runup, overtopping, and their impact forces acting on the coastal infrastructures (e.g., the seawall and the circular cylinder); the influence of focused transient wave groups on harbors has not yet been studied. In this study, the generation and propagation of focused transient wave groups and their interactions with the harbor are simulated using a fully nonlinear Boussinesq model, FUNWAVE 2.0. To this end, four elongated harbors with constant depth and a series of focused wave groups with various focused wave amplitudes, spectral width parameters, and incident directions are considered. Based on the Morlet wavelet transform and discrete Fourier transform techniques, the capability of focused transient wave groups to trigger the harbor resonance phenomenon is revealed for the first time. Subsequently, the influences of spectral width parameter, incident wave direction, and resonant mode on different resonant wave parameters (including maximum runup and resonant intensity of various resonant modes inside a harbor) are comprehensively investigated, and it is found that these three factors have significant effects on resonant wave parameters.

Single Gaussian wave groups with different initial wave steepness e0 and width N are produced in laboratory in finite depth to study the nonlinear evolution, the extreme events and breaking. The results show that wave groups with larger e0 will evolve to be several envelope solitons (short wave groups). By analyzing geometric parameters, a break in the evolution of the wave elevation and asymmetric parameters after extreme wave may be an indicator for the inception of refocus and the maximal wave moving to the middle, namely, wave down-shift occurs. The analysis of the surface elevations with HHT (Hilbert-Huang Transform), which presents the concrete local variation of energy in time and frequency can be exhibited clearly, reveals that the higher frequency components play a major role in forming the extreme event and the contribution to the nonlinearity. Instantaneous energy and frequency in the vicinity of the extreme wave are also examined locally. For spilling breakers, the energy residing in the whole wave front dissipates much more due to breaking, while the energy in the rear of wave crest loses little, and the intra-wave frequency modulation increases as focus. It illustrates that the maximal first order instantaneous frequency f1 and the largest crest tend to emerge at the same time after extreme wave when significant energy dissipation happens, and vice versa. In addition, it shows that there is no obvious relation of the CDN (combined degree of nonlinearity) to the wave breaking for the single Gaussian wave group in finite water depth.

Periodic undulating topographies (such as sandwaves and sandbars) are very common in coastal and estuarine areas. Normally incident water surface waves propagating from open sea to coastal areas may interact strongly with such topographies. The wave reflection by the periodic undulating topography can be significantly amplified when the surface wavelength is approximately twice the wavelength of the bottom undulations, which is often called as Bragg resonant reflection. Although the investigations on the hydrodynamic characteristics related to Bragg reflection of a region of undulating topography have been widely implemented, the effects of Bragg reflection on harbors have not yet been studied. Bragg resonant reflection can effectively reduce the incident waves. Meanwhile, however, it can also significantly hinder the wave radiation from the harbor entrance to the open sea. Whether Bragg reflection can be utilized as a potential measure to alleviate harbor oscillations is unknown. In the present study, Bragg reflection and their interactions with the harbor are simulated using a fully nonlinear Boussinesq model, FUNWAVE 2.0. For the purpose, an elongated harbor with constant depth is considered, and a series of sinusoidal bars with various amplitudes and numbers are deployed outside the harbor. The incident waves considered in this paper include regular long waves and bichromatic short wave groups. It is revealed for the first time that for both kinds of incident waves, Bragg resonant reflection can significantly alleviate harbor resonance. The influences of the number and the amplitude of sinusoidal bars on the mitigation effect of harbor resonance and on the optimal wavelength of sinusoidal bars that can achieve the best mitigation effect are comprehensively investigated, and it is found that the former two factors have remarkable influences on the latter two parameters. The present research provides a new option for the mitigation of harbor oscillations via changing the bottom profile, which is feasible as long as the navigating depth is guaranteed.

