Velocity Of Long Waves Research Articles

Physical Model Experiment for Estimating Wave Overtopping on a Vertical Seawall under Regular Wave Conditions for On-Site Measurements

Apart from implementing hardware solutions like raising the crest freeboard of coastal structures to efficiently counter wave-overtopping, there is a simultaneous requirement for software-driven disaster mitigation strategies. These tactics involve the swift and accurate dissemination of wave-overtopping information to the inland regions of coastal zones, enabling the regulation of evacuation procedures and movement. In this study, a method was proposed to estimate wave-overtopping by utilizing the temporal variation of wave heights exceeding the structure’s crown level, with the aim of developing an on-site wave measurement system for providing wave-overtopping information in the field. Laboratory model experiments were conducted on vertical seawall structures to measure wave-overtopping volumes and wave runup heights under different wave conditions and structural freeboard variations. By assuming that the velocity of water inundation on the top of the structure during wave-overtopping events is equivalent to the long-wave velocity, an overtopping discharge coefficient was introduced. This coefficient was utilized to estimate the rate of wave-overtopping based on the temporal changes in wave runup heights measured at the top of the structure. Upon reasonably calculating the overtopping discharge coefficient, it was verified that the estimation of wave-overtopping could be achieved solely based on the wave runup heights.

Fully nonlinear investigation on energy transfer between long waves and short-wave groups over a reef

Long waves are amplified as short-wave groups shoal and break over reefs, therefore, having significant impacts on coastal inundation, structure stability, and sediment transport. This study investigated the cross-reef variation of long-wave energy exchange with short-wave group over a reef using fully nonlinear analysis of simulation results by the non-hydrostatic model SWASH. The objective was to elucidate the mechanisms of long-wave transformation under nonlinear short-wave group forcing over a reef, and to assess the consequences of simplifications in linear and weakly nonlinear analyses in this problem. The energy transfer between short and long waves is the work done by wave radiation stress on long waves, which is the product of radiation-stress gradient and long-wave velocity. Unlike conventional linear and weakly nonlinear analysis, the Stokes transport and long-wave modulation of local water depth are included in the fully nonlinear analysis. It was found that only the long-wave energy flux gradient predicted by the fully nonlinear analysis was balanced by the work done by wave radiation stress over a shallow reef. The fully nonlinear analysis showed that strict mass conservation has to be used to extract long wave velocity properly. In contrast, in linear and weakly nonlinear analysis, the long-wave velocity is extracted from single-point velocity measurements. The fully nonlinear analysis demonstrated that the generation and growth of incoming breakpoint-forced long waves overcame the dissipation of bound long waves in the surf zone, leading to amplification of incoming long-wave energy flux. This phenomenon occurred even when short waves mainly broke over the horizontal reef flat with large submergence, indicating that long-wave evolution is not locally controlled but dependent on wave spatial evolution history. Outgoing breakpoint-forced long waves were dissipated considerably during de-shoaling over the forereef due to substantial energy transfer to incoming short waves, though both of them are free waves. The phase coupling between outgoing long waves and incoming short-wave groups occurred at all frequencies at the breakpoint, which was found to be the main driving mechanism for the energy transfer. According to the fully nonlinear analysis, the reef-flat submergence may affect the long wave in a complex fashion, i.e. , reducing the submergence may enhance the energy transfer from short waves to long waves or suppress long-wave growth by increasing its frictional dissipation at the same time. • Long-wave energy balance is achieved with fully nonlinear analysis, but not linear and weakly nonlinear ones. • Strict mass conservation has to be used to extract long-wave velocity properly. • Long wave amplitude is dependent on short-wave spatial evolution history. • Radiation stress does positive work on incoming long waves in both shoaling and surf zones. • Substantial energy is transferred from outgoing free long waves to incident short waves during de-shoaling.

