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Wind wave

A wind wave, also known as a wind-generated wave, is a surface wave on a body of water, such as an ocean or lake, formed by the transfer of energy from wind blowing across the water surface.^[1]^[2] These waves arise from friction between the moving air and the water, initially creating small disturbances called capillary waves or ripples that evolve into larger gravity waves as wind speed, duration, and fetch (the distance over which the wind blows) increase.^[3]^[4]^[5] Wind waves typically have periods ranging from 5 to 20 seconds and wavelengths of 40 to 600 meters, dominating sea surface motions for periods shorter than 300 seconds near their generation source.^[1]^[6]^[7] As wind waves mature, they can transition into swell when the generating wind subsides, allowing the waves to propagate freely across vast ocean distances with reduced energy loss, often traveling thousands of kilometers while maintaining organized, longer-period forms.^[8] This propagation is influenced by factors such as wave direction relative to prevailing winds and currents, with wind waves generally exhibiting irregular, chaotic patterns in active generation areas compared to the more uniform swell.^[8]^[9] The height and steepness of wind waves are limited by wind conditions; for instance, sustained winds of 20 meters per second over a sufficient fetch can produce waves up to 5-10 meters high, though extreme events in high winds can exceed this.^[5]^[10] Wind waves play a critical role in ocean-atmosphere interactions, facilitating momentum, heat, and gas exchange across the air-sea interface, which influences weather patterns, climate dynamics, and marine ecosystems.^[11] They also pose significant hazards to maritime navigation, coastal infrastructure, and offshore operations, with wave breaking and associated turbulence driving sediment transport and erosion along shorelines.^[6] In oceanography, understanding wind wave generation and evolution is essential for modeling sea states, predicting storm surges, and assessing renewable energy potential from wave power.^[11] Modern studies increasingly integrate physics-guided deep learning to improve wave forecasting accuracy, addressing challenges in capturing nonlinear interactions during high-wind events.^[11]

Formation and Generation

Mechanism of Formation

Wind blowing across the sea surface generates shear stress at the air-sea interface, which disturbs the otherwise calm water and initiates the formation of small waves. This stress arises from the frictional drag between the moving air and the water surface, creating initial disturbances that manifest as capillary waves—short waves (wavelengths less than about 1.7 cm) where surface tension provides the primary restoring force. As these capillary waves grow and interact with the wind, they transition into capillary-gravity waves and eventually dominate gravity waves, where buoyancy acts as the restoring force for longer wavelengths. The process begins with instabilities in the shear flow, leading to exponential growth of these initial wavelets.^[12]^[13] Turbulence within the wind flow is essential for the progression from these small ripples to larger, fully developed seas. Turbulent eddies in the atmospheric boundary layer impinge on the water surface, generating pressure fluctuations that resonate with the waves, thereby transferring momentum and energy efficiently. This turbulence enhances airflow separation over wave crests, creating regions of sheltering on the leeward sides that amplify wave growth. As waves steepen, micro-breaking events induced by turbulence further couple the air and water motions, sustaining the energy input until a mature sea state is reached, characterized by a balance between wind forcing and wave dissipation.^[12]^[13] The mechanism of wind wave formation has intrigued observers for centuries. Leonardo da Vinci documented early insights in the late 15th century, describing how wind-generated waves propagate across the water surface without transporting the underlying fluid mass, likening them to undulations in a field of grain. These qualitative observations were later substantiated through systematic field experiments in the 19th and 20th centuries, including ship-based visual measurements that quantified wave development under varying wind conditions. A seminal contribution came from Sverdrup and Munk in 1947, who analyzed extensive oceanographic data to establish empirical relationships for wave growth.^[14]^[15] In fetch-limited scenarios, where wind blows steadily over a finite distance, the significant wave height H_s grows according to the approximate empirical relation:

H_s \approx 0.0163 \left( \frac{U^2}{g} \right)^{0.5} F^{0.5}

Here, U is the wind speed at 19.5 m above the surface (in m/s), F is the fetch length (in km), and g is the acceleration due to gravity (9.81 m/s²). This formula, derived from mid-20th-century analyses of field data, illustrates how wave height scales with wind speed and the distance over which the wind acts, transitioning from young waves near the upwind boundary to more developed conditions farther downwind.^[15]

Influencing Factors

The generation and initial development of wind waves are primarily influenced by wind speed, duration, fetch length, water depth, and the direction of the fetch relative to the wind. Wind speed determines the amount of energy transferred to the water surface, with higher speeds producing larger waves, while duration refers to the time the wind blows steadily, allowing waves to grow progressively. Fetch length, the unobstructed distance over which the wind acts on the water, limits wave development if short, as it restricts the time available for energy accumulation. Water depth plays a key role in shallow environments, where it can constrain wave height and growth by introducing bottom friction, unlike in deep water where waves form freely without seabed interaction.^[16]^[5]^[17] The alignment of fetch direction with the prevailing wind is crucial, as a consistent wind direction maximizes effective fetch and enhances wave growth, whereas shifting winds reduce it by dispersing energy. A minimum wind speed of approximately 2.3 m/s is required to initiate capillary waves in clean water conditions, marking the threshold where wind shear overcomes surface tension to start wave formation.^[18]^[19] In fetch-limited environments, such as enclosed basins, wave heights and periods are significantly smaller than in the open ocean due to geographical constraints that cap the distance over which winds can act. For instance, in western Long Island Sound, easterly winds along the basin's axis yield up to 65% higher significant wave heights compared to westerly winds over shorter fetches, but overall waves remain underdeveloped relative to unlimited ocean conditions.^[20] Atmospheric stability further modulates wave generation through its effects on wind profiles, with unstable conditions (e.g., warm water relative to air) enhancing turbulence and increasing wave growth rates by up to 50% in energy terms, while stable conditions suppress mixing and reduce growth. Thermal stratification alters the boundary layer, influencing shear stress at the air-sea interface and thus the efficiency of energy transfer from wind to waves.^[21]