Forecasting coastal infragravity (IG) waves is crucial in preventing coastal disasters caused by IG waves. In this study, the performance of the unstructured WAVEWATCH Ⅲ (WW3) in predicting coastal IG waves is comparatively assessed using in-situ wave measurements, considering the effects of numerical schemes, time steps, and IG-source parameterization methods. The wave measurements are performed using three acoustic sensors installed at the coast of south Sri Lanka, with two being near or inside a harbor. Sensitivity analysis of the numerical schemes and time steps shows that the explicit scheme combined with a small time step leads to unexpected oscillations of the IG wave height. However, the time step has a negligible influence on the performance of the implicit scheme combined with the domain decomposition method. Generally, IG sources estimated from the empirical formula method (source M1) and the second-order spectrum method (source M2) perform similarly in reproducing the IG wave height. Regarding the IG frequency spectrum, both the IG sources properly predict the magnitude of the spectral density, but the spectral shapes are not adequately captured. M2 may provide a better directional spread of the IG wave energy. Attention is also paid to the IG waves inside the harbor. The IG waves within the harbor are underestimated, owing to the lack of some mechanisms related to IG waves in the simulation. M1 should be adopted when crudely predicting the magnitude of IG waves inside a harbor. Overall, the unstructured WW3 can effectively predict the coastal IG wave height in southern Sri Lanka, with significant correlation and acceptable errors, when compared with in-situ measurements. Finally, the IG source M1 is refined to remove the erroneous spectral gap between the IG and wind-wave frequency bands in the WW3 spectra when compared to the in-situ observed spectra.

A direct coupled method is established on the basis of the Boussinesq equations and Rankine source method to analyze the responses of floating bodies near islands and reefs. The detailed theoretical derivation, numerical solution method and the relevant calculation program THAFTS-BR are also completed. Then, the motions and connector loads of a three-module VLFS deployed near islands and reefs are investigated based on the model tests and the numerical method. Some useful conclusions are obtained, which can provide a reference for the analysis of dynamic responses, design and construction of the floating bodies near seashores and islands.

The statistical properties of obliquely incident irregular waves over a planar sloping bottom were investigated numerically by the well-known numerical wave model FUNWAVE 2.0. Irregular waves based on the averaged JONSWAP spectra with various wave heights and peak periods were simulated to propagate over planar bottoms. A wide range of incident angles from 0° though 75° were considered to study the influence of incident angles. It was found that incident angles have a significant influence on the wave nonlinearity. The wavelet-based bicoherence revealed that the degree of triad wave interactions of primary waves and the higher harmonics was apparently weakened by increasing wave incident angles. Importantly, using the simulated data, the Klopman wave height distribution was improved by incorporating the influence of obliquely incident angles. It was found that the improved Klopman distribution shows better performance for describing wave height distributions in shallow water depth. Moreover, two empirical formulae are recommended to reflect the relationship between the skewness and the asymmetry of waves and the Ursell number for obliquely incident waves on plane sloping bottoms.

A direct coupled method is established on the basis of the Boussinesq equations and RANKINE source method to analyze the hydroelastic responses of floating bodies near islands and reefs. The detailed theoretical derivation, numerical solution method and the relevant calculation program THAFTS-BR are also completed. A single module VLFS in uniform water depth is then taken as the research object, the accuracy and reliability of the THAFTS-BR program are validated by comparing the hydroelastic responses of VLFS calculated by the popularly-used software THAFTS. Finally, the motions and loads of the VLFS deployed near islands and reefs are investigated based on the model tests and the numerical method. Some useful conclusions are obtained, which can provide a reference for the analysis of dynamic responses, design and construction of the floating bodies near seashores and islands.

In this paper, the problem of linear long-wave scattering by a submerged circular cylinder or a circular pit located in a general idealized bottom topography is considered. Because of the generality of the bottom topography, the problem has various physical and engineering backgrounds. By skillfully using variable transforms, a close-formed analytical solution to the long-wave equation (LWE) in terms of Fourier-cosine series is developed, which finds several existing analytical solutions to be its degenerated cases. Based on the present solution, the influence of various factors, including the convexity or concavity of the bottom topography, the submergence of the circular cylinder or circular pit, and the wavelength of incident waves on wave amplification or attenuation is investigated.

