Wave Theory Research Articles

Second-order water wave properties over Roseau’s three-parameter bottom topography

Summary In his book Asymptotic Wave Theory (North-Holland 1976), Roseau develops the only known exact solution for linear wave propagation over a shoal. Remarkably, almost half a century later, the only properties of that solution which appear to have been used to validate more wide-ranging models are the limiting values of wave height in the far field. The primary aim of this work is to remedy this situation by providing a robust computational setting where the solution may be fully evaluated randomly both on the surface and the interior, speedily and to a high degree of accuracy. Such ‘benchmark’ solutions will facilitate a much more rigorous approach to validation of more general alternative approximation models. One such has some of its numerical results here examined and subsequently largely vindicated by the present calculations. Computation of the absolutely convergent solution integrals is supported by cubic spline approximation of an integrand component which satisfies a certain difference equation and the work is extended to cover bottom profiles with overhang including the extreme case where the overhang degenerates to a semi-infinite flat plate. The further development of solutions at second-order reveals details of mean Eulerian current and the attendant Stokes drift enabling mass transport computations; all illustrated through the entire depth with a range of contouring and arrows plots. The force field that drives a second harmonic component is calculated and shows excellent agreement with its equivalent in the more general modal expansion model applied to a non-identical but somewhat similar profile. This further validates the modelling approach and application described in that work.

A Lagrangian approach for variable speed limit implementation in C-ITS framework

With the development of Cooperative-Intelligent Transport System (C-ITS) technologies, new strategies based on embedded technologies have emerged to manage road networks. This paper focuses on adapting to this connectivity context a Variable Speed Limit (VSL) system to detect shockwaves and anticipate their propagation based on the kinematic wave theory to dampen them. We provide an alternative framework to adapt the VSL strategy, well-suited for Eulerian approaches, into a Lagrangian context. While the Eulerian approach is based on Loop Detector (LD) and macroscopic traffic indicators (e.g. flow, density), our Lagrangian approach relies on Road Side Units (RSUs) that record the GPS traces shared by Connected Vehicles (CVs). Based on the combination of CV trajectories, Fundamental and Space-Time Diagrams theory, shockwave estimation and prediction processes directly operate on congestion waves, which release the estimation issue for traffic density. The simulation-based analysis reveals that the performance of the Lagrangian approach is comparable to the Eulerian configurations.

High-Impact Dynamic Loading Method for Calibration of Triaxial Acceleration Sensors

Abstract In this study, a high-impact dynamic loading device was designed to generate a three-dimensional pulse excitation signal with high intensity shock acceleration and achieve triaxial synchronous calibration of a triaxial acceleration sensor. A light-gas gun interior ballistic model and a sensor mechanical response model were developed, and the relationships between the bullet impact velocity, barrel length, and initial chamber pressure were obtained. Additionally, the transformation relationship of the sensor’s triaxial acceleration in different coordinate systems was derived. Based on the stress wave theory and the finite element method, the influence of the bullet impact velocity on the variation pulse, and different slope and deflection angles on the triaxial acceleration were analyzed. By optimizing the parameter design, machining the prototype, and conducting high-impact dynamic loading tests, the results showed that the deviation between theoretical and measured values of the generated triaxial acceleration signal was small, and the maximum deviation was less than 4%. This indicated that the proposed high-impact dynamic loading device satisfied the calibration requirements for calibrating triaxial acceleration sensors, which can generate a three-dimensional acceleration with a peak value of not less than 700,000 m/s².

An Additional Note to “A New Insight into the Linear Theory of Magnetoacoustic Waves in the Homogeneous Flux Tubes with an Abrupt Boundary”

This is to address the criticism expressed in the article (Goedbloed & Poedts ) referred to as GP[24] about the work (Yelagandula ) referred to as Y[23]. It will be shown that the dispersion relation obtained in Y[23] is indeed correct within the framework of the ZSER model discussed in Y[23] despite the correct criticism of GP[24] about the electrodynamic boundary condition.

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

AbstractThis paper presents a novel higher‐order shear deformation beam theory for analyzing the stability and free vibration of functionally graded (FG) sandwich beams with emphasis on the effect of higher‐order moment (HOM) and cross‐sectional warping. The governing equation of axially loaded FG sandwich beams is derived from three‐dimensional equations of the theory of elastic waves in bodies with homogeneous initial stresses. The characteristic equations for typical end conditions are obtained exactly. The numerical results of the natural frequencies and critical loads are calculated and verified for special cases by comparing them with the existing solutions. The effects of gradient index, core thickness, HOM, and warping shapes on the natural frequencies and critical buckling loads are elucidated for different slenderness ratios and end constraints.