Characteristics and Classification

Types of Wind Waves

Wind waves are classified into primary types based on their dominant restoring forces and wavelengths. Capillary waves, also known as ripples, are the smallest wind-generated waves with wavelengths typically less than 1.7 cm, where surface tension provides the primary restoring force.^[22] These form under light winds and appear as fine disturbances on the water surface. Gravity waves, the most common type in oceans, have wavelengths from centimeters to hundreds of meters and are restored primarily by gravity, dominating the energy in wind-driven seas.^[23] Infragravity waves, with periods between 30 seconds and 5 minutes, arise from nonlinear interactions among shorter wind waves and occupy the low-frequency end of the wave spectrum.^[24] Wind waves evolve through distinct development stages influenced by wind duration and fetch. In the young sea stage, waves grow rapidly under accelerating winds, resulting in steep, irregular forms with heights increasing nonlinearly.^[9] Fully developed seas represent an equilibrium state where wave height reaches a maximum for a given wind speed and duration, producing a chaotic mix of waves across various sizes and directions.^[25] Following this, waves transition to decaying swell as they propagate away from the wind source, losing energy through dispersion and becoming more organized.^[1] A key distinction exists between sea and swell within wind wave classifications. Sea waves, or wind seas, are generated locally by current winds, exhibiting short periods (4-8 seconds), irregularity, and alignment with the wind direction.^[26] In contrast, swell consists of waves that have traveled far from their origin, featuring longer periods (over 10 seconds), greater regularity, and reduced steepness due to sorting by dispersion.^[27] Rogue waves represent extreme manifestations of wind waves, defined as individual waves exceeding twice the significant wave height—the average of the highest one-third of waves in a given period.^[28] These unpredictable events, often forming through wave superposition or focusing, can pose severe hazards to maritime activities. In severe storms, rogue waves occur approximately twice daily at any location, highlighting their relative frequency under intense conditions.^[29]

Wave Spectrum

The wave spectrum provides a two-dimensional representation of the energy distribution in wind wave fields as a function of frequency and direction, denoted as S(f, θ), where f is the wave frequency and θ is the propagation direction.^[30] This spectral description captures the irregular nature of ocean waves, integrating over all components to yield total variance in sea surface elevation.^[31] Key components of the wave spectrum include the peak frequency f_p, defined as the frequency corresponding to the maximum energy density; the spectral width, a measure of the bandwidth or spread of energy across frequencies, often quantified relative to f_p; and the directional spreading, which describes the angular distribution of wave energy around the mean direction, typically narrower in actively generated waves.^[31]^[32] These elements vary with sea state, influencing overall wave behavior such as propagation and energy transfer.^[30] In young seas, where waves are actively growing under local wind forcing, the spectrum exhibits a narrow peak with high energy concentration at f_p and limited directional spreading, often aligned closely with the wind direction.^[33] As waves evolve with increasing fetch and duration, the spectrum broadens, with f_p shifting to lower values, greater spectral width, and increased directional spreading, transitioning to the more dispersed characteristics of swells that propagate independently of local winds.^[33]^[34] Empirical observations from the Joint North Sea Wave Project (JONSWAP) in fetch-limited conditions, under stationary winds over distances up to 160 km, revealed spectra with a pronounced peak and an overshoot in energy relative to equilibrium forms, parameterized by a peak enhancement factor γ ≈ 3.3 to account for the sharper spectral shape during development.^[34] This factor highlights the heightened energy at f_p in developing seas compared to fully developed states, based on direct measurements of wave growth and directional properties.^[34]

Spectral Models

Spectral models for wind waves provide mathematical representations of the wave energy spectrum, evolving from early empirical formulations in the mid-20th century to more sophisticated parameterizations that account for sea state development and directionality.^[35] The foundational work began in the 1950s with spectral approaches introduced by Neumann (1952) and Pierson (1953), which shifted from descriptive wave statistics to frequency-domain analyses based on field observations, laying the groundwork for quantitative predictions of wave energy distribution.^[35] By the 1960s and 1970s, these efforts progressed to parameterized models fitted to extensive datasets, incorporating physical principles like similarity theory, and culminated in hybrid formulations that blend empirical fits with theoretical constraints for broader applicability.^[36] The Pierson-Moskowitz (PM) spectrum represents fully developed seas, where waves have reached equilibrium with sustained winds over long fetches.^[37] Derived from shipborne measurements in the North Atlantic, it assumes a one-dimensional frequency spectrum that peaks at a frequency inversely proportional to wind speed.^[38] For developing seas, where fetch or duration limits growth, the JONSWAP spectrum extends the PM form by introducing a peak enhancement factor to better capture the narrower spectral peak observed in fetch-limited conditions.^[39] This parameterization, based on data from the Joint North Sea Wave Project, adjusts the exponential tail to reflect higher energy concentration around the peak frequency during early wave growth stages.^[39] A key parameterization for the PM spectrum is given by

