Wave Equations

Collection of equations used to calculate wave quantities / descriptors

Wave Action Equation

$\frac{d}{dt}\left(\frac{E}{f}\right)+c_g\cdot \nabla \left(\frac{E}{f}\right)=S_{wind}-S_{brk}+S_{nl}$

Basic Equations

Phase speed: $c_p=\frac{g}{2\pi f_p}$ (deep water)

Wave age: $\frac{c_p}{U_{10}}$ Interpretation: lower wave age indicates younger, more actively forced wind sea; larger wave age indicates more mature waves or swell influence.

Deep water dispersion relation: $k=\frac{(2\pi f{)}^2}{g}$

Spectral Moments

Weighted by Frequency

Generalized: $m_n=\int \int f^{n}E(f,\theta )dfd\theta$

Zeroth moment: $m_0=\int \int E(f,\theta )dfd\theta$ - total surface elevation variance - Significant wave height: $H_s=4\sqrt{m_0}$

Weighted by Wave Number

Generalized: $M_p=\int k^{p}E(k)dk$ - This describes the geometry of the surface itself - sea surface $\eta (x)$ variance is $⟨{\eta }^2⟩$ = $\int E(K)dk$ is analogous to $m_0$

Surface slope: $\frac{d\eta }{dx}$ - a derivative multiplies a Fourier component by $ik$

Slope variance: $⟨{\left(\frac{d\eta }{dx}\right)}^2⟩=\int k^{2}E(k)dk$ This is mean square slope: $\text{MSS}=\int k^{2}E(k)dk$ = $\int \frac{(2\pi f{)}^{4}E(f)}{g^2}df$

Surface curvature: $\frac{d^2\eta }{d{x}^2}$
- curvature variance becomes: $\int k^{4}E(k)dk$

Friction Velocity

From the spectra $u_{\ast ,tail}=\frac{2{\pi }^3{A}_{tail}}{\beta I(p)g}$ where $I(p)={\int }_{-\pi /2}^{\pi /2}{\cos }^p\theta d\theta$ and p values range (1.9->3.1), $\beta =0.012$

Bulk parameterization $u_{\ast ,bulk}=\sqrt{C_D}{U}_{10}$