Wave Mechanics and Hydrodynamics - Theory & Concepts

Wave Mechanics and Hydrodynamics

The fundamental principles of wave generation, propagation, and transformation in coastal environments.

Wave Generation and Spectra

How ocean waves are formed by wind and mathematically represented as random sea states.

Wave Generation Mechanisms

Ocean waves are primarily generated by the transfer of energy from wind to the water surface. The characteristics of wind-generated waves depend on three key factors:

Wind Speed ( $U$ ): The velocity of the wind blowing over the water surface. Higher wind speeds create larger waves.
Wind Duration ( $t$ ): The length of time the wind has been blowing at a constant speed and direction.
Fetch ( $F$ ): The uninterrupted straight-line distance over water that the wind blows. A long fetch allows waves to build up significant energy.

When waves are actively being generated by wind, the sea state is called a "sea." When waves propagate away from their generation area and become more regular and organized, they are called "swell."

Wave Spectra and Random Waves

Real ocean waves are irregular and random, composed of many individual wave components with different heights, periods, and directions. A wave spectrum describes the distribution of wave energy over different frequencies.

Significant Wave Height ( $H_s$ or $H_{1/3}$ ): A statistical parameter defined as the average height of the highest one-third of waves in a record. It is widely used for structural design.
Peak Period ( $T_p$ ): The wave period associated with the maximum energy density in the wave spectrum.
Pierson-Moskowitz Spectrum: A mathematical model for a fully developed sea, where wind has blown over a long fetch for a long duration, reaching an equilibrium state.
JONSWAP Spectrum (Joint North Sea Wave Project): An extension of the Pierson-Moskowitz spectrum for fetch-limited seas, characterizing developing sea states common in coastal regions.

Wave Data Analysis and Forecasting

Methods used to predict wave conditions from wind data and statistically analyze recorded wave fields.

Wave Forecasting (SMB Method)

The Sverdrup-Munk-Bretschneider (SMB) method is a classic, semi-empirical approach used to predict significant wave height (

H_s

) and peak period (

T_p

) based on wind characteristics. It uses nomograms or empirical formulas derived from extensive field data.

The method inputs the three key wave generation factors: Wind Speed ( $U$ ), Fetch ( $F$ ), and Wind Duration ( $t$ ).
It determines whether a sea state is "fetch-limited" (the fetch is too short for the wind to generate the maximum possible waves) or "duration-limited" (the wind hasn't blown long enough). The limiting factor governs the final forecasted wave height and period.

Wave Data Analysis (Zero-Crossing Method)

When analyzing raw wave surface elevation data collected from a wave buoy, engineers must extract statistical parameters like

H_s

Zero-Upcrossing Method: A standard statistical technique where a single wave is defined as the portion of the record between two consecutive points where the water surface crosses the mean water level moving upwards.
The wave height is the vertical distance between the highest crest and the lowest trough between these two zero-crossings.
By isolating all individual waves in a record using this method, they can be ranked by height to calculate $H_{1/3}$ (significant wave height) or other statistical values like $H_{1/10}$ (average of the highest 10% of waves).

Linear Wave Theory

Also known as Airy Wave Theory, it is the fundamental mathematical description of ocean waves.

Wave Parameters

Understanding the basic components of a wave is crucial for coastal design.

Wave Height ( $H$ ): The vertical distance between the crest and the trough.
Wavelength ( $L$ ): The horizontal distance between two consecutive crests or troughs.
Wave Period ( $T$ ): The time it takes for two consecutive crests to pass a fixed point.
Water Depth ( $d$ ): The distance from the still water level to the seabed.

Wave Energy and Power

The energy contained within a wave is a critical parameter for evaluating coastal impact and designing structures. The total energy (

E

) of a wave is the sum of its kinetic energy (due to water particle motion) and potential energy (due to the elevation of the water surface). For linear waves, it is proportional to the square of the wave height.

Total wave energy per unit surface area

Wave Power (Energy Flux)

Wave Power (

P

) represents the rate at which wave energy is transmitted in the direction of wave propagation. It is a key metric for coastal processes like sediment transport and for determining the energy available for wave energy converters.

Wave power per unit crest length

Wave Celerity

Wave Celerity (

C

) is the phase speed of the wave.

Wave Celerity

Wave Dispersion Relationship

The dispersion relationship connects the wavelength, wave period, and water depth.

Dispersion relationship for linear waves

Deep and Shallow Water Approximations

Based on the dispersion relationship, simplifications can be made:

Deep Water: When $\frac{d}{L} > \frac{1}{2}$ , $\tanh\left(\frac{2\pi d}{L}\right) \approx 1$ . Thus, $L_0 = \frac{g T^2}{2\pi}$ .
Shallow Water: When $\frac{d}{L} < \frac{1}{20}$ , $\tanh\left(\frac{2\pi d}{L}\right) \approx \frac{2\pi d}{L}$ . Thus, $C = \sqrt{gd}$ .

Non-Linear (Higher-Order) Wave Theories

While Airy Wave Theory is mathematically simple, it assumes wave height is infinitesimally small compared to wavelength and depth. In reality, as waves move into shallower water, they become steeper, and their crests become sharper while their troughs become flatter.

Stokes Wave Theory (2nd to 5th Order): Used for steeper waves in deep to transitional water depths. It mathematically accounts for the sharpening of crests and flattening of troughs by adding higher-order mathematical terms to the linear solution.
Cnoidal Wave Theory: Used for relatively long waves in shallow water where Stokes theory breaks down.
Solitary Wave Theory: Models a single wave crest moving entirely above the still water level, often used to approximate waves at the exact point of breaking or tsunamis in shallow coastal areas.