The Impact of Baryonic Physics on the Kinetic Sunyaev–Zel’dovich Effect

Abstract Poorly understood “baryonic physics” impacts our ability to predict the power spectrum of the kinetic Sunyaev–Zel’dovich (kSZ) effect. We study this in a sample high-resolution simulation of galaxy formation and feedback, Illustris. The high resolution of Illustris allows us to probe the kSZ power spectrum on multipoles . Strong AGN feedback in Illustris nearly wipes out gas fluctuations at and at late times, likely somewhat underpredicting the kSZ power generated at . The post-reionization kSZ power spectrum for Illustris is well-fit by over 10,000, somewhat lower than most other reported values but consistent with the analysis of Shaw et al. Our analysis of the bias of free electrons reveals subtle effects associated with the multi-phase gas physics and stellar fractions that affect even linear scales. In particular, there are fewer electrons in biased galaxies, due to gas-cooling and star formation, and this leads to an electron bias of less than one, even at low wavenumbers. The combination of bias and electron fraction that determines the overall suppression is relatively constant, , but more simulations are needed to see if this is Illustris-specific. By separating the kSZ power into different terms, we find that at least 6% (10%) of the signal at ℓ = 3000 (10,000) comes from non-Gaussian connected four-point density and velocity correlations, , even without correcting for the Illustris simulation box-size. A challenge going forward will be accurately modeling long-wave velocity modes simultaneously with Illustris-like high resolution to capture the complexities of galaxy formation and its correlations with large-scale flows.

Nonlinear evolution and runup–rundown of long waves over a sloping beach

The initial value problem of the nonlinear evolution, shoreline motion and flow velocities of long waves climbing sloping beaches is solved analytically for different initial waveforms. A major difficulty in earlier work utilizing hodograph-type transformation when solving either boundary value or initial value problems has been the specification of equivalent boundary or initial condition in the transformed space. Here, in solving the initial value problem, the transformation is linearized in space at , then the full nonlinear transformation is used to solve the initial value problem of the nonlinear shallow-water wave equations. A solution method is presented to describe the most physically realistic initial waveforms and simplified equations for the runup–rundown motions and shoreline velocities. This linearization of the initial condition does not appear to affect the subsequent nonlinear evolution, as shown through comparisons with earlier studies. Comparisons with runup results from solutions of the boundary value problem suggest the same variation with the runup laws. The methodology presented here appears simpler than earlier work as it does not involve the numerical calculation of singular elliptic integrals.

Modelling the morning glory of the Gulf of Carpentaria

The morning glory is a meteorological phenomenon which occurs in northern Australia and takes the form of a series of roll clouds. The morning glory is generated by the interaction of nocturnal seabreezes over Cape York Peninsula and propagates in a south-westerly direction over the Gulf of Carpentaria. In the present work, it is shown that the morning glory can be modelled by the resonant flow of a two-layer fluid over topography, the topography being the mountains of Cape York Peninsula. In the limit of a deep upper layer, the equations of motion reduce to a forced Benjamin–Ono equation. In this context, resonant means that the underlying flow velocity of the seabreezes is near a linear long-wave velocity for one of the long-wave modes. The morning glory is then modelled by the undular bore (simple wave) solution of the modulation equations for the Benjamin–Ono equation. This modulation solution is compared with full numerical solutions of the forced Benjamin–Ono equation and good agreement is found when the wave amplitudes are not too large. The reason for the difference between the numerical and modulation solutions for large wave amplitude is also discussed. Finally, the predictions of the modulation solution are compared with observational data on the morning glory and good agreement is found for the pressure jump due to the lead wave of the morning glory, but not for the speed and half-width of this lead wave. The reasons for this are discussed.

The velocities of long waves in composites with interphase imperfections

We develop a model of elastic strain of a composite material with interphase imperfections. In the context of this model we construct a continuous model of first order (theory of effective models) for stratified and unidirectional fibrous compounds. We determine the velocities of propagation of long (compared with the size of the structure) waves in any direction. We exhibit the possible characteristic regularities of the influence of interphase imperfections on the value of the velocities. Bibliography: 6 titles.

Modulation of short wind waves by a long surface wave: a mechanism for feedback with an air flow

The paper concentrates on the evolution of a spectrum of short wind waves (SW) along the profile of a long surface wave (LW). Short wave spectral variations are considered in the relaxation approximation. The SW spectrum is modulated by the orbital velocities of long waves and by the variations of wind stress along the LW profile. The latter effect occurs due to wind flux perturbations induced by both the long wave proper and variations of the sea surface roughness induced by the SW modulations. To describe this effect, a feedback mechanism is introduces—the growth of energy of short waves results in the larger roughness of the sea surface, thereby contributing to the local wind stress, which facilitates, in turn, the growth of short waves. With moderate and strong winds being involved, this effect (aerodynamic feedback) is shown to be dominant in the short wave spectrum modulation. The mechanism becomes more efficient with intensification of the wind and decreasing of the long waves' frequency. Results of model calculations are in agreement with the known experimental data.