S(f) = \alpha \frac{g^2}{(2\pi)^4 f^5} \exp\left[-1.25 \left(\frac{f}{f_p}\right)^{-4}\right],

where \alpha is the Phillips constant (typically 0.0081), g is gravitational acceleration, f is frequency, and f_p is the peak frequency.^[37] The JONSWAP spectrum modifies this by multiplying the PM form by a peak enhancement function \gamma^{\exp\left[ -\frac{(f - f_p)^2}{2\sigma^2 f_p^2} \right]}, with \gamma \approx 3.3 for typical conditions, \sigma = 0.07 for f \leq f_p, and \sigma = 0.09 otherwise.^[39] To incorporate directionality, spectral models extend the one-dimensional form with spreading functions that distribute energy azimuthally, often using D(\theta) = \frac{2}{\pi} \cos^{2s}(\theta/2) for |\theta| \leq \pi/2, where \theta is the direction relative to the mean wave propagation and s (spreading parameter) increases with frequency to reflect narrower spreading at higher frequencies.^[40] Seminal directional models, such as those proposed by Donelan et al. (1985) from lake measurements, refine this cosine-based spreading to depend on both wavenumber and wind speed, improving representations of short-crested seas.^[41] These models have been validated against field data, showing good agreement with buoy measurements for moderate wind speeds and fetches, where PM spectra match significant wave heights within 10-20% in fully developed cases, and JONSWAP improves peak frequency predictions by up to 15% in developing seas.^[42] Satellite altimeter data, such as from Envisat, further confirm spectral shapes in open ocean settings, with root-mean-square errors in energy density below 0.5 m²/Hz for JONSWAP applications.^[43] However, limitations emerge in extreme conditions, such as hurricanes, where PM underestimates high-frequency tails due to unaccounted nonlinear interactions and swell interference, leading to overpredictions of peak periods by 20% or more.^[44] Modern hybrid models address these by integrating parametric forms with physics-based source terms, enhancing accuracy across a wider range of sea states.^[36]

Propagation and Transformation

Shoaling and Refraction

As wind-generated surface waves propagate from deep to intermediate and shallow water, they undergo shoaling, a process in which wave height increases due to the decrease in local phase speed and wavelength while conserving energy flux. This height amplification arises because the group velocity, which determines energy transport, diminishes as depth decreases, concentrating energy into a narrower crest. In the shallow-water regime, the shoaling coefficient K_s, relating local wave height H to a reference height H_0 via H = K_s H_0, approximates K_s \approx (h_0 / h)^{1/4}, where h_0 is the reference water depth; this form, known as Green's law, stems from the linear shallow-water approximation where c \propto \sqrt{gh} and h is water depth.^[45] The onset of significant shoaling effects occurs around the critical depth where water depth d = L/20, with L the local wavelength, defining the boundary for shallow-water dynamics in which bottom friction and nonlinearity become prominent.^[46] Refraction accompanies shoaling as waves encounter varying bathymetry, causing wave crests to bend and align more parallel to depth contours, thereby altering wave direction toward shallower regions. This directional change follows Snell's law adapted for water waves, expressed as \frac{\sin \theta}{c} = \frac{\sin \theta_0}{c_0} = \text{constant}, where \theta is the angle between the wave ray (perpendicular to the crest) and the normal to the depth contour, and subscript 0 denotes deep-water conditions; the relation holds because the intrinsic wave frequency remains constant while phase speed c varies with local depth.^[47] For irregular coastlines with complex bathymetry, shoaling and refraction interact nonlinearly, leading to wave focusing or spreading that can amplify or attenuate heights beyond simple plane-slope predictions. Ray tracing methods address these combined effects by numerically solving the ray equations—derived from the eikonal approximation of the wave dispersion relation— to track individual wave rays from offshore to the coast, enabling computation of refraction coefficients and transformed wave fields.^[48]

Wave Breaking

Wave breaking occurs when the amplitude of a wind-generated wave becomes sufficiently large relative to its wavelength or the local water depth, leading to instability and collapse of the wave crest. This process dissipates wave energy into turbulence, heat, and momentum transfer to the underlying water column, marking a critical transition from progressive wave motion to dissipative surf zone dynamics. In deep water, breaking is primarily driven by excessive steepness, while in shallow water, it results from the interaction with the seabed as waves shoal and heighten.^[49] The classification of breaking types depends on the surf similarity parameter \xi_0 = \tan \beta / \sqrt{H_0 / L_0}, where \beta is the bottom slope, H_0 the deep-water wave height, and L_0 the deep-water wavelength. Spilling breakers form for \xi_0 < 0.5 (typically gentle slopes and moderate to high steepness), where the crest becomes unstable and foam spills down the front face gradually. Plunging breakers occur for $0.5 < \xi_0 < 3.3 (moderate slopes and varying steepness), characterized by the crest curling over and plunging forward into the trough, creating an air cavity. Surging breakers develop for \xi_0 > 3.3 (steep slopes and low steepness), where the wave surges up the beach without full collapse, often reflecting partially. This classification, originally quantified through laboratory observations, highlights how the interplay of bottom topography and wave steepness influences the breaking mechanism.^[49] Breaking criteria provide thresholds for onset. In deep water (where depth d > L/2), the Miche limit establishes instability when the steepness exceeds H/L > 0.14, derived from nonlinear wave theory assuming the crest particle velocity approaches the phase speed. In shallow water (d < L/20), breaking initiates when the height-to-depth ratio surpasses H/d > 0.78, based on solitary wave stability analysis where orbital velocities at the crest equal the wave celerity. These limits are semi-empirical, validated across flume and field data, though actual breaking can vary by 10–20% due to irregularity. Shoaling increases wave height prior to breaking, often pushing steepness toward these limits. Each breaking event dissipates a significant fraction of the wave's kinetic energy, typically 10–20% for spilling and plunging types in controlled conditions, converting it into turbulent kinetic energy and bubble entrainment that enhances vertical mixing. This loss scales with the cube of wave height and occurs rapidly over one wave period, reducing subsequent propagation and contributing to overall spectral decay in wind wave fields. Plunging breakers exhibit higher dissipation (up to 20–25%) due to intense overturning, while spilling types spread the loss more gradually.^[50]^[51] Wind forcing modulates breaking by altering wave growth and stability; onshore winds enhance steepness and promote earlier breaking, while offshore winds suppress it by sheltering crests. Laboratory experiments since the 1920s, including early flume studies on wind-wave interactions, have demonstrated these effects, showing wind speeds above 7 m/s increase breaking probability by 15–30% through momentum transfer. These observations underpin modern parameterizations of wind-induced dissipation in wave models.^[52]

