Mathematical models for the stochastic statistics of ocean surface waves derived from first principles in Mathematical Geoenergy (GEM), Chapter 12 (Section 12.3 — Stochastic Aquatic Waves).
Note: Chapter 12 also covers deterministic tidal forcing (Section 12.1) and the El Niño/Southern Oscillation — ENSO (Section 12.2). The ENSO material is hosted in a separate repository: pukpr/GEM-LTE. The present subdirectory focuses exclusively on the stochastic aquatic waves content of GEM Ch. 12, Section 12.3.
Wind-driven ocean waves are the archetypal example of a maximum-entropy (MaxEnt) dispersion problem in an aquatic medium. The derivation in GEM Ch. 12.3 follows the same unifying thread as wind energy (Ch. 11) and earthquake magnitudes (Ch. 13):
Eq. 12-25 — MaxEnt CCDF for wave height
\[P(H) = e^{-aH^2}\]Applying the Principle of Maximum Entropy to the wave energy $E \propto H^2$ with only the mean energy $\bar{E} = 1/a$ known yields an exponential energy distribution; the change of variables $E = aH^2$ produces this Rayleigh-type complementary CDF.
PDF (derivative of Eq. 12-25)
\[p(H) = -\frac{d}{dH}P(H) = 2aH\cdot e^{-aH^2}\]The scale parameter $a = 8/H_s^2$ where $H_s = 4\sigma_H$ is the significant wave height (the mean of the highest third of waves).
Eq. 12-26 — Newton’s-law time-to-fall relation
\[t^2 \sim H\]A wave crest of height $H$ falls under gravity in time $t \sim \sqrt{H}$. Because wave period $T \sim t$ and frequency $f \sim 1/t$:
Eq. 12-28 — Height–frequency substitution
\[H \sim \frac{1}{f^2}, \qquad \frac{dH}{df} \sim -\frac{1}{f^3}\]Eq. 12-27 — Change-of-variables rule
\[p(f)\\,df = \left|\frac{dP}{dH}\right| \cdot \left|\frac{dH}{df}\right| df\]Eq. 12-29 — Pierson-Moskowitz wave-energy spectral density
\[p(f) \\;\propto\\; \frac{e^{-c/f^4}}{f^5}\]The constant $c = \alpha_{\mathrm{PM}}\left(\tfrac{g}{2\pi U}\right)^{4}$ where $\alpha_{\mathrm{PM}} \approx 8.1 \times 10^{-3}$, $g$ is gravitational acceleration, and $U$ is the wind speed at 19.5 m height.
The peak frequency is:
\[f_p = \left(\frac{4c}{5}\right)^{1/4}\]Significant wave height scales empirically as $H_s \approx 0.22\cdot U^2/g$.
Eq. 12-30 — Shifted-exponential wave-spacing CCDF
\[P(L) = \begin{cases} 0 & L < L_0 \\ e^{-\alpha(L-L_0)} & L \geq L_0 \end{cases}\]$L_0$ is the minimum inter-crest spacing and $1/\alpha$ is the mean excess spacing. The parameter ratio $L_0/\langle L\rangle$ measures coherence: $L_0 \to \langle L\rangle$ gives strongly ordered swell; $L_0 \to 0$ gives a fully random (Markov) sea state.
Eq. 12-31 — Fourier transform of spacing PDF
\[\tilde{P}(S_x) = \frac{e^{-iS_x L_0}}{1 + iS_x/\alpha}\]Eq. 12-32 — Semi-Markov power spectral density (two forms)
Form 1 (odd-harmonic — symmetric up/down steps):
\[I(S_x) = \frac{1}{\bigl(S_x - \alpha\sin(S_x L_0)\bigr)^2 + \alpha^2\bigl(1 + \cos(S_x L_0)\bigr)^2}\]Form 2 (all-harmonic — saw-tooth steps):
\[I(S_x) = \frac{1}{\bigl(S_x + \alpha\sin(S_x L_0)\bigr)^2 + \alpha^2\bigl(1 - \cos(S_x L_0)\bigr)^2} - \frac{1}{S_x^2(1 + \alpha L_0)^2}\]As $L_0 \to 0$ both forms recover the Markov (white-noise) flat spectrum.
Eq. 12-33 — Mean wavelength
\[\langle L \rangle = L_0 + \frac{1}{\alpha}\]| File | Purpose |
|---|---|
waves_symbolic.py |
Symbolic derivation of all equations (Eqs. 12-25 – 12-33) using SymPy |
waves_numerical.py |
Numerical implementation, validation, and composite figure (8 panels) |
waves_model_output.png |
Output figure generated by waves_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python waves_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python waves_numerical.py
Wave height is Rayleigh-distributed for the same reason wind speed is. Both obey the substitution $E \propto (\text{amplitude})^2$ under MaxEnt, making the Rayleigh distribution a universal result for squared-amplitude physical quantities.
The Pierson-Moskowitz spectrum is a direct consequence of MaxEnt. No empirical curve-fitting is required: the $f^{-5}$ power-law and the $e^{-c/f^4}$ cut-off both follow from the Rayleigh height distribution via the Newton’s-law change of variables $H \sim 1/f^2$.
Coherent and incoherent sea states are two extremes of one model. The shifted-exponential spacing distribution (Eq. 12-30) with parameter $L_0$ smoothly interpolates from a flat Markov spectrum ($L_0 = 0$) to a richly harmonic swell spectrum ($L_0 \to \langle L\rangle$), matching both open-ocean chop and laboratory wave-tank measurements.
Significant wave height scales as $H_s \propto U^2$. This quadratic wind-speed dependence follows from the PM coefficient $c \propto (g/U)^4$ and the peak-frequency scaling $f_p \propto g/U$, giving $H_s \approx 0.22\cdot U^2/g$ — a widely used engineering estimate.
Spatial and temporal spectra are related by the dispersion relation. Because wave phase velocity connects spatial wavenumber $S_x$ to temporal frequency $f$, the semi-Markov spatial PSD (Eq. 12-32) can be interpreted in either the spatial or temporal domain, unifying the wave-height and wave-spacing statistics within a single framework.