An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

Scattering of gravity waves by multiple surface-piercing floating membrane

Interaction of surface gravity waves with multiple vertically moored surface-piercing membrane breakwaters in finite water depth is analyzed based on the linearized theory of water waves. The study is carried out using least square approximation method to understand the effect of the vertical membrane as effective breakwater. Initially the problem is studied for a single membrane wave barrier but for the case of multiple membrane breakwaters the study is carried out using the method of wide-spacing approximation. In the present study, it is observed that the deflection of the membrane is reduced with the increase in the stiffness parameter of the mooring lines attached to the membrane. In the case of single surface-piercing membrane with moored and fixed edge conditions, the reflection and transmission coefficients are compared and analyzed in detail. The resonating pattern in the reflection coefficients are also observed for multiple floating membrane which can also be referred as Bragg's resonance. In the presence of the porosity constant the wave reflection is also observed to be decreasing and the change in the distance between the vertical floating breakwaters also helps in the attenuation of wave height. It is observed that the presence of multiple floating breakwater helps in the reduction of wave height in the transmitted region.     

台湾浅滩多尺度沙波地貌的地形傅里叶分解

海底沙波在全球广泛分布、成因复杂,但往往多种尺度的沙波叠加在一起形成复杂的沙波地貌体系,导致难以进行量化研究。针对该问题,本文提出一种实用的傅里叶分析方法,设计了巴特沃斯滤波器,将水深数据变换到频率域,进而将复杂沙波地貌分解成不同频率的单一类型沙波。并以台湾浅滩复杂的沙波地貌体系为例进行了分析研究,分解量化出3种空间尺度的沙波:巨型沙波(波长>100 m,波高>5 m)、中型沙波(波长5～100 m,波高0.4～5 m)和沙波纹(波长<5 m,波高<0.4 m)。本文提出的海底沙波地貌量化分析方法,有助于研究不同尺度海底沙波的成因与机理,对沙波区海洋工程的安全评估也具有实用价值。     

Improved explicit approximation of linear dispersion relationship for gravity waves

For water waves the transcendental dispersion relationship is solved by iterative methods when wave period and water depth are given and wavelength or wave number is required. A highly accurate explicit approximation to linear dispersion relationship is proposed based on Eckart's explicit relationship. While Eckart's expression is accurate to within 5%, the improved relationship has a maximum relative error of less than 0.05%. A simpler form of the relationship with 0.2% accuracy is also given.     

基于类海啸波型的岛礁水动力特性数值模拟研究

基于非静压单相流模型NHWAVE建立了高精度二维数值波浪水槽,采用日本2011年实测真实海啸波型系统研究了海啸波在岛礁上传播变形的规律,并且分析了波高、礁坪淹没水深和礁前斜坡坡度等因素对孤立波和真实海啸传播变形的影响。结果表明,相比孤立波,类海啸波的波长明显大于孤立波波长,在测点处引起的水面变化持续时间更长,同等波高情况下真实海啸波型比孤立波能够携带更多的能量,与岛礁的相互作用也更为复杂,在礁坪上形成的淹没水深约为孤立波的两倍。礁前斜坡坡度和礁坪淹没水深均对类海啸波的反射和透射系数有显著影响。随着礁前斜坡坡度的增加,反射系数和透射系数均逐渐增加。随着礁坪淹没水深的增加,反射系数逐渐减小,而透射系数逐渐增大。但是,反射系数和透射系数均随着入射波高的增加而逐渐减小。     

Calculating the Wave Force on Partially Immersed Large-Scale Horizontal Cylinders

Large-scale interceptors constitute the main structure of offshore self-driven floating marine litter collection devices,and the structural stability of such interceptors under the action of waves directly influences the overall safety of the device.When the ratio of the diameter of a horizontal cylinder in such interceptors to the incident wavelength is larger than 0.25,the wave force can be calculated by using the diffraction theory,by considering the problem as that of the interaction between the waves and a partially immersed large-scale horizontal cylinder.In this study,an analytical approach to calculate the wave force on a partially immersed large-scale horizontal cylinder was formulated by using the stepwise approximation method.Physical model tests were conducted to investigate the effects of different factors(wave height,period,and immersion depth)on the wave force on a large-scale horizontal cylinder under conditions involving short-period waves.The results show that both horizontal and vertical wave forces on the cylinder increase as the wave height(immersion depth)increases in most cases.The vertical wave force decreases with the decrease of the period.For the horizontal wave force,it increases with the decrease of the period when the wavelength is larger than the diameter of the cylinder and decreases with the decrease of the period when the wavelength is smaller than the diameter of the cylinder.     

