Gravity waves are a fundamental aspect of fluid dynamics in water bodies. They shape our oceans, influence weather patterns, and impact coastal environments. Understanding their properties, generation, and propagation is crucial for predicting and managing their effects on human activities and natural systems.
From wind-wave interactions to breaking waves, gravity waves exhibit complex behaviors. Their mathematical description, measurement techniques, and applications in forecasting, energy conversion, and coastal engineering highlight their significance in both scientific research and practical applications.
Properties of gravity waves
Wavelength and wave height
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Wavelength is the horizontal distance between two consecutive wave crests or troughs
Determines the size and period of the wave
Wave height is the vertical distance between a wave crest and the adjacent trough
Influenced by wind speed, duration, and fetch (distance over which the wind blows)
Longer wavelengths generally correspond to higher wave heights
Dispersion relation
Mathematical relationship between the wavelength, wave period, and water depth
Governs the propagation speed of waves with different wavelengths
In deep water, longer wavelengths travel faster than shorter wavelengths
In shallow water, the propagation speed depends on water depth rather than wavelength
Dispersion relation is given by ω2=gktanh(kh), where ω is the angular frequency, g is the acceleration due to gravity, k is the wavenumber, and h is the water depth
Phase and group velocity
Phase velocity is the speed at which an individual wave crest or trough propagates
Determined by the dispersion relation and varies with wavelength and water depth
Group velocity is the speed at which a group of waves with similar wavelengths propagates
Represents the speed of energy transport in the wave field
In deep water, group velocity is half the phase velocity
In shallow water, group velocity approaches the phase velocity
Generation of gravity waves
Wind-wave interaction
Waves are primarily generated by wind blowing over the water surface
Wind transfers energy to the water through pressure fluctuations and shear stress
As wind speed increases, wave height and wavelength increase
Fetch and duration of the wind also influence wave growth
Longer fetches and durations result in larger, more developed waves
Resonance and feedback mechanisms
Resonance occurs when the wind speed matches the phase speed of the waves
Leads to efficient energy transfer from the wind to the waves
Feedback mechanisms, such as wave-induced pressure fluctuations, further enhance wave growth
Miles' mechanism describes the resonant interaction between wind and waves
Phillips' mechanism explains the initial generation of waves by random pressure fluctuations
Propagation of gravity waves
Deep vs shallow water waves
Deep water waves occur when the water depth is greater than half the wavelength
Particle motion in deep water waves is circular and decreases exponentially with depth
Shallow water waves occur when the water depth is less than 1/20th of the wavelength
Particle motion in shallow water waves is elliptical and uniform throughout the water column
Transitional water waves occur between deep and shallow water conditions
Wave refraction and diffraction
Refraction occurs when waves propagate from deep to shallow water at an oblique angle
Causes waves to bend and align more parallel to the depth contours
Results in wave energy convergence or divergence, affecting wave height and direction
Diffraction occurs when waves encounter obstacles or pass through gaps
Allows waves to bend around obstacles and spread out behind them
Important in the sheltering of coastlines by headlands or islands
Energy transport and dissipation
Waves transport energy as they propagate across the ocean surface
Energy flux is proportional to the square of the wave height and the group velocity
Energy dissipation occurs through various mechanisms, such as whitecapping, bottom friction, and wave breaking
Dissipation rates depend on factors like wave steepness, water depth, and bottom roughness
Energy dissipation leads to a gradual decrease in wave height over time and distance
Breaking of gravity waves
Wave steepness and instability
Wave steepness is the ratio of wave height to wavelength
As waves propagate and interact with each other or the bottom, their steepness may increase
When wave steepness exceeds a critical value (typically around 1/7), waves become unstable and break
Breaking occurs as the particle velocity at the wave crest exceeds the phase velocity of the wave
Types of breaking waves
Spilling breakers occur on gently sloping beaches or in deep water
Characterized by a gradual breaking process, with foam and turbulence at the wave crest
Plunging breakers occur on moderately sloping beaches
Characterized by a steep wave front that overturns and plunges forward, creating a barrel-shaped crest
Surging breakers occur on steeply sloping beaches
Characterized by a rapid rise and fall of the water level at the shoreline, with minimal wave breaking
Energy dissipation in breaking
Wave breaking is a highly dissipative process, converting wave energy into turbulence and heat
Energy dissipation rates in breaking waves can be orders of magnitude higher than in non-breaking waves
Dissipation is concentrated near the surface, where turbulence and air entrainment are most intense
Breaking-induced dissipation is a key factor in limiting wave growth and maintaining equilibrium wave conditions
Dissipation also plays a crucial role in nearshore processes, such as sediment transport and beach morphology
Interaction of gravity waves
Wave-wave interactions
Waves can interact with each other through various nonlinear mechanisms
Resonant interactions occur when the frequencies and wavenumbers of interacting waves satisfy specific conditions
Can lead to energy transfers between waves and the generation of new wave components
Triad interactions involve three wave components and are important in shallow water