Physical Principles

Fundamental Physics

The fundamental physics of wind waves is described by linear wave theory, which assumes small-amplitude waves where the wave height is much smaller than the wavelength, allowing for a linearized approximation of the governing equations. This theory also posits irrotational flow, meaning the velocity field can be derived from a scalar potential satisfying Laplace's equation, and neglects viscosity, treating the fluid as inviscid and incompressible. These assumptions simplify the analysis to focus on gravity-driven surface waves, providing a foundational framework for understanding wave motion under moderate wind conditions.^[53]^[54] A key relation in this theory is the dispersion relation, which connects the angular frequency \omega of the wave to its wavenumber k and water depth d: \omega^2 = g k \tanh(k d), where g is the acceleration due to gravity. This equation indicates that wave speed depends on both wavelength and depth, leading to dispersive behavior where different wave components propagate at different velocities. In deep water, where k d \gg 1, the hyperbolic tangent approximates to 1, simplifying to the deep-water dispersion relation \omega^2 = g k, implying that longer waves travel faster than shorter ones.^[54]^[53]/11:Appendix_A-_Linear_wave_theory) Under linear theory, water particle motion follows closed orbits that vary with depth. In deep water, these orbits are nearly circular, with particles moving in a backward direction relative to the wave propagation at the surface and diminishing exponentially with depth. In shallower water, the orbits become elliptical, flattening horizontally and vertically as the bottom is approached, until in very shallow conditions they approximate straight-line back-and-forth motion. These kinematics illustrate how wave energy is carried forward while individual particles oscillate without net displacement./11:Appendix_A-_Linear_wave_theory)^[55] The phase velocity c = \omega / k represents the speed of individual wave crests, while the group velocity c_g = d\omega / dk governs the propagation of wave energy and information. In deep water, the group velocity is half the phase velocity, c_g = c / 2, meaning energy travels more slowly than the apparent wave crests, which overtake the group. This distinction arises directly from the dispersion relation and explains phenomena such as the spreading of wave packets generated by wind. Dispersion also influences the shape of wave spectra by separating frequency components over time.^[53]/16:_Ocean_Waves/16.1:_Linear_Theory_of_Ocean_Surface_Waves)^[56]

Energy and Momentum Transfer

Wind waves acquire energy primarily from the wind through two key mechanisms: the sheltering acceleration proposed by Jeffreys and the pressure perturbation mechanism developed by Miles. Jeffreys' mechanism (1925) posits that wind induces a tangential stress on the leeward side of wave crests due to airflow sheltering, accelerating the air and creating a pressure difference that transfers momentum to the waves. In contrast, Miles' mechanism (1957) emphasizes perturbations in the atmospheric boundary layer above the wave surface, where wave-induced pressure fluctuations interact with the wind shear to amplify wave growth, particularly for longer waves. These processes dominate the initial wave generation and sustained growth under varying wind conditions.^[57] The energy flux of wind waves, which governs their propagation and transformation, is given by the product of the group velocity C_g and the energy density E, representing the rate at which wave energy is transported. In deep water, C_g is half the phase speed, derived from the dispersion relation, allowing energy to propagate at this reduced speed relative to individual wave crests.^[58] In shallow water, dissipation occurs through bottom friction, often parameterized using quadratic friction laws that relate shear stress to the square of the near-bottom orbital velocity, with empirical drag coefficients typically ranging from 0.005 to 0.01.^[59] This friction becomes significant when water depth is less than half the wavelength, leading to energy loss that scales with wave height and bottom roughness.^[60] Momentum transfer from wind waves influences both ocean currents and the atmosphere. To the ocean, waves impart momentum via Stokes drift, a net Lagrangian transport in the direction of wave propagation, with surface velocities up to several percent of the phase speed for typical wind seas.^[61] This drift enhances upper-ocean mixing and Ekman transport, particularly under strong winds.^[62] Conversely, waves exert drag on the atmosphere through form drag and viscous stresses, where airflow separation over wave crests contributes substantially to the total wind stress; in storms, wave-supported stress accounts for approximately 50% of the momentum flux from wind to ocean.^[63] This bidirectional exchange modulates air-sea coupling during high-wind events.^[64]