On the wave excitation and the formation of recirculation eddies in an axisymmetric flow of uniformly rotating fluids

The inertial waves excited in a uniformly rotating fluid passing through a long circular tube are studied numerically. The waves are excited either by a local deformation of the tube wall or by an obstacle located on the tube axis. When the flow is subcritical, i.e. when the phase and group velocity of the fastest wave mode in their long-wave limit are larger than the incoming axial flow velocity, the excited waves propagate upstream of the excited position. The non-resonant waves have many linear aspects, including the upstream-advancing speed of the wave and the coexisting lee wavelength. When the flow is critical (resonant), i.e. when the long-wave velocity is nearly equal to the axial flow velocity, the large-amplitude waves are resonantly excited. The time development of these waves is described well by the equation derived by Grimshaw & Yi (1993). The integro-differential equation, which describes the strongly nonlinear waves until the axial flow reversal occurs, can predict the onset time and position of the recirculation eddies observed in the solutions of the Navier-Stokes equations. The numerical results and the theory both show that the flow reversal most probably occurs on the tube axis and also when the waves are excited by a contraction of the tube wall. The structure of the recirculation eddies obtained in the solutions of the Navier-Stokes equations at Re = 105 is similar to the axisymmetric or ‘bubble-type’ breakdown observed in the experiments of the vortex-breakdown which used a different non-uniform (Burgers-type) rotation. In uniformly rotating fluids the formation of the recirculation eddies has not been observed in the previous numerical studies of vortex breakdown where a straight tube was used and thus the inertial waves were not excited. This shows that the generation of the recirculation eddies in this study is genuinely explained by the topographically excited large-amplitude inertial ‘waves’ and not by other ‘instability’ mechanisms. Since the wave cannot be excited in a straight tube even in the non-uniformly rotating flows, the generation mechanism of the recirculation eddies in this study is different from the previous numerical studies for the vortex breakdown. The occurrence of the recirculation eddies depends not only on the Froude number and the strength of the excitation source but also on the Reynolds number since the wave amplitude generally decreases by the viscous effects. Some relations to the experiments of vortex breakdown, which have been exclusively done for non-uniformly rotating fluids but done in a ‘non-uniform tube’, are discussed. The flow states, which are classified as supercritical, subcritical or critical in hydraulic terminology, changes along the flow when the upstream flow is near resonant conditions and a non-uniform tube is used.

Long-Wave Component in Near-Bottom Velocities Under Random Waves on a Gentle Slope

This paper deals with characteristics of long-wave component in near-bottom velocities under random waves. Comparisons were made between measured and simulated time histories of long-wave velocity,...

Dissipative effects on the resonant flow of a stratified fluid over topography

The effect of dissipation on the flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit for the case when the flow is near resonance, i.e. the basic flow speed is close to a linear long-wave speed for one of the long-wave modes. The two types of dissipation considered are the dissipation due to viscosity acting in boundary layers and/or interfaces and the dissipation due to viscosity acting in the fluid as a whole. The effect of changing bottom topography on the flow produced by a force moving at a resonant velocity is also considered. In this case, the resonant condition is that the force velocity is close to a linear long-wave velocity for one of the long-wave modes. It is found that in most cases, these extra effects result in the formation of a steady state, in contrast to the flow without these effects, which remains unsteady for all time. The flow resulting under the action of boundary-layer dissipation is compared with recent experimental results.

Intrinsic frequency spectra of short gravity‐capillary waves obtained from temporal measurements of wave height on a lake

Intrinsic frequency spectra of water waves in the range of 6–17 Hz were obtained as a function of both wind speed and wind stress from point measurements of wave height. In a lake with a limited fetch there are two types of surface motions causing Doppler shift in the frequencies of short waves: orbital velocity of long waves and surface wind drift. The former was estimated from long‐wave amplitude by using a linear wave theory. The error in this estimate is of the order of the long‐wave slope, and for this work it is typically 10%. The latter was approximated by the friction velocity. The friction velocity could be either taken as 3% of the mean wind speed measured at a height of 10 m or obtained from our direct measurements of the wind stress. The surface drift velocities obtained by these two approaches were found to be in close agreement. However, the estimate based on the mean wind speed was preferred because of its simplicity. Doppler frequency shift can be corrected in either the frequency or the equivalent spatial domain. A comparison of these techniques from both a theoretical and a practical point of view was made. The two techniques were found to produce comparable results. Experimental results showed that the spectral energy of short waves rapidly increased in response to increasing winds and jumped up by an order of magnitude when wave breaking occurred.