Numerical study on cnoidal wave run-up around a vertical circular cylinder

A finite element model of Boussinesq-type equations was set up, and a direct numerical method is proposed so that the full reflection boundary condition is exactly satisfied at a curved wall surface. The accuracy of the model was verified in tests. The present model was used to further examine cnoidal wave propagation and run-up around the cylinder. The results showed that the Ursell number is a nonlinear parameter that indicates the normalized profile of cnoidal waves and has a significant effect on the wave run-up. Cnoidal waves with the same Ursell number have the same normalized profile, but a difference in the relative wave height can still cause differences in the wave run-up between these waves. The maximum dimensionless run-up was predicted under various conditions. Cnoidal waves hold entirely distinct properties from Stokes waves under the influence of the water depth, and the nonlinearity of cnoidal waves enhances rather than weakens with increasing wavelength. Thus, the variations in the maximum run-up with the wavelength for cnoidal waves are completely different from those for Stokes waves, and there are even significant differences in the variation between different cnoidal waves.     

Ephemeral sand waves in the hurricane surf zone

Airborne bathymetric LIDAR observations along the Florida panhandle after Hurricane Dennis (2005) show the first unequivocal observations of surf-zone sand wave trains.

These are found in depths of 5m along the trough of the hurricane bar, where hindcasts show strong longshore currents only during severe storms. The waves extend over tens of kilometers of coast after Dennis but are absent from the same area in four other datasets. Observed wavelength to water depth ratios are comparable to river dunes and tidal sand waves but height to depth ratios are smaller, with the largest wave heights around 0.1 times the water depth. The sand wave generation mechanism is hypothesized to be from wind-and-wave-induced longshore currents, which were hindcast to be large during Dennis, with destruction from water wave orbital velocities.     

Estimation of infragravity waves at intermediate water depth

The accuracy of nearshore infragravity wave height model predictions has been investigated using a combination of the spectral short wave evolution model SWAN and a linear 1D SurfBeat model (IDSB). Data recorded by a wave rider located approximately 3.5 km from the coast at 18 m water depth have been used to construct the short wave frequency-directional spectra that are subsequently translated to approximately 8 m water depth with the third generation short wave model SWAN. Next the SWAN-computed frequency-directional spectra are used as input for IDSB to compute the infragravity response in the 0.01 Hz–0.05 Hz frequency range, generated by the transformation of the grouped short waves through the surf zone including bound long waves, leaky waves and edge waves at this depth. Comparison of the computed and measured infragravity waves in 8 m water depth shows an average skill of approximately 80%. Using data from a directional buoy located approximately 70 km offshore as input for the SWAN model results in an average infragravity prediction skill of 47%. This difference in skill is in a large part related to the under prediction of the short wave directional spreading by SWAN. Accounting for the spreading mismatch increases the skill to 70%. Directional analyses of the infragravity waves shows that outgoing infragravity wave heights at 8 m depth are generally over predicted during storm conditions suggesting that dissipation mechanisms in addition to bottom friction such as non-linear energy transfer and long wave breaking may be important. Provided that the infragravity wave reflection at the beach is close to unity and tidal water level modulations are modest, a relatively small computational effort allows for the generation of long-term infragravity data sets at intermediate water depths. These data can subsequently be analyzed to establish infragravity wave height design criteria for engineering facilities exposed to the open ocean, such as nearshore tanker offloading terminals at coastal locations.     