Radio-Frequency Conductivity Evaluation Method Based on Surface/Interface Scattering of Metallic Coatings

Developing non-destructive evaluation methods for the radio frequency (RF) conductivity of conductive coatings can accelerate the performance evaluation and development of wireless communication devices. By using a split-resonator cavity to compare 800 nm copper/graphite and 1000 nm copper/graphite, we found that the RF conductivity increased by 45.5% and 82.7%, respectively, from 15 GHz to 40 GHz (pure copper was −7.2%), indicating that the bulk materials analysis method is not suitable for coating materials. Combined with electromagnetic wave theory, we believe that the critical factor lies in the additional losses of the electromagnetic waves at the copper/graphite interface and substrate. Based on the skin depth theory, the concept of triple skin depth is proposed to calculate the power loss of copper/graphite at different frequencies, considering rough Peff (including the power loss of the rough surface, copper coatings, copper/graphite interface, and graphite) compared with smooth pure copper Pc. Combined with the relationship between RF conductivity and electromagnetic wave power loss, the conductivity of copper coatings σCu at different frequencies is obtained by analyzing the measured σeff. Compared with the roughness model, the calculation error decreased from 30% to below 7%. Our study provides a theoretical basis for the regulation of the RF conductivity of metal coatings at different frequencies.

The development of implicit leadership theories during childhood: A reconceptualization through the lens of overlapping waves theory.

Implicit leadership theories (ILTs) are people's lay theories, definitions, or conceptualizations of leadership. In adults, they determine what actions we perceive as leadership, influence to whom we grant leadership status, and shape our own behaviors when we want to be seen as leader. Naturally, there has been an enduring interest in how these ILTs develop in children. Current theorizing on the development of leadership conceptualizations in children aligns with a stepwise progression mirroring Piaget's stage-based approach to cognitive development. However, contemporary approaches to cognitive development, such as Siegler's overlapping waves theory (OWT), acknowledge that children's development is linked to cognitive success and failure. This article integrates the findings from empirical studies into children's leadership conceptualizations and reinterprets them against OWT. This reinterpretation resolves findings that align poorly with a stepwise approach and demonstrates a strong fit with OWT. As such, children's leadership conceptualizations develop by generating and testing cognitive approaches-physical-spatiotemporal, functional, socioemotional, and humanitarian-and instead of progressing through these in order and according to age, they display variation and selection, that with experience and exposure, lay down selective combinations, which often engage multiple dimensions simultaneously. Consequently, the development of children's understanding of leaders is nonlinear, can be multidimensional, and is based on trial and error largely in response to their experiences. The article concludes with a discussion of the implications for future research and practice. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Experimental study on water replenishment holes in offshore monopiles

Adequate replenishment of water within the monopile foundations of offshore wind turbines is essential to ensure the effective performance of cathodic protection measures applied inside the monopile to combat internal corrosion. This study presents experimental results for the process of replenishing water within a monopile with one opening below the still water level. The results are presented as (1) the relative wave height and set-down inside the monopile under various regular wave conditions and (2) head loss coefficient for the flow in and out of the water replenishment opening in the monopile. The study demonstrates a clear link between the ratio of the wave height of the internal and external surface elevations, the relative wave height, and the dimensions of the opening in the monopile as well as the wave period. Furthermore, a simple predictive model is proposed based on linear wave theory and static energy loss considerations. The orientation of the opening with respect to the wave direction has limited effect on the relative wave height, however it has a distinct effect on the mean surface level (MSL) inside the monopile. The lowest MSL occurred when the opening hole was positioned perpendicular (at a 90-degree angle) to the direction of wave propagation. Geometric considerations suggest that the flow in and out of the cylinder is slightly asymmetric and the results also show a small difference in the head loss coefficient for the flow in and out of the monopile, respectively. On average, however, the head loss coefficient was found to be ζ=2.0.