Second-order low-frequency wave forces on a vertical circular cylinder

Using the multiple-scale perturbation method, the diffraction of a nonlinear nearly periodic wavetrain by a vertical circular cylinder is investigated. The envelope of the incident wavetrain is assumed to modulate slowly in the direction of wave propagation. The relationship between the envelopes of incident and scattered waves is derived. It is shown that second-order scattered set-down waves propagate only at the long-wave velocity (gh)½. The formula for low-frequency wave forces acting on the cylinder is presented. The low-frequency wave forces, which are second-order quantities, are caused by set-down waves beneath the wavetrain and the results of the self-interactions of the leading-order first harmonic wave components. Numerical solutions are presented for the case where the wave envelope varies sinusoidally.

A two‐scale model of short wind‐generated waves and scatterometry

The principle of conservation of action is applied to derive a spectral form for short, wind‐generated waves on the ocean which is valid to second order in the slopes of longer surface waves. Using this form, composite surface theory is applied to microwave backscatter from the ocean surface in the Bragg regime to derive an expression for the normalized radar cross section of the sea, σ0, which is also valid to second order in long‐wave slope. This expression is compared with previous data and empirical models of σ0 and is shown to account for the observed incidence angle, azimuth angle, wind speed, and polarization dependence. Comparisons are made at Ku and C band for incidence angles between 20° and 70°. The theory indicates the manner in which atmospheric stability, long‐wave spectra, and sea surface temperature may influence σ0. Upwind‐downwind asymmetry of σ0 is shown to occur because of the second‐order interaction of short‐wave modulation with the orbital velocities of long waves. Furthermore, the results suggest that the wind speed dependence of σ0 varies only slightly with microwave frequency, while the magnitude of σ0 increases somewhat with frequency until viscous or turbulent dissipation reduces short‐wave growth rates and thus amplitudes. Implications for scatterometer algorithm development are discussed.

Slow-drift motion of a two-dimensional block in beam seas

Because of the inherent nonlinearity in the boundary conditions on the free surface, water waves with frequencies from neighboring parts of the sea spectrum interact and force low-frequency oscillations at the second order. Since the physical phenomenon involves vastly different timescales, the perturbation method of multiple scales is applied here to a rectangular cylinder in beam seas. It will be shown that at the second order there are two kinds of long waves; one is locked to the envelopes of the incident, reflected and transmitted short waves, while the other propagates away from the body at the long-wave velocity [sq rt (gh)] . The latter contributes to the damping of slow-drift oscillations of the body. Analytical results for the displacement amplitudes of the slow sway and for the radiated long waves are derived. The transient evolution due to incident envelopes of finite and semi- finite duration is also predicted.

On a nonlocal theory of longitudinal waves in an elastic circular bar

After determining the values of the nonlocal moduli for longitudinal waves in an infinite space, Fourier transforms of the equations of axially symmetric longitudinal waves in an infinite circularly cylindrical rod are established and decoupled according to the Pochhammer procedure. Dispersion equation is obtained from the conditions of traction free surface of the rod, and compared with its classical counterpart. While the velocity of long waves coincides, as required, with that derived in the classical case, the velocity of short waves turns out to be about 36% less.

Measurements of wind velocity and pressure with a wave follower during Marsen

Near sea‐surface air pressure and wind velocities at several levels above the instantaneous water surface were measured using a wave follower during the Marine Remote Sensing (MARSEN) Experiment. The rate of momentum transfer from wind to waves is determined from the air pressure data. From the wind velocity measurements, the wave‐induced airflow and its coherence with waves are obtained for various wind velocities and phase speeds of the ocean waves. The pressure results indicate that momentum transfer to waves can be specified by the equation γi = 0.12 (μ − 1)2 in the ranges μ = 1 to 7.4, where γi is related to the imaginary part of the wave‐induced pressure, as defined by equation (2), μ = (8/3π)(U/c), U is wind speed at 10 m above the mean water level, and c is wave phase speed. The wind velocity results indicate that the wave‐induced airflow is much smaller than the mean wind speed. An empirical equation is proposed for the modulation of atmospheric transfer to short waves induced by the orbital velocity of long waves. Such modulation explains adequately the excess modulation of short waves by long waves as suggested by Wright et al. [1980] when the short waves are 13 cm long but not when they are 2.3 cm long. The agreement for the 13 cm waves supports their suggestion that the excess modulation is due to a modulation in atmospheric transfer. For the shorter 2.3 cm waves the less adequate agreement and the scatter in the results of Wright et al. suggest that other mechanisms such as short wave blocking by long wave orbital velocity, short wave breaking, and other strong nonlinear transfer among the short wave spectral components also may be contributing to the excess modulation of short waves by long waves.