An investigation of the effects of a uniform current strength direction (following or opposing wave propagation) on the nonlinear transformation of irregular waves over a submerged trapezoidal sill is carried out using SWASH, a non-hydrostatic numerical wave model. The nonlinear parameters (i.e., asymmetry, skewness, and kurtosis) are calculated, and the empirical formulas for these parameters are presented as a function of the local Ursell number based on the present numerical data measured. In the shoaling area of the submerged sill, the nonlinear characteristics of waves are more obvious when waves propagate in the same direction as the currents than when waves propagate in the opposite direction. Whereas nonlinear parameters grow with the strengthening of the following currents over the crest, they tend to decrease as the adverse current velocity increases over the crest area of the submerged sill.

In this article, effects of the variation of the offshore reef topography on the transient resonance induced by solitary waves with various wave heights are first investigated. The transient resonance is simulated by a fully nonlinear Boussinesq model, FUNWAVE-TVD. This paper focuses on investigating how the variations of the plane reef-face slope, the reef-ridge width, the lagoon width and the reef-face profile shape affect the wave energy distribution, the total wave energy and the maximum oscillation (runup) inside the harbor. Results indicate that all of the uniformity of the wave energy distribution, the total wave energy and the amplification factor of the incident solitary wave (defined as the ratio of the maximum runup to the incident wave height) are shown to gradually decrease with the plane reef-face slope. For the other three topographical parameters (i.e., the reef-ridge width, the lagoon width and the reef-face profile shape), their influences on the resonant wave parameters inside the harbor becomes more complicated. Not all of the resonant wave parameters inside the harbor (i.e., the uniformity of the wave energy distribution, the total wave energy and the amplification factor of the incident solitary wave) present monotonic changes with these three topographical parameters.

Infragravity (IG) waves significantly affect the operational efficiency of ports. Therefore, an accurate prediction of IG waves inside a harbor is necessary. In this study, the accuracy of the wave-group-resolving model XBeach Surfbeat (XB-SB, Delft University of Technology, Delft, The Netherlands) in predicting the IG waves inside a harbor was assessed by comparing its results with field measurements. Field measurements were performed at Hambantota Port in southern Sri Lanka. Three acoustic waves and current sensors were used to observe the wave characteristics inside and outside the harbor. First, the model was validated against observations outside the port. Next, the performance accuracy of XB-SB in modeling the hydrodynamics in the harbor was evaluated by comparing its results with the values measured inside the port. The results of the numerical simulations indicated that both the nearshore short and IG wave heights can be accurately reproduced by XB-SB in an open domain without many obstacles. However, the short wave heights in the harbor are severely underestimated by XB-SB. The IG waves inside the harbor are overestimated most of the time. Moreover, the natural periods of Hambantota Port are well calculated by XB-SB. In general, XB-SB is a reliable tool for predicting nearshore IG waves. However, it requires further improvement to reproduce the hydrodynamics in a well-sheltered harbor, such as Hambantota Port.

Free-surface oscillations within a square harbor subjected to solitary waves are investigated through physical experiments. The effects of various wave heights and water depths on the resonance within the harbor are examined. In this study, the technique of wavelet-based bicoherence spectra is employed for the first time to investigate the nonlinear wave–wave interaction of the transient harbor resonance induced by solitary waves. The results show that the relation between the maximum surface elevation, ηmax, within the harbor and the incident wave height of a solitary wave can be regarded as approximately linear. Moreover, for the same incident solitary wave height, the larger the water depth is, the greater the value of ηmax becomes. The analyses of the wavelet-based bicoherence indicate that the natural modes of the harbor are generated in the following two ways: (1) direct generation from the energy transfer of the solitary waves, and (2) the subsequent generation from the nonlinear interaction between various wave components. The first mechanism plays a decisive role in the generations of low-order natural modes, while the second mechanism also has a vital catalytic role in triggering higher-order modes. Finally, it is examined that the duration of harbor oscillations changes obviously with incident wave heights and water depths.

