Ocean Wave Spectrum Simulator

The core of linear wave theory is the dispersion relation. It defines how the angular frequency (related to period) is linked to the wavenumber (related to wavelength) and the water depth.

Where $\omega = 2\pi/T$ is the angular frequency, $k=2\pi/\lambda$ is the wavenumber, $g$ is gravity, and $d$ is water depth. The $\tanh$ function is the mathematical key—it smoothly transitions the physics from deep to shallow water.

From the dispersion relation, we derive the phase velocity ($C_p$), the speed of individual wave crests. This equation shows the two important limits you can explore in the simulator.

Deep water ($d \gt \lambda/2$): $\tanh(kd) \approx 1$, so $C_p \approx \sqrt{g/k}$. Speed depends on wavelength.
Shallow water ($d \lt \lambda/20$): $\tanh(kd) \approx kd$, so $C_p \approx \sqrt{gd}$. Speed depends only on depth.

Coastal Engineering & Harbor Design: Engineers use these exact equations to model how waves will propagate into a harbor. They need to predict where waves will refract, slow down, and potentially cause damaging resonance inside the harbor, which directly affects the placement of breakwaters.

Offshore Structure Design: Platforms for oil, gas, or wind energy must withstand wave forces. The particle orbits and wave energy equation $E = \frac{1}{8}\rho g H^2$ are used to calculate the total load on the structure's legs, which is critical for ensuring stability in storms.

Surf Forecasting: Forecast models use the dispersion relation to "back-track" waves seen by deep-water buoys to their source storm. Since long-period swell travels faster, its arrival time at a beach can be predicted days in advance, which is essential for surfers and maritime safety.

Sediment Transport & Beach Erosion: The shift from circular to elliptical particle orbits in shallow water is what moves sand along the seabed. Understanding this helps coastal managers predict how beaches will change after storms or where to place sand nourishments to combat erosion.

When you start using this simulator, there are a few key points to keep in mind. First, the "Significant Wave Height (Hs)" is not the "Maximum Wave Height". Significant wave height is the average of the highest one-third of wave heights. For example, even if Hs is 3m, there can occasionally be much larger waves mixed in, exceeding 5m. In design, it's crucial not to overlook these "maximum waves". Next, "Peak Period (Tp)" and "Average Period" are different. Tp corresponds to the period at the peak of the spectrum and is close to the interval of the "swell" you see in the wave profile. However, if you calculate the actual zero-up-crossing period from a wave record, it's typically shorter than Tp. For instance, even if you set Tp=10 seconds, the average period will often be around 7-8 seconds. Finally, remember that simulation results are just "one possible realization". The irregular waves generated by Fourier synthesis will produce a different wave profile every time for the same Hs and Tp, due to the randomness of the phases. When evaluating the response of an actual structure, the golden rule is to account for this "variability" by running calculations with multiple different wave profiles (using different random seeds).