Modeling and Applications

Numerical and Analytical Models

Analytical models for wind waves provide foundational descriptions of wave behavior under idealized conditions, focusing on periodic waves in deep water. The linear theory developed by Airy assumes small-amplitude waves where the wave height is much smaller than the wavelength, leading to sinusoidal surface elevations and neglecting nonlinear effects.^[65] This theory yields expressions for wave kinematics, such as the dispersion relation \omega^2 = gk for deep water, where \omega is angular frequency, g is gravity, and k is wavenumber.^[66] Extensions of Airy theory incorporate mild nonlinearities for improved accuracy in moderate sea states. For steeper waves, nonlinear extensions like Stokes waves account for higher-order terms in the wave steepness ka, where a is amplitude. Stokes theory, originally formulated to fifth order, describes progressive waves with asymmetric crests and troughs, capturing effects like wave-induced mean currents. These analytical solutions remain essential for validating numerical models and understanding fundamental wave dynamics, though they are limited to monochromatic waves and uniform conditions.^[67] Numerical models simulate the evolution of irregular wind wave fields by solving the wave action balance equation, which governs the spectral density of wave action N(\mathbf{x}, \mathbf{k}, t):

\frac{\partial N}{\partial t} + \nabla_{\mathbf{x}} \left( \mathbf{c}_g N \right) + \frac{\partial}{\partial \mathbf{k}} \left( \dot{\mathbf{k}} N \right) = S

Here, \mathbf{c}_g is the group velocity vector, \dot{\mathbf{k}} represents changes in wavenumber due to refraction or currents, and S denotes source terms including wind input S_{in}, whitecapping dissipation S_{ds}, and nonlinear wave-wave interactions S_{nl}.^[68] The nonlinear term S_{nl}, dominant for fully developed seas, redistributes energy across frequencies via resonant quadruplet interactions, as parameterized in the Hasselmann kinetic equation.^[69] These models discretize the spectrum on a directional-frequency grid and propagate it spatially, enabling predictions of wave height, period, and direction over large domains. Third-generation spectral models, which fully resolve nonlinear interactions without empirical closures, have become standard for operational forecasting. The WAM (Wave Model) integrates the action balance equation with parameterized source terms, achieving global wave predictions with biases under 0.5 m in significant wave height during validation against buoys.^[70] Similarly, WAVEWATCH III, developed for flexible grids including unstructured meshes, incorporates advanced propagation schemes and has been widely adopted for hindcasts and real-time forecasts, supporting applications in marine safety and offshore engineering. As of 2025, WAVEWATCH III has advanced to version 7.14, featuring improved high-resolution modeling for tropical regions and enhanced physics parameterizations.^[71] Modern updates to these models post-2000 address limitations in complex environments by incorporating ocean currents and sea ice effects. Currents modulate wave refraction, Doppler shifting, and energy transfer, with studies showing up to 20% variations in significant wave height in current-dominated regions like the Gulf Stream when included.^[72] For ice-covered areas, parameterizations in models like WAVEWATCH III account for damping by ice floes and partial ice concentration, improving simulations in polar seas where waves penetrate marginal ice zones, enhancing forecast accuracy for ice breakup and navigation.^[73]

Seismic Signals

Wind waves interacting with the ocean bottom generate seismic signals known as microseisms, which are low-amplitude ground vibrations detectable by seismometers worldwide. These signals arise primarily from two mechanisms: primary microseisms, caused by direct pressure fluctuations from waves impacting the seafloor, and secondary microseisms, resulting from nonlinear interactions between opposing wave trains that produce standing waves and enhanced pressure variations. Primary microseisms typically occur in the frequency range of 0.05 to 0.5 Hz, corresponding to the dominant periods of ocean swells, while secondary microseisms appear at roughly double that frequency, around 0.1 to 1 Hz, due to the beating of wave frequencies. The generation of these seismic waves involves acoustic coupling where dynamic seafloor pressure from passing waves propagates as P-waves through the solid earth. The amplitude of microseisms scales with the square of the ocean wave height, meaning stronger storms produce significantly more intense signals, as the pressure perturbation is proportional to the wave energy flux. For instance, during major extratropical cyclones, secondary microseism amplitudes can reach levels that dominate background seismic noise. Microseisms have been observed since the late 19th century, with early attributions to ocean waves by Timoteo Bertelli in the 1870s.^[74] Following the 1906 San Francisco earthquake, persistent low-frequency tremors were noted and further linked to Pacific Ocean wave activity. Today, these signals enable remote monitoring of storm intensity and track tropical cyclone development by correlating microseism amplitudes with wave model outputs, providing a global, land-based proxy for ocean conditions without direct instrumentation.