For the first time, transient seismic-induced oscillations inside an actual harbor are investigated using a fully nonlinear Boussinesq-type wave model. To this end, the open-source code FUNWAVE-TVD, developed by Shi et al. (2012), is extended to include seismic effects by adding a time-dependent forcing term to the governing equations. This extension is fully validated against analytical solutions and experimental data available in the literature. Uni-directional harmonic seismic excitation is considered as an approximation in the case study of Hambantota Port. Results reveal that the natural modes of Hambantota Port with periods of O (100–1000 s) can be triggered by seismic disturbances with periods of O (1 s). Some modes that are generated immediately under the excitation are found to be those of sloshing. Thus, a possible mechanism for the seismic-induced long-period harbor oscillations is proposed using a sloshing theory. Subsequently, it is determined that the seismic direction has significant effects on the excitation of natural modes and the energy decay rate. Lastly, the seismic influence on the wave motions inside the harbor including a consideration of the incident waves from the open sea is examined.

In order to theoretically study the buckle propagation of subsea pipelines with slip-on buckle arrestors, a two-dimensional ring model was set up to represent the pipeline and a nonlinear spring model was adopted to simulate the contact between pipeline’s inner walls and between pipeline’s outer wall and slip-on buckle arrestor’s inner wall during buckle propagation. In addition, some reverse springs are added to prevent the wall of left and right sides separating from the inner wall of slip-on buckle arrestors. Considering large deformation kinematics relations and the elastic-plastic constitutive relation of material, balance equations were established with the principle of virtual work. The variation of external pressure with respect to the cross-sectional area of pipelines was analyzed, and the lower bound of the crossover pressure of slip-on buckle arrestors was calculated based on Maxwell’s energy balance method. By comparing the theoretical results with experiment and finite element numerical simulation, the theoretical method is proved to be correct and reliable.

In typhoon risk assessment and warning, a critical component is a good representation of the typhoon wind field model. In this study, a new analytical model based on the logarithmic spiral trajectory model is developed to simulate the surface wind speed distribution of a typhoon. The logarithmic spiral trajectory model could overcome the limitation of the parametric gradient wind model. A slab surface layer of constant depth is used to solving the tangential equilibrium equation, and the frictional drag at the upper boundary of the surface layer is considered correctly. Consequently, the theoretical method for determining the Holland β parameter is derived from the logarithmic spiral trajectory model. It is concluded that β increases with the surface layer depth or decreases with the radius to maximum winds. By analyzing the change in kinetic energy of the air particle, the interpretation of the relationship between β and the influencing factors is provided. The models are applied to the 17th typhoon NESAT and the 19th typhoon NALGAE of 2011. Through comparisons between the observed wind records and the simulation results, the logarithmic spiral trajectory model proposed in this study could accurately simulate the wind speeds.

The harbor resonance triggered by double solitary waves (DSWs) with different wave parameters (including various wave heights and relative separation distances) is simulated based on the fully nonlinear Boussinesq model, FUNWAVE-TVD. A long and narrow harbor with different topographies is adopted. In the current study, effects of incident wave height, relative separation distance and bottom profile on hydrodynamic characteristics related to the transient oscillations are mainly investigated. The hydrodynamic characteristics considered include the evolution of the maximum free-surface elevation, the maximum runup, the wave energy distribution and the total wave energy inside the harbor. Results show that Green's law can accurately estimate the evolution of the maximum free-surface elevation in most part of the harbor area. The impacts of the topography on the maximum runup exhibit a strong dependence on the incident wave height. The smaller mean water depth inside the harbor, the larger relative separation distance, and the higher incident wave height tend to result in greater uniformity of the wave energy distribution. The normalized total wave energy is always shown to decrease gradually with the incident wave height, and to increase remarkably at first and then decrease slightly with the increase of the mean water depth.