Observations of coastal infragravity wave characteristics under swell-dominated conditions

This study analyzed the characteristics of infragravity (hereafter IG) waves under a microtidal and swell-dominated wave climate based on the field observations collected in the nearshore waters (water depth: ∼ 13 m) of southern Sri Lanka. The main focus is on the dependence of nearshore IG waves on the incident short wave parameters, with particular attention to the influence of the short wave spectral shape on the IG wave characteristics. The observation results indicate that IG waves are primarily locally excited by incident swell wave energy, and accurate prediction of the IG wave height can be achieved using the parameter HS2Tm−1,0, which represents the short wave energy flux. Bound IG wave energy is much smaller than the free IG wave under conventional sea states but becomes significant under high-energy swell sea states.Higher levels of IG wave energy are found to be forced under narrower spectral shapes and smaller directional spreading short wave conditions, and these conditions typically correspond to stronger coupling strengths in the bound IG wave theory. Compared to the single-peaked spectra wave condition, the forced IG wave energy decreases under the double-peaked spectra wave condition. This is likely mainly because the latter tends to have a broader spectrum and a larger directional spreading. On the other hand, under the narrow-banded spectral short wave conditions, IG wave energy is inclined to be distributed over a lower and smaller frequency range. This is noteworthy because lower-frequency IG waves tend to be more difficult to break. In addition, the bound IG wave energy was found to increase at low tide, but for the intermediate water depths in which the observation site is located in this study, the influence of the small tidal range in the study area on the IG wave energy was not significant.

Performance Assessment Of Two Active Absorption Systems In A Large Wave Flume

The quality of most coastal or ocean laboratory experiment depends on the accuracy, uniformity and stability of the generated waves. Generation of waves in the laboratory seems simple, but it should consider the spatio-temporal variability of the wave profile. As they propagate, waves interact with the boundaries, i.e. changes in water depth, lateral walls, test specimens, instrument supporting structures, passive absorbers, etc., as well as with varying hydrodynamic conditions (wave-induced currents, wave-wave interactions, or nonlinear energy transfer, among others). Reflected, scattered and radiated waves may propagate back to the wave generator and reflect back. Re-reflections from the wave generator may be a non-realistic representation of the intended wave field, to the extent that the total energy content in the testing facility may increase significantly and uncontrollably. Theoretically, for weakly nonlinear unidirectional waves propagating over a relatively small number of wavelengths, and as long as the still water depth and facility width does not change, the wave properties should remain constant in space and time, which can be identified as the uniformity and stability of the waves. Under these conditions, nonuniform or unsteady wave properties can be associated to an inadequate selection of the wave theory (Airy, Stokes n-order, Cnoidal, Stream Function, etc., e.g. Fenton 1999), the approximation of the wave generation technique (first or second order, nonlinear - see e.g. Mohtat et al., 2020), formation of wave instabilities (e.g. Benjamin-Feir, 1967), wave reflection, scattering or radiation, and the performance of the active wave absorption system. It can be said that the performance of the active wave absorption control system is one of the responsible factors that can affect directly the quality of the generated waves. Many laboratories have implemented different passive and active wave absorption systems, but the performance on the uniformity and stability of the generated waves, as well as the performance of the active wave absorption system used, have not been assessed in a systematic way, and related published data and information is quite scarce, with few interesting exceptions, e.g. (Lykke-Andersen, 2016; Spinneken 2010; Schäffer and Skourup 1996). However, a direct comparison among the different available active wave absorption control systems is still missing. This paper presents a thorough experimental study on the performance assessment of two different active wave absorption control systems (MTS and Awasys7) as a means to establish the quality of the experiment in terms of the uniformity (space) and stability (time) of generated wave parameters (descriptors) in the laboratory.

Effect of static compression on near-field tsunami waves: Three-dimensional solution

An analytical solution of three-dimensional surface wave profiles due to arbitrary spatio-temporal disturbance of a circular ocean bottom in a compressible ocean is obtained by incorporating the influence of the static ocean background compression under the assumptions of linearized water wave theory. Time-domain simulations of the surface profile in three dimensions and the pressure distribution within the water column for a circular uniform rise and tilt are shown. The corresponding animation movies depict the temporal evolution of the surface profile and pressure field inside the water column eloquently, which was not shown in earlier literature. The impact of static compression is also discussed through the simulations. A novel analytical expression of the potential function for a generic tilted motion (rmcos(mθ),m∈Z) of the circular ocean floor is derived. An efficient numerical code is developed to find surface elevation and pressure distribution, implementing the inverse Fourier integral as matrix multiplication. Validation is performed for the specific case of a rising flat ocean floor, showing the oscillations due to acoustic-gravity modes. Initially, a simplified problem of a flat rising ocean bottom is solved using the eigenfunction matching method, which involves finding a particular solution for the nonhomogeneous ocean bottom condition and the solution for its homogeneous counterpart. Solutions are obtained using a newly developed inner product between the depth-dependent functions. Later, a Green function technique is used to incorporate the impact of the arbitrary spatio-temporal motion of the circular portion of the ocean bed. The solution obtained from the eigenfunction matching method is utilized to obtain the analytical form of Green’s function and, eventually, an expression of surface elevation and pressure distribution inside the ocean water column.

