10.3 Gravity waves

Properties of gravity waves

Gravity waves are surface waves where gravity acts as the restoring force pulling displaced water back toward equilibrium. They're the waves you see on oceans, lakes, and rivers, and they govern everything from coastal erosion to ship design. This section covers their core physical properties, how they're generated and propagate, and the math and measurement techniques used to describe them.

Wavelength and wave height

Wavelength is the horizontal distance between two consecutive crests (or troughs). Wave height is the vertical distance from trough to crest. Together, these two parameters define the basic geometry of a wave.

• Wave height is controlled by three wind factors: speed, duration, and fetch (the uninterrupted distance over water that the wind blows).
• Greater wind speed, longer duration, and longer fetch all produce larger waves.
• Wavelength determines the wave's period (the time between successive crests passing a fixed point). Longer wavelengths have longer periods.

Dispersion relation

The dispersion relation connects a wave's frequency to its wavelength and the water depth. It's the single most important equation for gravity waves:

$\omega^2 = gk\tanh(kh)$

where $\omega$ is angular frequency, $g$ is gravitational acceleration, $k = 2\pi/\lambda$ is the wavenumber, and $h$ is water depth.

This equation tells you how fast waves of different wavelengths travel:

• Deep water ($h \gg \lambda$): $\tanh(kh) \to 1$, so $\omega^2 \approx gk$. Longer wavelengths travel faster, which is why swell from a distant storm arrives with long-period waves first.
• Shallow water ($h \ll \lambda$): $\tanh(kh) \approx kh$, so $\omega^2 \approx gk^2h$, giving a phase speed of $c = \sqrt{gh}$. Speed depends only on depth, not wavelength, so shallow-water waves are non-dispersive.

Phase and group velocity

These two velocities describe different things, and confusing them is a common mistake.

• Phase velocity ($c_p$) is the speed of an individual crest. From the dispersion relation: $c_p = \omega / k$.
• Group velocity ($c_g$) is the speed at which a packet of waves travels. This is the speed at which energy moves through the water.

The relationship between them shifts with depth:

• In deep water: $c_g = \frac{1}{2}c_p$. Energy travels at half the speed of individual crests.
• In shallow water: $c_g \approx c_p$. Energy and crests travel together.

Generation of gravity waves

Wind-wave interaction

Wind is the primary energy source for ocean surface waves. The transfer happens through two mechanisms working together:

1. Wind exerts shear stress on the water surface, dragging it along.
2. Pressure fluctuations in the turbulent airflow push down on the water unevenly, creating small disturbances.

Once small ripples form, they present a rougher surface to the wind, which increases the energy transfer. Waves grow as wind speed, fetch, and duration increase. A fully developed sea occurs when energy input from the wind balances energy lost to dissipation and wave-wave interactions.

Resonance and feedback mechanisms

Two classical theories explain wave generation:

• Phillips' mechanism describes the initial stage. Random turbulent pressure fluctuations in the air create small surface disturbances. These grow linearly with time.
• Miles' mechanism takes over once waves have some amplitude. When the wind speed at a critical height matches the phase speed of a wave component, resonant coupling occurs. The wave extracts energy from the mean airflow efficiently, and growth becomes exponential.

These mechanisms work in sequence: Phillips gets waves started, Miles amplifies them.

Propagation of gravity waves

Deep vs. shallow water waves

The ratio of water depth to wavelength determines the wave regime:

In deep water, orbital motion is negligible below about half a wavelength. In shallow water, the bottom constrains vertical motion, flattening the orbits into ellipses that extend all the way to the seabed.

Wave refraction and diffraction

Refraction occurs when waves approach shallower water at an angle. The part of the wave in shallower water slows down while the deeper portion keeps moving faster, causing the wave crest to bend toward alignment with the depth contours (bathymetric lines).

• Refraction can converge wave energy on headlands (increasing wave height) or diverge it in bays (decreasing wave height).

Diffraction occurs when waves encounter an obstacle (like a breakwater) or pass through a gap. Wave energy spreads laterally into the sheltered region behind the obstacle. This is why harbors still experience some wave activity even behind protective structures.

Energy transport and dissipation

Waves carry energy across the ocean. The energy flux (power per unit crest length) is:

$P = E \cdot c_g$

where $E$ is the wave energy per unit area (proportional to $H^2$, the square of wave height) and $c_g$ is the group velocity.