Quadruplet interactions involve four wave components and are dominant in deep water
Wave-current interactions
Currents can modify the propagation and characteristics of waves
Opposing currents increase wave steepness and height, while following currents have the opposite effect
Currents can also refract waves, changing their direction of propagation
Wave-current interactions are important in tidal inlets, river mouths, and coastal regions with strong currents
Can lead to phenomena such as wave blocking, wave breaking, and the formation of rogue waves
Wave-bottom interactions
As waves propagate into shallower water, they interact with the bottom topography
Bottom friction dissipates wave energy, leading to a gradual decrease in wave height
Bottom topography can also refract and diffract waves, affecting their direction and height
In very shallow water, waves can induce sediment transport and reshape the bottom profile
Wave-bottom interactions are crucial in nearshore processes, such as beach erosion and accretion
Mathematical description of gravity waves
Linear wave theory
Assumes small-amplitude waves and irrotational, inviscid flow
Governed by the Laplace equation for the velocity potential and appropriate boundary conditions
Provides analytical solutions for wave properties, such as surface elevation, velocity, and pressure fields
Valid for a wide range of wave conditions, particularly in deep water
Limitations include the inability to describe wave breaking and nonlinear effects
Nonlinear wave theories
Account for the nonlinear effects that become important in shallow water and for large-amplitude waves
Stokes wave theory is a perturbation approach that includes higher-order terms in the wave steepness
Cnoidal wave theory describes highly nonlinear waves in shallow water, characterized by sharp crests and flat troughs
Solitary wave theory describes a single, non-periodic wave propagating in shallow water
Nonlinear theories provide more accurate descriptions of wave properties and kinematics, especially near the surface
Numerical modeling of waves
Used to simulate wave propagation, transformation, and interaction in complex environments
Spectral wave models (e.g., WAM, SWAN) solve the wave action balance equation to predict the evolution of the wave spectrum
Phase-resolving models (e.g., Boussinesq, mild-slope) solve the governing equations for the surface elevation and velocity fields
Computational fluid dynamics (CFD) models solve the full Navier-Stokes equations, capturing detailed wave-structure interactions
Numerical models are essential tools for wave forecasting, coastal engineering design, and understanding wave dynamics in realistic conditions
Measurement of gravity waves
In-situ wave measurements
Wave buoys are the most common in-situ wave measuring devices
Measure surface elevation time series using accelerometers, GPS, or other sensors
Directional wave buoys can measure both wave height and direction
Pressure sensors deployed on the seafloor can measure wave-induced pressure fluctuations
Current meters (e.g., ADCPs) can measure wave orbital velocities in addition to currents
Remote sensing of waves
Satellite altimeters (e.g., Jason, Sentinel-3) measure wave height along their ground tracks
Synthetic Aperture Radar (SAR) can measure wave height, wavelength, and direction over large areas
High-frequency (HF) radar systems can measure nearshore wave parameters and surface currents
Stereo video systems can measure wave properties and breaking characteristics in the surf zone
Remote sensing techniques provide spatial coverage and long-term monitoring capabilities
Wave spectra and statistics
Wave measurements are often analyzed in terms of their frequency or directional spectra
Spectra describe the distribution of wave energy across different frequencies or directions
Statistical parameters, such as significant wave height (Hs), peak period (Tp), and mean direction, are derived from the spectra
Wave spectra can be used to identify different wave systems (e.g., wind sea, swell) and their characteristics
Spectral analysis is crucial for understanding wave climatology, extreme events, and long-term trends
Applications of gravity waves
Ocean wave forecasting
Operational wave forecasting systems predict wave conditions for maritime safety, coastal activities, and engineering applications
Based on numerical wave models (e.g., WAM, WaveWatch III) driven by wind fields from atmospheric models
Provide short-term forecasts (up to 10 days) and long-term projections (seasonal to climate scales)
Assimilate satellite and in-situ wave measurements to improve forecast accuracy
Wave forecasts are essential for ship routing, offshore operations, and coastal hazard assessment
Wave energy conversion
Waves are a promising source of renewable energy, with high power density and global availability
Wave energy converters (WECs) extract energy from the motion of waves and convert it into electricity
Different WEC types include oscillating water columns, point absorbers, and overtopping devices
WEC design and performance depend on the local wave climate and the device's response characteristics
Challenges include efficiency, survivability, and environmental impact
Wave energy has the potential to contribute to the global renewable energy mix and support coastal communities
Coastal engineering considerations
Waves are a primary driver of coastal processes, such as sediment transport, erosion, and flooding
Coastal structures (e.g., breakwaters, seawalls) are designed to protect shorelines from wave action
Design of coastal structures requires knowledge of extreme wave conditions and long-term wave climate
Beach nourishment and nature-based solutions (e.g., artificial reefs) can mitigate erosion and enhance coastal resilience
Wave-structure interactions are critical for the stability and functionality of coastal infrastructure (ports, harbors, offshore platforms)
Integrated coastal zone management considers the multiple uses and impacts of waves on coastal resources and communities