Numerical investigation of the effect on the erosion effect under the action of solitary wave

Local scour is a factor that cannot be ignored in hydraulic engineering, which seriously threatens the safety of hydraulic structures. There is a lack of consideration of periodic hydraulic conditions in the existing research on erosion. In this paper, the application range of various wave theories is analysed, and the isolated wave which is suitable for offshore is selected to study. Based on the Euler multiphase flow model, it is developed by using the user definition, and the generation and extinction modules are introduced to simulate the isolated waves, and then the cyclic waves in the offshore area are simulated by the superposition of multiple isolated waves. The hydrodynamic characteristics of isolated waves, including the pressure distribution of isolated waves, are analysed.

Analytical solutions for hydrodynamic responses of arrays of floating truncated cylinders using multi-term Galerkin method and its application to a new wave energy converter device

In the context of linear water wave theory, the analytical solutions for the diffraction and radiation of a truncated cylinder array are developed in the presence of ambient incident waves. Each cylinder in the array can oscillate with five degrees of freedom (DOFs), i.e., surge, sway, heave, roll, and pitch. This paper adopts the multi-term Galerkin method to expand the fluid velocity at the interface of different regions into a set of basis functions containing Gegenbauer polynomials, which accurately and efficiently characterizes the cube root singularity of the fluid velocity near the edges of the truncated cylinders. Using the dynamic equilibrium equations, the amplitudes of each DOF of the cylinders in the array are solved. The analytical solution presented in this paper converges rapidly, and high-precision hydrodynamic response results can be obtained using just a few truncated terms (e.g., the upper bounds of m0 = 5 and p0 = 22 can yield results of five-figure accuracy). For the 4-cylinder array, under the same accuracy conditions (the error less than 1%), the computation time of the conventional method developed by Zeng et al. [“Hydrodynamic interactions between waves and cylinder arrays of relative motions composed of truncated floating cylinders with five degrees of freedom,” J. Fluids Struct. 115, 103785 (2022d)] based on the exact algebraic method [Kagemoto and Yue, “Interactions among multiple three-dimensional bodies in water waves: An exact algebraic method,” J. Fluid Mech. 166, 189–209 (1986)] is 3.9 times longer than that of the present method. As the number of cylinders increases, the advantage of the present method in terms of convergence speed becomes more apparent, e.g., for the 16-cylinder array, the conventional solution takes 6.3 times longer than the present solution. To extract wave energy more efficiently, a new 5DOF wave energy converter (WEC) device that can extract energy in 5DOFs is proposed. The present method is adopted to investigate the hydrodynamic performance of the 5DOF WEC arrays. Compared with the traditional 1DOF (heave) WEC, the 5DOF WEC can significantly improve the energy capture performance of arrays, especially in the high-frequency wave region.

Geometrization of the Huygens–Fresnel principle: Applications to Fraunhofer diffraction

The classical wave theory can trace its historical origins to the seminal works of Christian Huygens, Thomas Young, and Augustin Fresnel. To explain some of light’s observed properties, such as rectilinear propagation, reflection, and refraction, Huygens proposed a simple geometrical construction of secondary spherical wavelets with centers of disturbance located on a primary wavefront. More than a century later, Young formulated the law of interference to both predict the formation of fringes in his now famous double slit experiment and also to estimate the wavelengths associated with different colors. A decade after that, Fresnel combined Huygens’ construction with Young’s interference law to qualitatively and quantitatively describe diffraction, which is the bending of light upon encountering an obstacle or an aperture. This grand synthesis, called the Huygens–Fresnel principle, acts as a powerful pictorial aid and conceptual tool that can describe a wide variety of complicated optical phenomena. However, the applications of the principle and its later developments, such as the Kirchhoff–Fresnel integral, are strewn with several simplifying assumptions and approximations that are aimed at minimizing the mathematical challenges involved. Consequently, two distinct formalisms are necessary to account for diffraction effects when the source of light or observation screen is placed nearby and far away from the aperture or obstacle. Recently, a hyperbola framework for analyzing wave interference at a multi-slit barrier was shown to successfully circumvent all conventionally imposed ad hoc conditions. The method commences directly from the Huygens–Fresnel principle and the ensuing predictions pertaining to the distribution of fringe characteristics, namely, positions, widths, and intensities on a detection screen can, therefore, justifiably claim accuracy in both the near field (Fresnel regime) and the far field (Fraunhofer regime). In this paper, the analysis that was previously carried out for the special case of slits of negligible widths is further extended to encompass slits of finite widths as well.