Energy is lost through several mechanisms:

• Whitecapping: breaking at the crests of steep wind waves in deep water
• Bottom friction: significant in shallow water, especially over rough or soft seabeds
• Wave breaking: the dominant dissipation mechanism near shore

These losses cause wave height to decrease gradually over distance.

Breaking of gravity waves

Wave steepness and instability

Wave steepness is the ratio $H/\lambda$ (wave height to wavelength). As waves shoal or interact, their steepness increases. A wave becomes unstable and breaks when the particle velocity at the crest exceeds the phase velocity of the wave itself.

The theoretical maximum steepness for a deep-water wave is approximately $H/\lambda = 1/7$ (about 0.14). Beyond this, the crest can no longer maintain its shape.

Types of breaking waves

The type of breaker depends on the beach slope and the wave's offshore steepness. The Iribarren number (ratio of beach slope to wave steepness) is often used to classify them:

• Spilling breakers: Occur on gentle slopes. Foam and turbulence cascade down the front face gradually. Most of the energy dissipation is spread over a wide surf zone.
• Plunging breakers: Occur on moderate slopes. The crest curls over and plunges forward, forming the classic "barrel" shape. Energy dissipation is intense and concentrated.
• Surging breakers: Occur on steep slopes. The wave front stays relatively intact and surges up the beach face with minimal breaking. Most energy is reflected back offshore.

Energy dissipation in breaking

Wave breaking converts organized wave energy into turbulent kinetic energy and ultimately heat. Dissipation rates in breaking waves can be several orders of magnitude higher than in non-breaking waves.

• Turbulence and air entrainment are concentrated near the surface.
• Breaking limits wave growth, maintaining an equilibrium between wind input and dissipation.
• In the surf zone, breaking drives important nearshore processes: longshore currents, sediment transport, and beach morphology changes.

Interaction of gravity waves

Wave-wave interactions

Waves don't just pass through each other cleanly. Nonlinear interactions transfer energy between wave components:

• Triad interactions involve three wave components and dominate in shallow water. They're responsible for the generation of harmonics and infragravity waves.
• Quadruplet (four-wave) interactions dominate in deep water. They redistribute energy within the spectrum, shifting it toward lower frequencies and higher frequencies simultaneously. This is the primary mechanism that shapes the deep-water wave spectrum.

For resonant interactions to occur, the frequencies and wavenumbers of the participating waves must satisfy specific matching conditions.

Wave-current interactions

Currents modify wave behavior in significant ways:

• An opposing current compresses wavelengths and increases wave steepness and height. In extreme cases, this can cause wave blocking, where waves cannot propagate against a sufficiently strong current.
• A following current stretches wavelengths and reduces steepness.
• Currents also refract waves by altering their propagation speed differently across the wave crest.

These interactions matter most at tidal inlets, river mouths, and regions with strong ocean currents. They can contribute to the formation of abnormally large rogue waves.

Wave-bottom interactions

As waves move into shallower water:

1. Bottom friction removes energy, reducing wave height.
2. Shoaling changes wave height (initially decreasing, then increasing as depth decreases further).
3. Refraction bends wave crests to align with bottom contours.
4. In very shallow water, wave orbital velocities reach the seabed, mobilizing sediment and driving morphological change (erosion, accretion, sandbar formation).

Mathematical description of gravity waves

Linear wave theory

Also called Airy wave theory, this is the foundation for most gravity wave analysis. It assumes:

• Small wave amplitude relative to wavelength and depth
• Irrotational, inviscid, incompressible flow
• A flat, impermeable bottom

The velocity potential satisfies the Laplace equation ($\nabla^2 \phi = 0$) with kinematic and dynamic boundary conditions at the free surface and a no-flow condition at the bottom. Solutions give analytical expressions for surface elevation, velocity field, pressure, and energy.

Linear theory works well for small-steepness waves in deep to intermediate water. It breaks down for steep waves, shallow-water waves, and near-breaking conditions.

Nonlinear wave theories

When linear assumptions fail, several nonlinear theories provide better accuracy:

• Stokes wave theory: A perturbation expansion in powers of wave steepness. Higher-order Stokes theories (2nd, 3rd, 5th order) capture asymmetric crest shapes and the net forward mass transport (Stokes drift). Best suited for deep to intermediate water.
• Cnoidal wave theory: Describes periodic waves in shallow water with sharp crests and flat, broad troughs. Based on the KdV (Korteweg-de Vries) equation.
• Solitary wave theory: A limiting case of cnoidal theory describing a single hump of water propagating without change of form in shallow water. No trough exists.