Measurement Techniques

Measurement of wind waves has evolved significantly since the 19th century, beginning with subjective visual observations by mariners using scales like the Beaufort scale to estimate wave heights based on wind force, such as 0 feet for Force 0 and up to 41 feet for Force 10.^[75] These reports, collected along shipping lanes, provided qualitative data on average wave trains but suffered from inconsistencies due to observer variability. By the early 20th century, ship-based photography and stereophotogrammetry introduced quantitative analysis of wave elevations and spectra, while mid-century advancements included pressure transducers on ship hulls and sea bottoms for spectral corrections using linear wave theory.^[76] The development of accelerometer-equipped buoys in the 1960s and 1970s enabled routine in-situ measurements, and late 20th-century remote sensing via radar altimeters marked a shift to global coverage. Post-2010, integration of artificial intelligence in processing satellite imagery, such as neural networks for synthetic aperture radar (SAR) data, has enhanced accuracy and real-time analysis of complex sea states.^[77] In-situ techniques primarily rely on moored instruments to capture local wave properties like height, period, and direction. Wave buoys, such as the Datawell Directional Waverider MkIII, measure vertical acceleration using a single-axis accelerometer, which is double-integrated to derive surface displacement for significant wave height over periods of 1.6 to 30 seconds; horizontal accelerometers, pitch-roll sensors, and a fluxgate compass simultaneously determine directional spectra.^[78] Pressure sensors, often bottom-mounted in shallow waters, detect wave-induced hydrostatic pressure fluctuations, which are transformed to surface elevations via linear dispersion relations and depth-dependent corrections like the hyperbolic cosine factor.^[76] Current meters, including acoustic Doppler current profilers (ADCPs), infer wave orbital velocities from Doppler shifts in backscattered acoustic signals, providing integrated wave spectra alongside current profiles.^[79] These methods target key characteristics such as significant wave height and peak period, offering high temporal resolution (e.g., 1 Hz sampling) but limited spatial coverage. Remote sensing methods enable broad-scale observations, complementing in-situ data with global or regional insights. Satellite altimetry, exemplified by the Jason series (e.g., Jason-3), employs Ku-band radar to emit pulses and analyze the returned echo's leading-edge slope, yielding significant wave height with along-track resolution of about 7 km.^[80] High-frequency (HF) radar systems operate at frequencies like 4.5 MHz, transmitting radio waves that interact with ocean surface currents via Bragg scattering; the Doppler spectrum of backscattered signals resolves radial currents and, through inversion techniques like MUSIC, derives directional wave spectra over ranges up to 150 km with 5 km grid spacing, particularly valuable for coastal monitoring.^[81] SAR instruments on satellites like Sentinel-1 use C-band imagery in interferometric wide swath mode to estimate wave parameters via empirical algorithms, such as CWAVE_S1A, which decomposes image spectra into orthogonal components for significant wave height retrieval.^[77] Accuracy of these techniques varies by method and conditions, with calibration and environmental factors influencing reliability. Buoy systems like the Datawell Waverider achieve wave height accuracy better than 0.5% of the measured value immediately post-calibration, remaining under 1% after three years, and overall errors typically below 5% with proper maintenance.^[78] Pressure sensors and current meters exhibit similar precision in controlled deployments but can degrade in strong currents or biofouling. Satellite altimetry from Jason missions underestimates significant wave height in long-period swells due to footprint averaging and signal attenuation, with biases up to 0.5 m in such cases.^[80] HF radar directional spectra have root-mean-square errors around 20-30 degrees in direction, while Sentinel-1 SAR achieves 0.5-0.7 m RMSE for wave height against buoy validations, aided by AI-driven neural networks trained on model data.^[77] Constellations like Sentinel-1 mitigate real-time global coverage gaps by providing frequent revisits (e.g., 6-12 days), enabling near-continuous monitoring despite sparse in-situ networks.^[77]