The energy of surface water waves is a very important concept. The most common method for calculating the wave energy is based on the linear or first-order Stokes wave theory. This study investigated the energy density (i.e., the potential, kinetic, and total energy) of very nonlinear waves for a finite water depth and deep water over horizontal bottoms. Analytical expressions for the wave energy density were derived based on the nonlinear Stokes wave theories using the perturbation expansion method and Taylor expansion. The energy density was accurate to the fourth order and sixth order for the second- and third-order Stokes wave theories, respectively. As for the fifth-order Stokes wave theory, higher-order energy density expressions were proposed, which were accurate to the tenth order. Compared with previous findings, the present fifth-order Stokes wave energy density was more consistent with the numerical integration result. Then, the homotopy analysis method (HAM) was applied to obtain the energy density. According to the expressions for the nonlinear wave energy density, the energy density was no longer a single-variable function of wave height and varied with wave parameters such as the wave height and wave number. In addition, the kinetic energy and potential energy were not equal, and the former was larger than the latter.

An artificial neural network is developed to estimate the wave conditions within a port efficiently. Wave motions within the harbor subject to incident waves with varying combinations of offshore parameters (i.e., significant wave height, peak period, and peak direction) are simulated using the Boussinesq-type wave model to obtain a pre-calculated dataset. The outputs of the network are significant wave height and low-frequency wave height within the port. The neural network is trained on the dataset to ensure that efficient estimations for unknown cases with incident parameters similar to those in the pre-calculated cases can be provided. This is done to avoid the high computational cost of the wave model during real-time prediction. A k-fold cross-validation method is utilized to determine the optimum network among a variety of architectures. As a first verification step, the statistical results show that the network performs well when estimating unknown cases that are not included in the training process. Subsequently, the network is successfully validated against the observed in-situ data, proving the reliability of both the wave model and neural networks. Once the forecasting of offshore wave parameters is provided, the wave conditions within the port can be estimated quickly to realize the real-time prediction.

The low-frequency (1/400- 1/30 Hz) oscillations within the Hambantota Port, Sri Lanka, are investigated based on the in-situ measured water surface elevations in and outside the harbor. By combining analyses of spectra derived from the in-situ measurement and modal analyses solving the extended mild-slope equation, four natural frequencies of the port are identified and their modal structures are obtained. It is found that the low-frequency oscillations within the port were dominated by its natural frequencies (i.e., the eigen oscillations) instead of the frequencies of external waves. Subsequently, the correlation analysis is carried out to determine the generation of the low-frequency oscillations. Although both the offshore low-frequency and short waves contributed to the low-frequency oscillations, the low-frequency energy spectrum within the port is found to be more significantly correlated with the offshore total low-frequency wave energy than with the offshore total short-wave energy. Moreover, the empirical formula and artificial neural network models are presented to efficiently estimate the low-frequency waves inside the harbor in terms of the offshore wave parameters. Although both methods give excellent results, the artificial neural network performs slightly better.

Dispersive focusing technology is one of the most widely used methods to simulate extreme waves in the laboratory. However, such methods that are based on the linear dispersion relation are not robust. The focusing waves are always generated at unexpected positions owing to nonlinear effects. Hence, in this paper, an improved focusing method named the All Phase Correction Method (APCM) is developed to produce accurate focusing waves at a predefined position. An improved phase estimation algorithm and second-order wavemaker theory is employed to iteratively correct the phases and amplitudes of the generated waves. Different types of experiments were conducted in a wave flume using this method. The results indicate that the present method works well for accurately generating extreme events. Both the component phases and the focal positions match well with the target values. The influences of wave steepness, spectral bandwidth and spectral type on the correction procedure were investigated extensively. The analysis revealed that with the increase in wave steepness and narrowness of spectral width, the phase deviation increased. Thus, more correction iterations are needed to achieve convergence of iteration. In addition, the spectral type also influences the wave correction procedure. The focal position deviations for the constant-steepness spectrum and the constant-amplitude spectrum are significantly smaller than that of the JONSWAP spectrum, and the correlation coefficient converges faster.