In-Plane crushing performance of bionic glass Sponge-Type honeycomb structures

Honeycomb structures are commonly adopted due to their superior energy absorption capacity. In this study, a new bionic glass sponge–type honeycomb structure (BSH) with a quadrilateral octagonal mesh microstructure inspired by the sea sponge structure was proposed. The in-plane crushing performance of the BSH with different geometrical parameters under different crushing speeds was investigated by ABAQUS/Explicit. The numerical findings suggested that the BSH displayed stronger energy absorption in contrast to the square, hexagonal and hierarchical honeycombs at both quasi-static and dynamic crushing conditions. More plastic hinges and more unit walls involved in deformation resulted in a high energy absorption capacity. In addition, three typical deformation modes of the BSH under different loading speeds were discussed, and the empirical model to predict the plateau stress of the BSH was established based on the shock wave theory. Finally, the effect of boundary segmentation parameter m on crushing performance was also illustrated. The energy absorption capacity reaches a maximum at m = 3 for quasi–static loading, whereas at m = 5 for dynamic loading. These findings provide valuable insights into the optimization of bionic honeycombs.

Analysis of Hydrodynamic Loads by a Vertical Hollow Cylindrical Structure in A Two-Layer Fluid of Finite Depth

This paper aims to study the effect of surge radiation by a vertically hollow cylinder floating in a two-layer fluid of finite depth. Since most of the important characteristics of the oscillating water column are preserved by the hollow cylindrical structure, so the proposed device can be considered as an oscillating water column (OWC) which is one of the important wave energy device. In this two-layer fluid system, the upper layer is bounded above by a free surface and the lower layer is bounded below by a solid impermeable bottom, and the submerged hollow cylindrical structure allows it to oscillate in a surge mode of motion in the upper layer. On the assumption that wave amplitudes are small in comparison to wavelengths, hence the linear water wave theory is applied to formulate boundary value problems by dividing whole domain into 2 sub-domains. The formulated boundary value problems for surge radiated potentials in each sub-domain; we execute to obtain solutions in each sub-domain by using variable separation and matched eigenfunction expansion methods. With the help of obtained surge radiated potentials, we derive the radiation forces by the device in terms of added mass and damping coefficients. A set of influences of different parameters of the device on the added mass and damping coefficients are demonstrated in both surface and internal wave modes of motion. Hydrodynamic coefficients oscillate highly near a particular frequency, which leads to the resonance phenomena. In addition to that, the 3D plots of the free surface elevation in surface and internal wave modes are presented. A density ratio γ = 0.97 has a higher oscillation than a density ratio γ = 0.93, 0.95. High oscillations occur around a particular frequency ω = 3.83 due to the resonance phenomenon in which the waves incoming match with the natural motion of the object. HIGHLIGHTS A floating hollow vertical cylindrical structure is considered in a two-layer fluid system having finite depth The separation of variables and matched eigenfunction expansion methods are utilized to obtain the solutions to the boundary-value problems The diffracted velocity potentials under the surge mode of motion are evaluated in the identified regions The added mass, damping coefficients, free surface and interface elevations in surface and internal wave modes are evaluated with different set of parameters of the device A number of results are presented regarding the effect of parameters of the device on the added mass, damping coefficients, free surface and interface elevations It is observed that the density ratio of the two-layer fluid has significant effect on the added mass, damping coefficient and the elevations GRAPHICAL ABSTRACT

Exploring Dynamic Spalling Behavior in Rock–Shotcrete Combinations: A Theoretical and Numerical Investigation