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Shallow-water wave theory - Coastal Wiki
May 12, 2025 · This article explains some theories of periodic progressive waves and their interaction with shorelines and coastal structures.
[19]
A laboratory study of spilling breakers in the presence of light‐wind ...
Feb 24, 2016 · In clean water with wind speeds lower than 2.3 m/s (the minimum wind speed of wind-generated waves for clean water in our tank), the breaking of ...
[20]
Waves in Western Long Island Sound: A Fetch‐Limited Coastal Basin
Jan 20, 2021 · We demonstrate that the significant wave height in the western Sound is larger when the wind is from the east, along the long axis of the basin, and smaller ...
[21]
https://doi.org/10.1016/S0378-3839(98)00011-8
[22]
10.2 Waves at Sea – Introduction to Oceanography
As the energy of the wind increases, so does the size, length and speed of the resulting waves. There are three important factors determining how much energy is ...Missing: depth | Show results with:depth
[23]
Oceanography » Ocean Waves: surface and interface waves
Definition of a wave. A wave is a disturbance that propagates through a medium, such as space, or along an interface between two media.
[24]
Wayne Crawford: Infragravity Waves
Infragravity waves are low frequency ocean surface gravity waves that are generated by non-linear interactions between ocean surface wind waves.
[25]
Wave Types - Institute for Water Resources
The growth of wind-generated oceanic waves is not indefinite. The point when waves stop growing is termed a fully developed sea condition.Missing: stages young authoritative
[26]
How are estimates of wind-seas and swell made from NDBC wave ...
Estimates of wind-sea and swell are made by selecting a separation frequency that partitions the wave spectrum into its wind-sea and swell parts.
[27]
3.5.4: Sea versus swell waves - Geosciences LibreTexts
Dec 19, 2021 · Swell waves are long, regular, and uni-directional, while sea waves are irregular, short-crested, and have a larger directional spread.
[28]
Rogue Waves - National Geographic Education
Oct 19, 2023 · A rogue wave is usually defined as a wave that is two times the significant wave height of the area. The significant wave height is the average ...Missing: >2x 300 authoritative
[29]
Study Finds Massive Rogue Waves Aren't as Rare as Previously ...
Mar 8, 2017 · Rogue waves are not rare as previously thought and occur roughly twice daily at any given location in a storm.
[30]
The Ocean Wave Directional Spectrum - The Oceanography Society
Oct 2, 2015 · The directional spectrum S(k) [or S(f,θ)] measures the distribution of wave energy in wave number, k, (or frequency, f) and direction.Missing: definition | Show results with:definition
[31]
Statistical description of wave parameters - Coastal Wiki
Oct 18, 2025 · The significant wave height, H_s, is the mean of the highest third of the waves; instead of H_s the notation H_{1/3} is also often used. H_s ...Missing: 0.0163 | Show results with:0.0163
[32]
Directional Spreading - an overview | ScienceDirect Topics
Wind generated waves are not uni-directional. Typical wind wave spectra have a directional spread about the wind direction.
[33]
On Evolution of Young Wind Waves in Time and Space - MDPI
The mechanisms governing the evolution of the wind-wave field in time and in space are not yet fully understood. Various theoretical approaches have been ...Missing: 20th | Show results with:20th
[34]
[PDF] Jonswap final report - Air-Sea Interaction Lab
Both of these experiments utilized a single mobile wave station to determine the evolution of the wave spectrum in space and time. Page 10. Hasselmann et al., ...
[35]
Reminiscences on the study of wind waves - PMC - PubMed Central
The wind blowing over sea surface generates tiny wind waves. They develop with time and space absorbing wind energy, and become huge wind waves usually referred ...
[36]
Wind waves: a review of research during the last twenty-five years
Summary. Surface gravity waves generated by the wind have been studied extensively over the last twenty-five years. Advances in the theories of wave genera.
[37]
[PDF] A Proposed Spectral Form for Fully Developed Wind Seas Based on ...
The data for the spectra of fully developed seas obtained by. Moskowitz (1963) for wind speeds from 20 to 40 knots as measured by anemometers on two weather ...
[38]
[PDF] A Proposed Spectral Form for Fully Developed Wind Seas Based on ...
tests applied by Moskowitz [1964] for the pre- cise determination of the spectrums over an im- portant range of frequencies for fully developed wind ...
[39]
[PDF] Measurements of Wind-Wave Growth and Swell Decay during the ...
The JONSWAP data is in general consistent with Kitaigorodskii's scaling hypothesis and confirms many of the spectral featui-es summarized by these authors:.
[40]
A unified directional spectrum for long and short wind‐driven waves
Jul 15, 1997 · The spectrum's directional spreading function is symmetric about the wind direction and has both wavenumber and wind speed dependence. A ...
[41]
[PDF] Directional Spectra of Wind-Generated Waves M. A. Donelan
Jun 30, 2007 · This paper attempts to provide a carefully documented description of the purely wind-generated wave directional spectrum derived from lake data ...
[42]
Validation of in situ wave spectrum with JONSWAP in the Indian ...
The buoy spectra were compared to the Joint North Sea Wave Project (JONSWAP) spectrum in deep- and shallow-water locations in the Bay of Bengal, and also the ...
[43]
A global evaluation of the JONSWAP spectra suitability on coastal ...
Dec 15, 2022 · The JONSWAP spectrum suitability is evaluated over the coastal regions worldwide. Results show a peak enhancement factor lower than the standard (ie 3.3) ...
[44]
Revisiting the Pierson–Moskowitz Asymptotic Limits for Fully ...
The time-honored topic of fully developed wind seas pioneered by Pierson and Moskowitz is revisited to review the asymptotic evolution limits of integral ...
[45]
Green's law and the evolution of solitary waves - AIP Publishing
Mar 1, 1991 · An exact solution to the linear shallow‐water wave equation is presented for solitary waves first evolving over constant depth and then over a sloping beach.
[46]
[PDF] Waves and Water Dynamics
Waves in which depth (d) is less than 1/20 of the wave- length (L/20) are called shallow-water waves, or long waves (Figure 8-6b). Shallow-water waves are said ...
[47]
[PDF] Linear Wave Theory - Organization of American States
A Shoaling Coefficient, Kd, can be calculated so that. H = Kd Ho. 5.58. From linear wave theory: Kd = [ tanh(2πd/L) ( 1 + {(4πd/L) /(sinh (4πd/L)}]. 1/2. 5.59.
[48]
Regional Swell Transformation by Backward Ray Tracing and ...
Feb 1, 2019 · Here, two numerically different linear methods are compared: backward ray-tracing and stationary linear SWAN simulations. The methods yield ...