The connector loads of a 3-module VLFS (Very Large Floating Structure) deployed in the shallow water areas have been investigated by experimental and numerical methods. For the numerical method, a simplified analysis procedure has been built based on 3D hydroelasticity theory and RMFC (Rigid Module Flexible Connector) model, and the effect of the uneven seabed can be included in this procedure. The model tests of the 3-module VLFS considering the complicated seabed topography and wave environment have been finished. Some key technologies for the model tests have been solved, such as the simulations of the complicated seabed topography and the connector model. A lot of the first-hand model test results have been discussed and compared with the numerical results. Some useful conclusions were obtained, which can provide reference basis for the preliminary and conceptual design of VLFS and its connectors.

The main purpose of this article is to decompose the response amplitudes of different resonant modes and to further study the wave energy distributions systematically inside long and narrow rectangular harbors with different bottom slopes when harbor oscillations are induced by normal-incident solitary waves. A series of numerical experiments are carried out using the FUNWAVE 2.0 model. The analysis procedure is mainly based on the improved normal mode decomposition method. Results show that when the incident wave height is small, the resonant wave energy inside harbors is dominated by the lowest few modes, and the higher modes only possess a very small proportion of the resonant energy; when the incident wave height increases, the relative energy distribution becomes uniform, and the proportion of energy in the higher modes increases. In addition, for the same incident wave height, the change of the bottom slope inside the harbor has a negligible effect on the relative energy distribution within the ranges of the variation in bottom slopes and the incident wave heights studied in this article.

This paper considers the nonlinear transformation of irregular waves propagating over a mild slope (1:40). Two cases of irregular waves, which are mechanically generated based on JONSWAP spectra, are used for this purpose. The results indicate that the wave heights obey the Rayleigh distribution at the offshore location; however, in the shoaling region, the heights of the largest waves are underestimated by the theoretical distributions. In the surf zone, the wave heights can be approximated by the composite Weibull distribution. In addition, the nonlinear phase coupling within the irregular waves is investigated by the wavelet-based bicoherence. The bicoherence spectra reflect that the number of frequency modes participating in the phase coupling increases with the decreasing water depth, as does the degree of phase coupling. After the incipient breaking, even though the degree of phase coupling decreases, a great number of higher harmonic wave modes are also involved in nonlinear interactions. Moreover, the summed bicoherence indicates that the frequency mode related to the strongest local nonlinear interactions shifts to higher harmonics with the decreasing water depth.

In the article by Sobey (Rodney J. Sobey, 2006. Normal mode decomposition for identification of storm tide and tsunami hazard. Coastal Engineering 53, 289–301), the author proposed a normal mode decomposition method to calculate the eigenfrequencies, the eigenmodes and the response amplitudes of different resonant modes in natural harbors that are subjected to storm tides and tsunamis. However, the numerical method to address the no-flow boundary condition in that article is imprecise, which would lead to inexact eigenfrequencies and eigenmodes. In this article, the mirror-image method was proposed to improve this handling process. The accuracy of the improved normal mode decomposition method was verified using three verification tests. With a set of numerical experiments, it was determined that during the process of decomposing the response amplitudes of different resonant modes, the numerical fitting error between the simulated free surfaces and the corresponding fitted ones gradually increases with the wave nonlinearity inside the harbor. This article sought to identify the critical wave condition under which the normal mode decomposition method can accurately decompose the response amplitudes of different modes.