Spalling is a widespread dynamic disaster during blasting excavation in underground engineering. To clarify the coupled dynamic response and spalling behavior of an underground tunnel with a spray anchor, an investigation based on the rock–shotcrete combination was conducted using theoretical and numerical methods. The mathematical representation of stress wave propagation between rock and shotcrete was deduced based on the elastic stress wave theory. A novel method for predicting the location and time of initial spalling in a rock–shotcrete combination was proposed. A numerical simulation was conducted to verify the validity of the proposed theoretical method. In addition, the effect of the material’s tensile strength, the loading amplitude, and the thickness of shotcrete on the stress evolution and spalling characteristics was studied. The results demonstrate that the initial spalling locations are sensitive to the relationship between the normalized tensile strength of the rock, shotcrete, and interface. A high incident amplitude can cause the initial spalling in rock, and the shotcrete or rock–shotcrete interface can cause initial spalling due to a low incident amplitude. The stress evolution and spalling characteristics are sensitive to the thickness of shotcrete. The location of the initial spalling failure changes with the thickness of the shotcrete. An appropriate increment in thickness and normalized strength of the shotcrete is beneficial to the dynamic stability of underground engineering.

Partial dam-break wave characteristics due to partial gate opening

Understanding how to predict the characteristic height of a dam-break wave is crucial for effective risk management in dam-break scenarios. This study introduces an experimental framework to simulate partial dam-break waves triggered by partial gate openings. We conducted numerous experiments to explore how different factors, including water-level differences (hi), reservoir widths (L), dam-break openings (e), and submerged depths (hw) influence the characteristic heights of the main-wave peak (h1c), subwave peak (h2c), and main-wave trough (h1t). Our analysis led to the development of an equation for predicting h1c. Furthermore, we compared the partial dam-break wave profile with the theoretical profile of a solitary wave, identifying optimal conditions for solitary wave generation and examining the velocity characteristics of partial dam-break waves. The results reveal that: (1) h1c, h2c, and h1t have positive correlations with hi and L, but negative correlations with e and hw; (2) notably, the main-wave pre-peak profile aligns more closely with the solitary wave theory; and (3) the solitary wave theory accurately predicts the main-wave peak velocity. This study enhances our understanding of wave characteristic evolution during partial dam-breaks and can be used for validation in numerical simulations.

Generation Of Highly Nonlinear Waves In A Short Wave Flume

The typical approach for generating nonlinear waves in physical models involves employing first- or second-order wave theory, requiring a large water depth at the wavemaker. When the prototype bathymetry shows a gentle slope, a large facility is required. However, practical constraints often make this unfeasible, leading to the use of steep transition slopes to obtain sufficient water depth at the generator. Incorporating a transition slope may generate unwanted free waves beyond the transition point, significantly impacting the wave parameters. The presence of these free waves causes the response of the tested structure to deviate from that found in the prototype. This paper offers guidelines for using transition slopes effectively while avoiding the generation of undesired free waves after the transition point.

Spatial energy evolution of focused waves generated in numerical wave tank

This paper numerically investigates the energy evolution of the focused wave and its correlation with input parameters for the wave generation. The focused wave is numerically generated based on the fully nonlinear potential wave theory which is solved by the Quasi Arbitrary Lagrangian Eulerian-Finite Element Method. The investigation is performed by dividing the generated wave energy into three categories according to their frequency intervals, i.e., the initially assigned frequency interval [f1fN], the frequencies lower (f < f1) and higher (f > fN) than the initial interval. The amount of the generated energy falling into three energy categories and the energy distribution in the initial frequency interval are analysed against amplitude parameters and frequency bands under three amplitude spectra. Four indicators are proposed to indicate the actual energy variations comparing to the initial design. It is found that for all the three frequency intervals the energies increase when the amplitude parameters increases or the given frequency shifts towards the lower frequency domain. The choice of wave amplitude spectrum plays a significant role to maintain the distribution of the energy in the design frequency range and to minimise the energies out of the design frequency range. The results from the second-order wave group theory are used to assist analyses of the mechanism of the energy evolution alongside the focused wave generation. It is found that the lower-frequency energy in the vicinity of the focusing point is dominated by the second-order difference waves, whereas the higher-frequency energy is mainly produced by the wave flap. The waves which are higher than the second-order and the complex interactions not considered by the second-order wave theory are found significantly disturbing the energy distribution in the originally assigned wave-frequency interval.