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💨Fluid Dynamics

💨fluid dynamics review

10.3 Gravity waves

8 min readLast Updated on August 20, 2024

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)\omega^2 = gk\tanh(kh), where ω\omega is the angular frequency, gg is the acceleration due to gravity, kk is the wavenumber, and hh 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 (HsH_s), peak period (TpT_p), 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


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.