Miguel A. Losada

REFLECTION AND TRANSMISSION FROM POROUS STRUCTURES UNDER OBLIQUE WAVE ATTACK

The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined. For normal wave incidence, the reflection and transmission from a porous breakwater has been studied many times using eigenfunction expansions in the water region in front of the structure, within the porous medium, and behind the structure in the down-wave water region. For oblique wave incidence, the reflection and transmission coefficients are significantly altered and they are calculated here. Using a plane-wave assumption, which involves neglecting the evanescent eigenmodes that exist near the structure boundaries (to satisfy matching conditions), the problem can be reduced from a matrix problem to one which is analytic. The plane-wave approximation provides an adequate solution for the case where the damping within the structure is not too great. An important parameter in this problem is Γ 2 = ω 2 h ( s - i f )/ g , where ω is the wave angular frequency, h the constant water depth, g the acceleration due to gravity, and s and f are parameters describing the porous medium. As the friction in the porous medium, f , becomes non-zero, the eigenfunctions differ from those in the fluid regions, largely owing to the change in the modal wavenumbers, which depend on Γ 2 . For an infinite number of values of ΓF 2 , there are no eigenfunction expansions in the porous medium, owing to the coalescence of two of the wavenumbers. These cases are shown to result in a non-separable mathematical problem and the appropriate wave modes are determined. As the two wavenumbers approach the critical value of Γ 2 , it is shown that the wave modes can swap their identity.

3-D non-breaking regular wave interaction with submerged breakwaters

Abstract Modelling of the transformation and interaction of regular wave trains with submerged permeable structures is carried out. The existing literature, is summarized relevant theories presented, and theoretical results are compared with existing laboratory data. Special attention is paid to wave reflection. The influence of wave characteristics including oblique incidence, structure geometry and porous material properties on the kinematics and dynamics over and inside the breakwater is considered. Two different models are presented: an eigenfunction expansion 3-D model and a 2-D model based on a mild-slope equation for porous media to account for breakwater slope.

PROPAGATION OF OBLIQUE INCIDENT WAVES PAST RIGID VERTICAL THIN BARRIERS

The linear theory for waves impinging obliquely on a vertical thin barrier is examined. Three types of structures are analysed: a barrier descending from the water surface to a distance, d, an underwater barrier, and a barrier with a submerged horizontal slit. Theoretical solutions are obtained by an eigenfunction expansion method for the transmission and reflection coefficients. The plane wave approximation is further examined and compared to the full solution. The distribution of the pressure along the barrier is also calculated and the maximum horizontal force is compared to the case of a vertical wall.

Harmonic generation past a submerged porous step

Experiments were conducted in a wave flume to study the differences between harmonic evolution of monochromatic waves as they propagate over a submerged impermeable or porous step under non-breaking conditions. Results are used as a preliminary analysis to establish some engineering design criteria on harmonic generation on submerged porous structures. The root-mean-squared wave height evolution is also studied and compared to linear models as a first approximation. It is shown that porous structure increases the effective relative depth and decreases the relative wave height, resulting in a lower Ursell number and a lower chance to generate harmonics. The effective water depth over a step as defined in the paper, provides information to evaluate the potential harmonic generation.

An analytical method to evaluate the efficiency of porous screens as wave dampers

Abstract The linear theory for water waves impinging obliquely on dissipative multilayered media is used to evaluate the reflection and transmission coefficients. The case of periodic medium consisting of alternating layers of upright porous walls and water of equal or different thicknesses is examined. Wave propagation in these media exhibits Bragg reflection. Using a plane wave assumption, which involves neglecting the evanescent eigenmodes that exist near the structure boundaries (to satisfy matching conditions), the problem can be reduced to a matrix equation which relates the complex amplitudes of the incident and reflected plane waves to the complex amplitude of the transmitted wave. The problem of wave propagation is analysed for an infinite channel, and for a wave flume in which a paddle generates the waves and a backwall limits waves propagation. The variation of the magnitude of the reflection coefficient, | R |, with k 1 Λ is discussed, where k 1 is the wavenumber in the water and Λ is the width of a unit cell consisting of two layers, one porous and another water. Increasing the porous layer width or decreasing the wave period broadens the range of k 1 Λ values for which resonance occurs. Furthermore, with increase in the angle of wave incidence, θ, the value of | R | decreases and the dependence of | R | on k 1 Λ is damped; for large angle of incidence the reflection is almost constant and negligible. On increasing the number of absorber units, an overall decrease in the reflection coefficient is achieved, but an increase in the number of oscillations between resonant peaks and in the peak amplitude also occurs.