It is demonstrated that offshore wavenumbers of edge waves change from imaginary wavenumbers in deep water to real wavenumbers in shallow water. This finding indicates that edge waves in the offshore direction exist as evanescent waves in deep water and as propagating waves in shallow water. Since evanescent waves can stably exist in a limited region while propagating waves cannot, energy should be released from nearshore regions. In the present study, the instability region is predicted based on both the full water wave solution and the shallow-water wave approximation.

New experiments focusing on the evolution characteristics of nonlinear wave trains were conducted in a large wave flume. A series of wave trains with added sidebands, varying initial steepness, perturbed amplitudes and frequencies, were physically generated in a long wave flume. The experimental results show that the increasing wave steepness, increases the speed of sidebands growth. To study the frequency and phase modulation, the Morlet wavelet transform is adopted to extract the instantaneous frequency of wave trains and the phase functions of each wave component. From the instantaneous frequency, there are local frequency downshifts, even an effective frequency downshift was not observed. The frequency modulation increases with an increase in amplitude modulation, and abrupt changes of instantaneous frequencies occur at the peak modulation. The wrapped phase functions show that in the early stage of the modulation, the phase of the upper sideband first diverges from that of the carrier waves. However, at the later stage, the discrepancy phase from the carrier wave transformed to the lower sideband. The phase deviations appear in the front of the envelope`s peaks. Furthermore, the evolution of the instantaneous frequency exhibits an approximate recurrence-type for the experiment with large imposed sidebands, even when the corresponding recurrence is not observed in the Fourier spectrum.

A numerical model, which can simulate wave generation and propagation is developed to simulate oscillations induced by seafloor movements inside a harbor of constant slope, and once verified and then validated through comparison with experimental results, the numerical results are used to examine the analytic solutions presented in Wang et al. (Wang, G., Dong, G., Perlin, M., Ma, X., Ma, Y., 2011. An analytic investigation of oscillations within a harbor of constant slope. Ocean Engineering 38, 479–486). Small-scale seafloor movement usually induces small longitudinal oscillations, but evident larger transverse oscillations. These transverse oscillations are sensitive to the location of the moveable seafloor. The numerical result of each transverse eigen frequency compares well with the theoretical solution; in addition the spatial structure of each mode is also well-captured by the theory. Furthermore, evident/larger longitudinal oscillations induced by large-scale seafloor movements are simulated, and the numerical resonant frequencies agree favorably with the analytical solutions. These longitudinal oscillations are sensitive to the horizontal location of the moveable seafloor.

An existing 2D time-domain method for separating irregular incident and reflected waves by wavelet transform [Ma et al., 2010. A new method for separation of 2D incident and reflected waves by the Morlet wavelet transform. Coastal Eng., 57(6):597–603] is extended to account for obliquely incident irregular waves propagating over sloping bottoms. The linear shoaling and refraction coefficients are adopted to determine the amplitude and phase changes of waves. The optimal central frequency of the Morlet wavelet is determined by the minimum Shannon wavelet entropy. Numerical tests show that the present method can accurately separate waves over horizontal depths. For waves at sloping bottoms, however, the separation errors increase as bottom slope increases and are significant for waves with incident angle larger than π /3.

Longitudinal and transverse oscillations within a harbor of constant slope are analyzed. Based on the linear shallow water approximation, longitudinal oscillations are described with Bessel equations. Ignoring friction, oscillations are forced using the period of the incident perpendicular wave field by the method of matched asymptotics. The analytic results show that the varying depth shifts the resonant wave numbers to lower values than those for the same geometric harbor with constant depth. Furthermore, we extend the shallow water equations to a linear, weakly dispersive, Boussinesq-type equation by modifying the offshore velocity component, and then use it to investigate possible existing transverse oscillations in the harbor of constant slope. These oscillations are types of standing edge waves. Their character is quite sensitive to the boundary condition at the backwall of the harbor.

