Unit
- Wind wave generation: Waves form when wind transfers energy to the water surface. Key factors include wind speed, fetch (distance over water), and duration. For a given fetch, longer durations produce higher waves. The JONSWAP spectrum describes developing seas, while the Pierson-Moskowitz model represents fully developed seas.
- Deep-water wave dispersion: In water deeper than half the wave length, wave celerity depends only on period. The dispersion relation is c = gT/(2pi). Longer period waves travel faster. This property causes wave groups to separate as they propagate from a storm.
- Shallow-water wave propagation: When water depth is less than 1/20 of the wave length, waves feel the bottom. Celerity reduces to sqrt(gd), making waves slow down and refract toward shallower regions. Wave height increases as energy concentrates, leading to breaking.
- Wave energy and power: Wave energy density per unit crest width is proportional to the square of height. Energy flux (power) is the product of energy density and group velocity. Deep-water wave power is significant: a 2 m, 8 s wave carries about 40 kW per meter of crest.
- Group velocity and wave trains: Wave energy travels at group velocity, which in deep water is half the phase speed. This explains why individual waves appear to move through a group. In shallow water, group velocity equals phase speed.
- Wave spectra representation: Real seas consist of many superimposed wave components. A wave energy spectrum S(f) shows energy distribution across frequencies. Significant wave height Hs is derived from the spectrum as 4sqrt(m0), where m0 is the total variance.
- Wave forecasting methods: Operational wave models like WAVEWATCH III use wind fields to predict wave spectra globally. For local applications, simple SMB (Sverdrup-Munk-Bretschneider) methods estimate height and period from fetch, wind speed, and duration.
- Wave hindcasting and historical data: To design coastal defenses, engineers rely on hindcast wave data spanning decades. Reanalysis products (e.g., ERA5) provide hourly wave parameters at grid points, enabling extreme value analysis for return period estimation.
- Wave breaking and surf zone dynamics: As waves approach the shore, they break when wave steepness exceeds a critical limit (typically H/L greater than 1/7) or when depth becomes too shallow (H/d greater than 0.78). Breaking dissipates energy and drives nearshore circulation, including longshore currents and rip currents. Different breaker types (spilling, plunging, surging) affect coastal erosion and sediment transport.
- Wave setup and runup: Wave setup is the superelevation of mean water level due to radiation stress gradients in the surf zone, raising water levels against the shore. Wave runup is the maximum vertical extent of wave uprush on a beach or structure. Both are critical for designing coastal defenses and estimating flood risk during storms.
- Coastal sediment transport: Waves and currents mobilize and transport sediment along the coast. Longshore transport moves sand parallel to the shore, driven by oblique wave approach. The CERC formula estimates volumetric transport rate from wave energy flux and breaking angle. Understanding sediment budgets is essential for beach nourishment and inlet management.
Code Example
Python
import numpy as np
def wave_length(T, d, g=9.81):
# Linear wave theory: solve dispersion relation iteratively
L0 = g * T**2 / (2 * np.pi) # deep-water length
L = L0
for _ in range(10):
L = L0 * np.tanh(2 * np.pi * d / L)
return L
# Example: T=8 s, d=10 m
print('Wave length:', round(wave_length(8, 10), 1), 'm')This function solves the implicit dispersion equation for wave length given period and depth. It uses a fixed-point iteration with a deep-water initial guess. For typical coastal depths, convergence occurs in a few iterations.
Quick Revision
- Wind waves are generated by energy transfer from wind to water; key factors are wind speed, fetch, and duration.
- Deep-water wave speed depends only on period; shallow-water waves slow down and refract.
- Wave energy density scales with height squared; power is energy flux times group velocity.
- Wave spectra characterize real sea states; significant wave height is derived from spectral variance.
- Hindcast data from reanalysis products supports long-term wave climate analysis.