Interaction of flexural gravity waves with shear current in shallow water

In the present study, the effect of shear current on the propagation of flexural gravity waves is analyzed under the assumptions of linearized shallow-water theory. Explicit expressions for the reflection and transmission coefficients associated with flexural gravity wave scattering by a step discontinuity in both water depth and current speed are derived. Further, trapping and scattering of flexural gravity waves by a jet-like shear current with a top-hat profile are examined and certain limiting conditions for the waves to exist are derived. The effects of change in water depth, current speed, incident wavelength and the angle of incidence on the group and phase velocities as well as on the reflection and transmission characteristics are analyzed through different numerical results.     

Profiles of fully developed (Airy) waves in different water depths

Airy waves have a sinusoidal profile in deep water that can be modeled by a time series at any point x and time t, given by η(x,t) = (H_o/2) cos[2πx/L_o − 2πt/T_w], where H_o is the deepwater height, L_o is the deepwater wavelength, and T_w is the wave period. However, as these waves approach the shore they change in form and dimension so that this equation becomes invalid. A method is presented to reconstruct the wave profile showing the correct wavelength, wave height, wave shape, and displacement of the water surface with respect to the still water level for any water depth.     

Coastal Bathymetric Studies from Space Imagery

Abstract

Studies of coastal bathymetry are important where littoral drift has implications on the planning of fishing and dredging operations. Also, there is a possibility of finding hitherto unknown bottom features in relatively less explored regions of the shallow seas around the globe. High resolution satellite imagery over oceans provides us with quantitative methods for estimating depth in shallow parts of the seas. One of the methods is the analysis of the refraction of coastal gravity waves observed on satellite imagery. A panchromatic image acquired by SPOT with 10 m resolution on March 22, 1986, over Bay of Bengal near Madras Coast, was used for this analysis. The image was enhanced to clearly bring out the wave structure seen on the sea surface. The image was then superimposed with a 1 km × 1 km grid. For each grid cell, 64 × 64 pixels at the center were considered for getting a Fast Fourier Transform to determine the wave spectrum and the dominant wavelength present there. The classical theory of gravity waves was used to relate the shallow water wavelengths obtained as above with the corresponding wavelengths in the deep water. The deep‐water wavelength was estimated to be 110 m using the known chart depths at a set of control points. The resulting depth estimates, when compared with standard bathymetric charts, were found, in general, to be well in agreement up to a depth of 30 m in the sea, with an r.m.s. error of 2.6 meters. The method seems to be very useful for remotely sensed bathymetric work. However, further research is required to reduce the error margin and operationalize the method.     

Experimental determination of wavelengths in the presence of a mean current

This paper describes a simple method for determining the wavelength of small amplitude waves under laboratory conditions where reflected wave components are present both with and without a mean current flow superimposed. It assumes a locally horizontal bed but requires no a priori assumption concerning the form of the dispersion relation with a coexisting current. Synchronous measurements of the water surface recorded along any straight line are analysed to yield Fourier coefficients at each location. It is then shown that for all practical conditions excluding a perfect standing wave, the average rate of change of wave phase in the chosen direction can be related directly to the component of incident wave number in that direction, irrespective of reflection coefficient or relative current strength. The technique has been applied to regular and bichromatic waves in a flume with an absorbing wave generator, and can also be applied in 3-D wave basins where waves and currents intersect at arbitrary angles. In combined wave–current experiments, by assuming the linear dispersion relation, it is also possible to estimate the effective current velocity.     

长江口沙波统计特征及输移规律

利用多波束测深系统对长江口南港、北槽、横沙通道和北港水下地貌进行测量,对沙波波高、波长、迎流倾角、背流倾角、水深和沉积物特征进行统计,并根据流速、沉积物粒径及水深估算其潮周期内净位移。结果表明:长江口沙波基本都为大型沙波,且沙波大小与其所在区域沉积物粒径呈正比关系;长江口各个区域沙波的对称性不同,涨、落潮优势流越明显,则沙波对称性越差,净位移越大;沙波对称性能反映此区域水动力强弱及潮周期内沙波净位移大小。     