Numerical modeling of waves

For realistic ocean conditions, analytical solutions aren't sufficient. Numerical models fall into three broad categories:

• Spectral models (WAM, SWAN, WaveWatch III): Solve the wave action balance equation for the evolution of the energy spectrum. They handle generation, propagation, and dissipation over large domains but don't resolve individual wave crests.
• Phase-resolving models (Boussinesq, mild-slope equation models): Solve for the actual surface elevation and velocity fields. They capture refraction, diffraction, and nonlinear shoaling in nearshore regions but are computationally expensive over large areas.
• CFD models: Solve the full Navier-Stokes equations. These capture detailed wave-structure interactions, breaking, and turbulence but are limited to small domains due to computational cost.

Measurement of gravity waves

In-situ wave measurements

• Wave buoys are the standard instrument. They measure surface elevation using accelerometers or GPS. Directional buoys also resolve the direction of wave propagation.
• Pressure sensors on the seafloor record wave-induced pressure fluctuations, from which surface elevation can be inferred (with depth-dependent attenuation corrections).
• Acoustic Doppler Current Profilers (ADCPs) measure orbital velocities throughout the water column and can derive wave parameters from velocity spectra.

Remote sensing of waves

• Satellite altimeters (e.g., Jason, Sentinel-3) measure significant wave height along narrow ground tracks with global coverage.
• Synthetic Aperture Radar (SAR) images wave patterns over large areas, providing wavelength, direction, and (with more processing) wave height.
• High-frequency (HF) radar measures surface currents and wave parameters in coastal zones, typically out to 100-200 km.
• Stereo video systems capture detailed wave fields and breaking statistics in the surf zone.

Remote sensing provides spatial coverage that point measurements (buoys) cannot, but typically at lower temporal resolution.

Wave spectra and statistics

Raw wave measurements (time series of surface elevation) are transformed into frequency spectra using Fourier analysis. The spectrum $S(f)$ shows how wave energy is distributed across frequencies.

Key statistical parameters derived from spectra:

• Significant wave height ($H_s$): Defined as $4\sqrt{m_0}$, where $m_0$ is the zeroth moment (total area) of the spectrum. It approximates the average height of the highest one-third of waves.
• Peak period ($T_p$): The period corresponding to the spectral peak (the frequency with the most energy).
• Mean direction: The energy-weighted average direction of wave propagation.

Directional spectra $S(f, \theta)$ separate wind sea (locally generated, broad-banded) from swell (remotely generated, narrow-banded), which is critical for forecasting and engineering design.

Applications of gravity waves

Ocean wave forecasting

Operational forecasting systems run spectral wave models (WAM, WaveWatch III) forced by wind fields from numerical weather prediction models. These systems:

1. Produce forecasts out to about 10 days for maritime safety, offshore operations, and coastal hazard warnings.
2. Assimilate satellite altimeter and buoy data to correct model errors.
3. Generate wave climatologies and extreme value statistics for long-term planning.

Accurate wave forecasts are essential for ship routing, search and rescue, and coastal flood warnings.

Wave energy conversion

Ocean waves carry substantial energy. A typical Atlantic swell might deliver 30-70 kW per meter of wave crest. Wave energy converters (WECs) aim to capture this energy, and the main device types include:

• Oscillating water columns (OWCs): Waves push air through a turbine inside a partially submerged chamber.
• Point absorbers: Floating buoys that move with the waves, driving a power take-off system.
• Overtopping devices: Waves run up a ramp into a reservoir; water flows back to the sea through turbines.

The main engineering challenges are efficiency across a range of sea states, survivability in extreme storms, and minimizing environmental impact.

Coastal engineering considerations

Waves drive sediment transport, erosion, and coastal flooding. Engineers must account for wave forces when designing:

• Breakwaters and seawalls to protect shorelines and harbors
• Beach nourishment projects that add sand to eroding beaches
• Nature-based solutions like artificial reefs or restored wetlands that attenuate wave energy

Design requires knowledge of the local wave climate, including extreme conditions (e.g., the 100-year return period wave height). Wave-structure interaction analysis ensures that ports, harbors, and offshore platforms can withstand the forces they'll encounter.