Past studies of harbor resonance have mainly been restricted to the quasi-steady oscillations induced by steady wave conditions. This paper investigates the response of a rectangular harbor to subaerial landslide-generated impact waves based on physical models, in order to compare the oscillations induced by steady and transient waves. In response to steady incident waves, oscillations within the harbor need to experience a long process to obtain their maximum value before the input energy and the losses are balanced. Landslide-generated impact waves usually include components with solitary wave characteristics and also components with dispersive wave characteristics. Each component travels with a different celerity. Usually, solitary wave components propagate faster, and arrive in the harbor first. Oscillations attain their maximum status as soon as these components arrive. The subsequently arriving components with dispersive characteristics do not enhance the resonance oscillations. So the waves with solitary characteristics are considered to play an important role in harbor resonance. Numerical experiments, using the FUNWAVE model, were conducted in order to further verify these conclusions.

A new method based on the Morlet wavelet transform for separating a 2D wave field into incident and reflected waves is proposed in this paper. The principle of this method, first, is derived for constant depths. Then, using the linear shoaling theory, the method is extended to an arbitrary sloping bathymetry. Owing to the time-frequency characteristic of wavelet transform, the present method can separate waves in the real time domain and is not confined by the stationary assumption of waves. The efficiency and accuracy of this method are demonstrated using numerical simulated data.

The effect of currents on the variation of cross-shore bound long waves forced by bichromatic waves over a plane slope was investigated in the laboratory. In still water the growth rate of the shoaling bound long waves over the slope is proportional to h – 5/2 ( h is still-water depth). It was found that the opposing current makes the amplitudes of the bound long waves greater than those of still water for all cases. However, the amplitudes of bound long waves in a following current are reduced in the weakly modulated cases but are enhanced in the fully modulated case.

The wavelet-based bicoherence, which is a new and powerful tool in the analysis of nonlinear phase coupling, is used to study the nonlinear wave–wave interactions of breaking and non-breaking gravity waves propagating over a sill. Two cases of mechanically generated random waves based on J onswap spectra are used for this purpose. Values of relative depth, kph (kp is the wave number of the spectral peak and h is the water depth) for this study range between 0.38 and 1.22. The variations of wavelet-based total bicoherence for the test cases indicate that the degree of quadratic phase coupling increases in the shoaling region consistent with a wave profile that is pitched shoreward, relative to a vertical axis as seen in the experiments, but decreases in the de-shoaling region. For the non-breaking case, the degree of quadratic phase coupling continues to increase until waves reach the top of the sill. Breaking waves, however, achieve their highest level of quadratic phase coupling immediately before incipient breaking and the degree of phase coupling decreases sharply following breaking. In addition the wavelet-based bicoherence spectra provide evidence of the harmonics' growth which is reflected in the energy spectra. The bicoherence spectra also show that quadratic phase coupling between modes within the peak frequency as well as between modes of the peak frequency and its higher harmonics are dominant in the shoaling region, even though there are relatively high levels of quadratic phase coupling occurring between other frequencies. Furthermore, using the temporal resolution property of the wavelet-based bicoherence, we find that the quadratic wave interactions occur more readily during segments of time with large change of wave amplitude, rather than those segments having large wave amplitudes, but small gradients in amplitude.

This paper proposes a new definition of groupiness factor (GF) based on the local wavelet energy density of the wave time series to describe the groupiness degree of waves. The main advantage of this new GF is that the effect of the operational definition on it is smaller than that on SIWEH-based GF or envelop-based GF. Then, the new GF is used to study the groupiness variations of mechanically generated irregular waves in a wave flume propagating on a slope of 1:45. The results of present study show that the decrease of groupiness in the coast is triggered by breaking. And energy distribution along the record time for the first harmonics of waves in the surf zone, which becomes more uniform than that out the surf zone, is the main reason causing the decrease of groupiness.