[49]
Breaker index - Coastal Wiki
The breaker index concept has been introduced to estimate the location on the shoreface where wave breaking occurs based on empirical formulas.
[50]
Laboratory study of energy transformation characteristics in breaking ...
Oct 13, 2024 · However, as the number of waves increased, the rate of energy dissipation owning to wave breaking was approximately 20%. This indicates a ...
[51]
Correlation of Breaking Wave Characteristics with Energy Dissipation
Energy loss of about 21% is found to occur in a good plunging breaker. An attempt has been made to correlate the dissipation rate with one of the commonly used ...
[52]
Influence of Wind on Breaking Waves | Vol 116, No 6 - ASCE Library
The breaker location, geometry, and type were found to depend strongly upon the wind direction. Onshore winds cause waves to break earlier, to break in deeper ...
[53]
[PDF] Linear Wave Theory
At a given depth, waves of different frequencies (hence of different wavelengths) travel at different speeds. This phenomenon is called dispersion. For a given ...Missing: seminal paper
[54]
[PDF] Water Waves - MIT
Linearized (Airy) Wave Theory. Consider small amplitude waves: (small free ... Dispersion relationship (6) uniquely relates ω and k, given h: ω = ω(k;h) ...
[55]
[PDF] LINEAR WAVE THEORY Part A - NTNU
These notes give an elementary introduction to linear wave theory. Linear wave theory is the core theory of ocean surface waves used in ocean and coastal ...
[56]
[PDF] Introduction to Ocean Waves - University of California San Diego
Dec 7, 2015 · For deep water gravity waves, the phase velocity exceeds the group velocity, but for capillary waves the opposite is true (as you will see in a ...
[57]
The Dynamical Coupling of Wind-Waves and Atmospheric Turbulence
Oct 18, 2021 · Phillips (1957) and Miles (1957) proposed two mechanisms which aim at explaining the initial and later stage of wave growth, respectively. The ...2.1 Wave-Induced Motions · 2.2 Wave-Induced Stress · 4 Turbulence In The Wave...<|control11|><|separator|>
[58]
[PDF] Lecture 1
Wave packets move with the group velocity, and they transport energy at a rate ! F = E ! Cg where E(x,y,z,t) is the average energy density of the wave in ...Missing: C_g | Show results with:C_g
[59]
Energy balance of wind waves as a function of the bottom friction ...
Waves propagating in shallow water dissipate energy in a thin, turbulent boundary layer near the bottom. This friction can be estimated with a simple quadratic ...
[60]
A Simple Model for Random Wave Bottom Friction and Dissipation in
They used a quadratic drag law to relate the bottom shear stress to the wave bottom velocity and assumed a constant drag coefficient and zero phase shift ...
[61]
The Stokes drift and wave induced‐mass flux in the North Pacific
Aug 18, 2012 · [2] The Stokes drift is one of the manifestations of ocean surface waves, and impacts mass and momentum transports near the surface. An accurate ...
[62]
Characterizing wind, wave, and Stokes drift interactions in the upper ...
Feb 2, 2025 · Overall, these studies indicate that Stokes drift significantly influences upper ocean mixing, wind stress distribution, and material transport ...
[63]
Experimental Study on Wind-Wave Momentum Flux in Strongly ...
The main goal of the paper is to quantify the effect of wave shape and airflow sheltering on the momentum transfer and wave growth. Primary results are ...
[64]
A Global Climatology of Wind–Wave Interaction in - AMS Journals
Generally, ocean waves are thought to act as a drag on the surface wind so that momentum is transferred downward, from the atmosphere into the waves. Recent ...
[65]
Airy, G.B. (1845) Tides and Waves. B. Fellowes. - References
Oct 29, 2024 · The objective of this paper is to present a new method for designing absorbing or non-reflective boundary conditions (ABC) or (NRBC), illustrated by the case ...Missing: seminal theory
[66]
[PDF] 1 Brief History and Overview of Nonlinear Water Waves - Elsevier
Both Airy (1845) and Stokes. (1847) provided summaries of the theory of linear and nonlinear waves and tides. One of the most important contributions of the ...
[67]
(PDF) Wave Modeling --the State of the Art - ResearchGate
A comprehensive theory for. monochromatic linear and nonlinear wave propagation was presented Airy (1845) and Stokes (1847),with. nonlinear eﬀects speciﬁc ...
[68]
[PDF] Progress in ocean wave forecasting - ECMWF
Jun 19, 2007 · Progress in ocean wave forecasting is described in the context of the fundamental law for wave prediction: the energy balance equation.
[69]
Kinetic equations in a third-generation spectral wave model
Jan 22, 2021 · The Hasselmann kinetic equation (HKE) forms the cornerstone of present-day spectral wave models. It describes the redistribution of energy ...
[70]
The WAM Model—A Third Generation Ocean Wave Prediction ...
A third generation wave model is presented that integrates the basic transport equation describing the evolution of a two-dimensional ocean wave spectrum.
[71]
Small‐scale open ocean currents have large effects on wind wave ...
Apr 13, 2017 · The wave model includes all four direct effects of current presented above, namely refraction, wave action conservation with a change in wave ...2 Wave Heights Over The Gulf... · 2.1 Current And Wave Model... · 3 Wave Heights In Drake...<|separator|>
[72]
Comparison of ice and wind-wave modules in WAVEWATCH III® in ...
The global wave model WAVEWATCH III® provides several modules to include the wind and ice effects. In this study, we first use data from a set of Inertial ...
[73]
[PDF] i'HE THEORY AND APPLICATIONS 3F OCEAN WAVE MEASURING ...
I subtitle of this report could be ?The Study of Ocean Waves: l3zom spark Plugs and Ferris Wheels to Spacecraft". Nearly all of the know.
[74]
Significant wave heights from Sentinel‐1 SAR: Validation and ...
Dec 26, 2016 · Two empirical algorithms are developed to estimate integral wave parameters from high resolution synthetic aperture radar (SAR) ocean images.
[75]
Directional Waverider MkIII - Datawell
### Summary of Datawell Directional Waverider MkIII
[76]
Comparative Wave Measurements at a Wave Energy Site with a ...
Options for commonly utilized in situ measurements include wave buoys, echo sounders, current meters, and pressure transducers (Holthuijsen 2007).
[77]
Jason Satellite Products
Jason satellites collect data on ocean circulation, sea level rise, and wave height, used for climate monitoring, oceanography, and forecasting. NCEI provides ...Missing: remote sensing
[78]
[PDF] Joint analysis of coastal altimetry and high-frequency (HF) radar data
In this study, a HF radar system and two Jason-2 satellite altimetry ... HFRs are remote-sensing instruments that send radio waves to the ocean ...