Everything this page computes, term by term — with the equation behind each and where it shows up on screen. Surface elevation is $\eta$, wavenumber $k$, angular frequency $\omega$, water depth $h$, gravity $g$, wave height $H$, phase speed $c$, and particle velocity components $u$ (horizontal) and $w$ (vertical), with $z$ measured from still water level down to the seabed at $z=-h$.

Linear (Airy) wave theory

Linear theory assumes the wave height is small compared to both the wavelength and the depth, which lets the free-surface boundary conditions be linearized:

$$\eta(x,t) = \frac{H}{2}\cos(kx - \omega t)$$

That's the wave drawn above when "Linear (Airy)" is selected.

The dispersion relation

Linear theory's boundary-value problem reduces to one equation linking frequency, wavenumber, and depth — the dispersion relation:

$$\omega^2 = gk\tanh(kh)$$

In deep water ($kh\gg1$, $\tanh\to1$) this simplifies to $\omega^2\approx gk$; in shallow water ($kh\ll1$, $\tanh(kh)\approx kh$) it becomes $\omega^2\approx ghk^2$. There's no closed form for $k(\omega)$ in between — this page solves it numerically with Newton–Raphson, every time the period or depth slider moves.

Particle orbits and depth attenuation

Linear potential-flow theory gives the horizontal and vertical particle velocity at any depth as:

$$u(x,z,t)=\frac{H\omega}{2}\cdot\frac{\cosh(k(z+h))}{\sinh(kh)}\cos\theta,\qquad w(x,z,t)=\frac{H\omega}{2}\cdot\frac{\sinh(k(z+h))}{\sinh(kh)}\sin\theta$$

with $\theta=kx-\omega t$. The depth-attenuation factor $\cosh(k(z+h))/\sinh(kh)$ is what the arrows above are showing: in deep water it collapses to $e^{kz}$ — orbits shrink exponentially within about one wavelength of the surface — while in shallow water ($kh\ll1$) it flattens out toward 1, so orbits barely shrink at all between the surface and the seabed. That contrast is exactly why the arrows under a deep-water theory (Linear, Stokes, Trochoidal) taper sharply, while the arrows under a shallow-water theory (Cnoidal, Solitary) stay almost the same length all the way down.

Beyond linear: Stokes 2nd- and 5th-order

Real waves aren't perfectly sinusoidal: crests are sharper and troughs are flatter than $\cos$ allows. Stokes expansion adds higher-frequency corrections on top of the linear term. The 2nd-order correction, at twice the wavenumber and frequency:

$$\eta_{\text{Stokes-2}}=\frac{H}{2}\cos\theta+\frac{\pi H}{8}\cdot\frac{H}{\lambda}\coth(kh)\left[3\coth^2(kh)-1\right]\cos(2\theta)$$

This page's "Stokes 5th-order" extends that pattern with three further harmonics ($3\theta$, $4\theta$, $5\theta$), each one's amplitude a fixed fraction of the one before it. This is a simplified, illustrative truncation, not the full closed-form Fenton (1985) fifth-order coefficient set — it produces a visibly sharper, more peaked crest than 2nd-order, in the right qualitative direction, but isn't engineering-grade. Velocities for both follow the same depth-attenuation pattern as linear theory, applied separately to each harmonic (each one decays with depth using its own wavenumber, $\cosh(nk(z+h))/\sinh(nkh)$).

Shallow water: cnoidal and solitary waves

In shallow water, nonlinearity reshapes a wave into tall, narrow crests separated by long, flat troughs — the cnoidal wave, built from the Jacobi elliptic function $\mathrm{cn}$:

$$\eta(x,t)=\eta_{\text{trough}}+H\,\mathrm{cn}^2\!\left(\frac{2K(m)}{\lambda}(x-ct),\,m\right)$$

where $K(m)$ is the complete elliptic integral of the first kind. This page maps the wave's Ursell number to $m$ with a simple monotonic approximation tuned for the right qualitative shape, not the full cnoidal/KdV dispersion closure — see the caveats below. The "Solitary" option pins $m$ close to its $m\to1$ limit: a single tall, isolated hump with long flat stretches on either side, the cnoidal wave's solitary-wave limit.

Both use the long-wave (Boussinesq) approximation for particle velocity, valid when $kh\ll1$: horizontal velocity is nearly uniform with depth, $u(x,z,t)\approx c\,\eta(x,t)/h$, with the small vertical velocity $w$ recovered from continuity. This is why the arrows under these two theories barely taper toward the seabed — the direct contrast with the deep-water theories above.