Experimental study of wave-induced flow in a porous structure

Abstract The kinematics and dynamics of oscillatory flow in porous media are experimentally studied in an idealized porous structure. The concept of seepage velocity, extensively used in literature for the study of porous media, is analyzed. Spatial and temporal fluctuations due to irregularities in the porous structure are evaluated; it is shown that, quantitatively, spatial fluctuations are always more important. Measurements of the velocity, pressure field and wave height inside the structure reveal that the flow behaviour changes, resulting in two characteristic regions: transition and transmission regions. The relative width of the structure, B/L, associated with the formation of a standing wave and resonant conditions inside the structure, was found to be an important parameter to establish the location of the two regions. The transition region is characterized by very irregular records with secondary peaks, little or no dissipation and important higher harmonics in the velocity spectra. Dissipation is the most important factor in the transmission region where the flow tends to become more regular. The porous structure works as a filter, filtering out the higher frequencies as the oscillation propagates towards the leeside of the structure.

Temporal and spatial relationship between sediment grain size and beach profile

Abstract Sediment samples and beach profile evolution data collected along one profile line at “El Puntal” Spit, Santander, Spain, are used to analyze the spatial and temporal structure of the grain size distribution variability and its relationship with the beach profile changes. Standard principal component analysis (PCA) and three-way PCA is applied to determine the temporal and spatial scales of variability of the data. Results indicate that the sediment grain size distribution varies markedly along the beach profile both spatially and temporally. These variations are shown to be strongly related to morphological changes in the beach profile. The spatial eigenvectors determined from the profile data and those from the sediment data exhibit similar patterns with their maxima and minima located at the same position. Since eigenvectors may be regarded as representative of uncorrelated modes of variability it is concluded that the spatial variability of both sediment and profile data are strongly related. In particular, it is shown that the location of the highest variability of grain size corresponds to that of the beach profile. Also, different grain sizes are shown to exhibit a distinct degree of variability which leads to the conclusion that each sediment size responds to the same hydrodynamics differently. The temporal eigenvectors determined from the profile and the sediment data shown a seasonal dependency. However, their maxima and minima are not located at the same position. It is shown that this temporal shift is due to the different response of each sediment size to the hydrodynamics, and in particular, that the recovery of the profile starts with fine material from the bar. It is inferred that models for beach profile evolution which do not take into account the sorting processes involved in the sediment transport cannot be fully succesful. A “master” grain size sample, constructed by adding all the grain samples taken over the profile, is used to further examine the cross-shore redistribution of the sediment. The following working hypothesis is suggested: “For a beach profile within a physiographic unit the master grain size does not depend on time”.

Bottom-boundary-layer measurements on the continental shelf off the Ebro River, Spain

Abstract Measurements of currents, waves and light transmission obtained with an instrumented bottom tripod (GEOPROBE) were used in conjunction with a theoretical bottom-boundary-layer model for waves and currents to investigate sediment transport on the continental shelf south of the Ebro River Delta, Spain. The current data show that over a 48-day period during the fall of 1984, the average transport at 1 m above the seabed was alongshelf and slightly offshore toward the south-southwest at about 2 cm/s. A weak storm passed through the region during this period and caused elevated wave and current speeds near the bed. The bottom-boundary-layer model predicted correspondingly higher combined wave and current bottom shear velocities at this time, but the GEOPROBE optical data indicate that little to no resuspension occurred. This result suggests that the fine-grained bottom sediment, which has a clay component of 80%, behaves cohesively and is more difficult to resuspend than noncohesive materials of similar size. Model computations also indicate that noncohesive very fine sand in shallow water (20 m deep) was resuspended and transported mainly as bedload during this storm. Fine-grained materials in shallow water that are resuspended and transported as suspended load into deeper water probably account for the slight increase in sediment concentration at the GEOPROBE sensors during the waning stages of the storm. The bottom-boundary-layer data suggest that the belt of fine-grained bottom sediment that extends along the shelf toward the southwest is deposited during prolonged periods of low energy and southwestward bottom flow. This pattern is augmented by enhanced resuspension and transport toward the southwest during storms.

Propagation of oblique incident modulated waves past rigid, vertical thin barriers

Abstract Rigid thin barriers can be used to reduce energy of incoming short waves in coastal areas. The eigenfunction expansions method is used to study the incidence of monochromatic waves. The scattering of the long waves generated due to oblique incident bichromatic waves are also of great importance and have to be considered in the design of sheltered areas. It is shown that maximum short wave reflection is associated with maximum long wave transmission. Possible resonant effects may be attenuated choosing a barrier orientation which may cause minimal long wave transmission for the dominant incident waves.

Wave spectrum scattering by vertical thin barriers

The linear theory for waves impinging obliquely on vertical thin barriers is extended to evaluate the scattering of irregular waves, described by a TMA directional wave spectrum. A very good agreement is obtained when comparing the spectral approach with experimental data by other authors. The dependence of the transmission and reflection coefficients on the directional spreading function and on the angle of wave incidence is analyzed. Moreover, the transfer function of the pressure on the barrier under normal and oblique incidence wave is evaluated.