Excitation of tsunami waves by a traveling displacement of the ocean bottom

The problems of wave excitation in the ocean by a vertical displacement occurring in a rectangular region of the bottom and traveling along this region with the arbitrary velocity are considered. The analysis is made in the framework of the linear theory of long waves, which correspond well, in general, to tsunami waves in the open ocean. Two models of displacement are discussed: (1) rectangular, and (2) sign‐variable in the direction transverse to its propagation. The main characteristics of radiated waves (the form and amplitude of impulses, the direction dependence of the wave energy, the dependence of the full energy of radiation E on duration T and the propagation velocity V of the break) are calculated. In all cases E grows with the approximation of V to the velocity of long waves (synchronism), the radiation having a sharp directivity. Generally speaking the second‐type displacement radiates the waves less effectively than the first one, except for the case of synchronism, however. In the longitudi...

The stability of small amplitude Rossby waves in a channel

The breakdown of Rossby waves in a bounded system is studied for the case in which the wave amplitude is small. In a very long, laterally bounded, channel all waves are unstable via second-order resonant interactions except those of wavenumber π/L in the cross-channel direction (where L is the channel width), which are stable if their longitudinal wavenumber is greater than 0·681π/L. These waves are, however, unstable to weaker side-band interactions, so that all waves with non-zero longitudinal wavenumber are unstable. The transition from sideband to triad instability occurs where the group velocity of the basic wave is equal to the velocity of long waves.

Intermittent ‘full’ upwelling in Lake Ontario

A strong enough longshore impulse leaving the coast to the left (looking downwind) generates ‘Full’ upwelling in which the thermocline comes to intersect the free surface. This problem is beyond the scope of linearized (small displacement) theory, but it may be treated as a simple geostrophic adjustment problem in which the wind stress is supposed to be exerted impulsively on a two-layer fluid. A minimum impulse is found to be necessary for a full upwelling to develop from hydrostatic equilibrium, the magnitude of which is close to the product of top layer depth and propagation velocity of long waves on the thermocline. When the upwelling-favoring wind impulse is greater than the minimum, the upwelled front moves offshore by a distance proportional directly to the extra impulse (above the minimum) and inversely to top layer depth times Coriolis parameter. Upwelling episodes observed in Lake Ontario during the International Field Year for the Great Lakes (IFYGL) and feasibility studies before IFYGL show a frontal behavior in good quantitative agreement with the simple theoretical model. The offshore component of the impulsively exerted wind leaves the position of the front unaffected. However, a sustained offshore wind increases somewhat the offshore displacement due to a prior longshore impulse. The effect is relatively weak, and an offshore wind alone rarely produces full upwelling. A sustained longshore wind acting on an already upwelled thermocline generates considerable negative potential vorticity. Adjustment following such a ‘second’ longshore impulse produces a more complex thermocline shape and under certain conditions a frontal countercurrent. In a closed basin, full upwelling following a wind stress impulse occurs over only a portion of the shoreline. Given quiescent conditions, the upwelled front may be expected to propagate alongshore, somewhat as an internal Kelvin wave. Observations in October 1972 in Lake Ontario show frontal motions resembling internal Kelvin waves. However, only one half of the wave propagates, that half in which particle velocities have the same direction as the wave propagation velocity. Furthermore, the propagating warm zone is much narrower than the cold upwelled zone which developed under a succession of previous wind impulses.

Selective withdrawal from a layered fluid

The selective withdrawal of a layered fluid from a reservoir has interesting properties which are not present if the stratification is continuous. The problem investigated is that of finding the ratios of the discharges from each flowing layer when the total discharge from all layers is regulated at the outlet. It is shown that the ratios are determined by the requirement that the flow be smooth at critical points in the flow, each critical point being a point at which the long-wave velocity on one of the interfaces is zero. If there are too many critical points, the flow is over-determined and becomes unsteady. It follows therefore that if the external conditions change slowly, such as in the slow draining of a reservoir, the flow alternates between intervals in which the flow ratios also change slowly and intervals in which the flow ratios oscillate unsteadily. An experiment is described in which the theoretical conclusions were tested for two-layer flow. Good agreement was obtained between theory and experiment.