Wave Transformation and Breaking

Changes in wave characteristics as waves move from deep to shallow water, and the physics of wave breaking.

Shoaling, Refraction, Diffraction, and Reflection

Waves undergo physical changes when interacting with the seabed and obstacles.

Shoaling: The change in wave height as waves move into shallower water, slowing down and becoming steeper. Calculated using the shoaling coefficient ( $K_s$ ).
Refraction: The bending of wave crests to align parallel to the underwater depth contours. Calculated using the refraction coefficient ( $K_r$ ).
Diffraction: The lateral transfer of wave energy along a wave crest when waves encounter an obstacle like a breakwater. Calculated using the diffraction coefficient ( $K_d$ ). Engineers often use standardized Diffraction Diagrams (e.g., Penny and Price curves) to estimate the attenuated wave height in the "shadow zone" or lee of the breakwater.
Reflection: The bouncing back of wave energy when a wave strikes an obstacle, such as a seawall or steep breakwater. Reflected waves can combine with incoming waves to form standing waves (clapotis), significantly increasing the total wave height and resulting structural loads. The Reflection Coefficient ( $K_{ref}$ ) is the ratio of reflected wave height to incident wave height.
Wave Setup and Setdown: As waves break and transfer their momentum to the water column, they cause localized changes in the mean water level. Wave setdown is a slight decrease in water level just seaward of the breaker zone, while wave setup is a significant increase in water level at the shoreline, acting additively with storm surge.

General wave transformation equation

Wave Breaking Types

As a wave moves into shallower water, it steepens until it becomes unstable and breaks. The type of breaking depends primarily on the seabed slope and the initial wave steepness.

Spilling Breakers: Occur on very gentle, flat beaches. The wave crest spills down the front face, dissipating energy gradually over a long distance.
Plunging Breakers: Occur on moderately steep slopes. The crest curls over and plunges downwards with significant force, forming a characteristic tube. They cause the most intense impact forces on coastal structures.
Surging Breakers: Occur on very steep slopes (e.g., seawalls or cliff faces). The wave surges up the face without fully breaking or curling over.
Collapsing Breakers: A transitional state between plunging and surging breakers where the lower part of the wave front collapses.

A common criterion for depth-limited wave breaking is when the wave height reaches approximately 78% of the water depth (

H_b \approx 0.78 d_b

Wave Forces on Structures

Determining the hydrodynamic loads exerted by waves on coastal infrastructure.

The Morison Equation

When a wave hits a slender structural member, like a vertical steel or concrete pile supporting a wharf or dolphin, it exerts a horizontal force. If the diameter of the pile (

D

) is small compared to the wavelength (

L

) (typically

D/L < 0.2

), the pile does not significantly reflect or diffract the wave. In this regime, the Morison Equation is used to calculate the total inline wave force (

F_T

) per unit length of the pile.

The total force is the sum of two components:

Drag Force ( $F_D$ ): The force exerted by the water particles flowing past the cylinder, proportional to the square of the horizontal particle velocity ( $u$ ).
Inertia Force ( $F_I$ ): The force required to accelerate the mass of water surrounding the cylinder as the wave passes, proportional to the horizontal particle acceleration ( $\frac{du}{dt}$ ).

Morison Equation for total inline wave force per unit length

Application of the Morison Equation

The Morison Equation is highly dynamic because both particle velocity (

u

) and acceleration (

\frac{du}{dt}

) change continuously as the wave crest and trough pass the pile, and they vary with depth (

z

) from the surface down to the seabed.

Velocity ( $u$ ) is maximum under the wave crest (creating maximum drag force).
Acceleration ( $\frac{du}{dt}$ ) is maximum where the water surface crosses the still water level (creating maximum inertia force).
Because drag and inertia peak at different times in the wave cycle, the maximum total force ( $F_T$ ) must be found by evaluating the equation over the entire wave period.
To find the total force on the entire pile, $F_T$ must be integrated from the seabed up to the instantaneous water surface.

Key Takeaways

Wave generation is governed by wind speed, duration, and fetch, and can be forecasted using the semi-empirical SMB Method. Real ocean waves are modeled using spectra like JONSWAP or analyzed statistically using the Zero-Upcrossing Method.
Wave energy ( $E$ ) is proportional to the square of the wave height ( $H^2$ ), and Wave Power ( $P$ ) is the rate of energy transport governed by the group celerity ( $C_g$ ).
Airy Wave Theory provides the linear foundation for determining wave celerity, length, and particle kinematics, while Higher-Order Theories (e.g., Stokes, Solitary) account for steepening waves in transitional and shallow waters.
In shallow water, wave celerity depends exclusively on water depth ( $C = \sqrt{gd}$ ).
As waves approach the shore, they undergo shoaling (height changes), refraction (bending due to depth), diffraction (bending around obstacles—often analyzed via Diffraction Diagrams), and reflection (bouncing off structures). Breaking waves also cause localized changes in mean water level, known as Wave Setup and Setdown.
Wave breaking occurs when waves become too steep, and the breaking type (spilling, plunging, surging) strongly influences structural design forces.
The Morison Equation calculates horizontal wave forces on slender marine structures (like piles) by summing dynamic drag and inertial forces.

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