Trochoidal (Gerstner) waves

The Gerstner wave is an exact nonlinear solution for deep water, first found by Gerstner (1809). Each particle, labeled by its mean position $(x_0,z_0)$, traces a perfect circle:

$$x(x_0,t)=x_0-\frac{H}{2}e^{kz_0}\sin\varphi,\qquad z(x_0,t)=z_0+\frac{H}{2}e^{kz_0}\cos\varphi,\qquad\varphi=kx_0-\omega t$$

The free surface ($z_0=0$) traced by this family of circles is a trochoid — peaked crests, broad troughs, with no small-amplitude assumption needed. Differentiating gives the orbital velocity directly: $u=\frac{H\omega}{2}e^{kz_0}\cos\varphi$, $w=\frac{H\omega}{2}e^{kz_0}\sin\varphi$ — the same $e^{kz}$ deep-water decay as linear theory's limit, but exact rather than approximate. This page draws the Gerstner wave at whatever depth is set on the slider, even though the solution is strictly valid only for deep water; accuracy degrades as $kh$ drops into the shallow/intermediate regime.

Irregular seas and the JONSWAP spectrum

Every theory above is a regular sea: one wave height and period. Real seas are irregular — many frequencies superposed with random phases. The JONSWAP spectrum describes how much energy sits at each frequency $f$, peaking at $f_p=1/T_p$:

$$S_0(f)=\frac{\alpha g^2}{(2\pi)^4f^5}\exp\!\left[-1.25\left(\frac{f_p}{f}\right)^4\right]\gamma^{\,r(f)},\qquad r(f)=\exp\!\left[-\frac{(f-f_p)^2}{2\sigma^2f_p^2}\right]

with $\sigma=0.07$ below $f_p$ and $0.09$ above it. The peakedness $\gamma$ sharpens the spectrum around $f_p$ for a developing sea; $\gamma=1$ flattens it back into the Pierson-Moskowitz spectrum for a fully developed sea. Rather than transcribe the usual approximate formula relating $\alpha$ to $H_s$ and $T_p$, this page numerically integrates $S_0(f)$ and rescales it so the resulting significant height comes out exactly $H_s$ by construction — shown live in the corner of the panel as a cross-check.

The sea below is built from $N\!\approx\!56$ frequency bins, each a tiny linear wave with amplitude $a_i=\sqrt{2S(f_i)\Delta f}$, wavenumber $k_i$ solved from $\omega_i$ by the same dispersion solver used everywhere else on this page, and a fixed random phase $\varphi_i$:

$$\eta(x,t)=\sum_i a_i\cos(k_ix-\omega_it+\varphi_i)$$

"Randomize sea" redraws a fresh set of phases; moving the $H_s$/$T_p$/$\gamma$ sliders reuses the same phases so the sea reshapes smoothly instead of jumping. This is a single, one-directional snapshot — no directional spreading, a finite (not continuous) set of components, and phases that are pseudo-random rather than statistically faithful over long runs.

What's simplified

The dispersion solver and the Linear/Stokes-2nd-order/cnoidal elevation formulas are established results, each checked against a standard reference during implementation (Dean & Dalrymple, Water Wave Mechanics for Engineers and Scientists, for the Stokes terms). Stokes 5th-order here is a deliberately simplified harmonic truncation rather than the full Fenton (1985) coefficients. The cnoidal/solitary elliptic parameter $m$ is fit with an approximate Ursell-number mapping rather than solved from the full KdV dispersion relation. Cnoidal/solitary particle velocities use the leading-order long-wave (Boussinesq) approximation, not exact potential flow. The Gerstner wave is shown regardless of depth, though it's only exact in deep water.

None of these theories are valid to add together the way linear waves are — each one describes a single monochromatic wave train on its own, which is why this page always shows one theory at a time driving the wave-and-arrows view, and shows all six only as separate overlaid curves below, never summed.

The seabed is assumed flat and uniform, the sea is unidirectional (no directional spreading), and there's no wave breaking, wind input, or dissipation modeled anywhere on this page. The wave's vertical scale in the top panel is exaggerated relative to the depth axis for visibility — the exaggeration factor in use is shown in the corner of that panel.

Ocean Waves

01 Wave

02 Water depth

03 Theory & arrows

04 Theory comparison

05 Playback

06 Irregular sea

The physics, end to end