孤立波特性的实验室模拟

孤立波是浅海水域中经常出现的一种波动现象,常用来描述海啸和风暴等引起的巨浪以及波长较长的表面波的某些特性。采用"水体瞬间坍塌"的方法产生孤立波,在二维波浪水槽内进行系列实验。实验结果表明,产生的孤立波波高与水深之比可达1.29;箱体宽度及箱内水体高度对波高影响较大;得到孤立波波高计算公式,可较好地反映孤立波波高与箱体宽度、水深和箱内水体高度之间的变化关系,并给出了公式的适用范围。     

Investigations on the reflection behaviour of a slotted seawall

Based on the linear wave theory and the eigenfunction expansion method, the interaction between waves and a slotted seawall is studied analytically as well as experimentally. The analytical investigations show that the reflection characteristics of a slotted seawall depend mainly on the porosity of the slotted plate and the incident wave height. It is found that the reflection coefficient reaches its minimum value as the chamber width is about a quarter of the incident wavelength. The reflection behaviour becomes optimal when the porosity is moderate (about 0.2 in the present study). The immersed depth has some effect on the reflection coefficient of the seawall structure, but the improvement is small when the immersed depth is larger than the half of the water depth. A series of physical model experiments are carried out to examine the analytical results. Comparisons between the analytical and experimental results of reflection coefficients are made for some comparable cases and show good agreement provided certain parameters of this model are suitably chosen. This demonstrates that the analytical model is able to account adequately for energy dissipation by the slotted plate and can provide instruction for the design of such seawalls.     

Depth of activation on a mixed sediment beach

The relationship between wave height and depth of sediment activation is evaluated on an estuarine beach to determine whether activation depth is less in pebbles than sand. Rods with washers were used to monitor three excavated beach plots filled with (1) pebbles with mean grain size of 11.5 mm; (2) sand and granules; and (3) sand, granules and pebbles. Plots were monitored for 26 events over 27 days. Significant wave heights ranged from 0.18 to 0.40 m and activation depths from 0.02 to 0.12 m. Activation depths in the pebble plot were less than the other two plots when waves reworked sediment not activated during previous tidal cycles. Proportionality coefficients for activation depth to wave height, when net change was < 0.02 m, were 0.24 in the pebble plot and 0.30–0.31 in the other plots when experimental fill sediments remained and 0.22 to 0.23 in all three plots over the entire monitoring period, which included activation of newly deposited native sediment. Results suggest that for similar wave heights, activation depths in pebbles is lower than in sand, granules and pebbles or sand and granules, but once waves have reworked the sediment there is little difference in activation depths.     

Side-scan sonar investigation into temporal variation in sand wave morphology: Helwick sands,Bristol Channel

The morphology and mobility of bedforms within a sand wave field having a water depth of 30 to 40 m have been studied by side-scan sonar surveys at different tidal stages and under various wave climates. Large sand waves with heights of 4 to 7 m retained their orientation throughout the survey period, Small sand waves with heights less than 2 to 3 m changed their height over a tidal cycle and their location (relative to larger sand waves) between surveys. The maximum change appeared to be related to ebb current acceleration. Megaripple wavelengths were reduced under surface wave action.     

Discussion of “A function to determine wavelength from deep in to shallow water based on the length of the cnoidal wave at breaking” by J.P. Le Roux

An empirical modification to the Airy equation for wave celerity reduces to the expression for solitary waves in shallow water whilst retaining its usual form for deep water. The equation yields celerities in reasonable agreement with those for cnoidal waves in intermediate water depths. In this aspect, it is similar to the work described by Le Roux [Le Roux, J.P., 2007. A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking. Coastal Engineering 54, 770–774]. The empirical modification has been widely applied in computer programs over the past 30 years.     

Interaction of current and flexural gravity waves

The interaction between current and flexural gravity waves generated due to a floating elastic plate is analyzed in two dimensions under the assumptions of linearized theory. For plane flexural gravity waves, explicit expressions for the water particle dynamics and trajectory are derived. The effect of current on the wavelength, phase velocity and group velocity of the flexural gravity waves is analyzed. Variations in wavelength and wave height due to the changes in current speed and direction are analyzed. Effects of structural rigidity and water depth on wavelength are discussed in brief. Simple numerical computations are performed and presented graphically to explain most of the theoretical findings in a lucid manner.