BEGIN WITH SOMETHING YOU KNOW.A wave approaching shore.What changes as the water gets shallower?
M.I.L.O. / SYSTEMS
MATHEMATICS
MADE PHYSICAL.
A wave approaches shore. The water gets shallower, and the wave changes. Here you can change the conditions yourself and follow the mathematics behind what you see. Start with a wave.
Follow a wave as the water gets shallower. Its motion changes, and the model has to change with it.Scroll to follow the wave. Drag a number to see what changes.
01 / DEEP WATER · depth 80 m
What travels when the water mostly moves in circles?
The water circles. The wave travels.
In this linear model, the dots trace circles while wave energy travels ahead. Period T is the time between crests; offshore height H₀ is crest to trough; offshore wavelength λ₀ is the distance between crests. Crest speed c follows a crest; group speed c_g tracks energy. The half-speed rule applies only when the water is deep relative to this wave's length.
TRY IT · DRAG A NUMBER
Set the period to 8.0s and a height of 1.20m . Compare wavelength and speed. The half-speed rule applies when depth exceeds half the wavelength.
Explore the mathematics
T is period; λ is wavelength; h is depth; g is gravity. The wave number k = 2π/λ and angular frequency ω = 2π/T.ω² = g k tanh(kh)
λ₀ WAVELENGTH
—m
c CREST SPEED
—m/s
c_g ENERGY SPEED
—m/s
H₀/λ₀ STEEPNESS
—limit 1/7
h/λ₀ DEPTH RATIO
—half-speed rule when > ½
DERIVATION + SOURCES
Potential flow: the velocity is the gradient of a potential φ that satisfies Laplace's equation ∇²φ = 0 in the water, with linearised conditions at the free surface and no flow through the bed. Solving on a flat bed gives a travelling sinusoid and the dispersion relation above. The energy travels at the group speed dω/dk, which is c/2 in deep water. Airy 1845 · Dean & Dalrymple, Water Wave Mechanics for Engineers and Scientists, 1991, ch. 3–4.
CLASSICAL · ANALYTIC · AIRY 1845 · FINITE-DEPTH DISPERSION · DEEP-WATER HALF-SPEED RULE DEPENDS ON h/λ · THE DOTS ON THE WALL ARE LINEAR-THEORY ORBITS
02 / THE SHELF · depth 6.0 m
What changes as the water gets shallower?
The wave slows in shallower water.
As the seabed rises, the wave slows and its crests move closer together. In sufficiently shallow water it grows taller and sharper. The time between arriving crests stays the same.
TRY IT · DRAG A NUMBER
Set the shelf depth to 6.0m. Compare height and length: linear shoaling sets the height, and a Stokes correction sharpens the crest.
Explore the mathematics
Energy-flux shoaling and a second-order Stokes shape: E is wave energy, c_g group speed, η surface displacement, a amplitude, θ phase, k wave number, and h depth.E cg = const · η = a cos θ + a²k/4 · cosh kh (2 + cosh 2kh) / sinh³ kh · cos 2θ
K_s SHOALING
—× H₀
H ON THE SHELF
—m
λ ON THE SHELF
—m
kh
—shallow when ≪ 1
Ur = Hλ²/h³
—Stokes holds < 26
√(gh) vs c
+—%
DERIVATION + SOURCES
Between two depths no energy is lost, so E cg stays constant; with E ∝ H² that fixes H at every depth, and in the shallow limit it becomes Green's law, H ∝ h−¼. The Korteweg–de Vries equation, ηt + c₀ηx + (3c₀/2h) η ηx + (c₀h²/6) ηxxx = 0, is the operator that balances steepening against dispersion on a shelf like this, and the Ursell number measures that balance. The lesson renders the KdV regime with Airy shoaling and a Stokes correction because a KdV solver cannot run a wave up a beach; the shallow-water solver takes over at the seam. Korteweg & de Vries 1895 · Ursell 1953 · Dean & Dalrymple 1991, ch. 11.
CLASSICAL · ANALYTIC · GREEN 1838 · STOKES 1847 · KdV IS THE FULL OPERATOR HERE; THE DISPLAY DOES NOT INTEGRATE IT
03 / THE BREAK · MODEL BREAKS IN — m OF WATER
Why can’t the wave keep its shape?
The crest outruns its own base.
A breaking wave overturns and loses energy to turbulence, sound, and heat. This model uses a moving surge of water to study that loss. It does not simulate the overturning face.
TRY IT · DRAG A NUMBER
Set the beach slope to 1:25 . A bore is a moving jump in depth. Compare where the shallow-water model forms one with the reference prediction.
Explore the mathematics
Saint-Venant shallow-water equations: h is depth, u horizontal speed, t time, x position, g gravity, z_b bed height, and τ_b/ρ bed friction per unit mass.∂h/∂t + ∂(hu)/∂x = 0 · ∂(hu)/∂t + ∂(hu² + ½gh²)/∂x = −gh ∂zb/∂x − τb/ρ
h_b McCOWAN
—m · H/h = 0.78
h_b MODEL
—m
H_b/h_b MODEL
—
ξ₀ IRIBARREN
—
ΔE ACROSS THE BORE
—m of head
DERIVATION + SOURCES
This lesson does not solve the two-phase flow needed to track an overturning surface. Surf-zone engineering models can replace the breaker with a bore in the shallow-water equations: mass and momentum are conserved across the jump, energy is not, and the loss is fixed by the jump's two depths, ΔE = (h₂ − h₁)³ / 4h₁h₂. The Iribarren number ξ₀ = tan β / √(H₀/L₀) sorts breakers: spilling below 0.5, plunging up to 3.3, surging above. Brocchini & Dodd 2008 · Battjes 1974 · Hibberd & Peregrine 1979.
COMPUTED · SHOCK STAND-IN · McCOWAN 1894 · BATTJES 1974 · BÉLANGER 1828 · THE SOLVER BREAKS LATE: IT HAS NO DISPERSION AND NO ROLLER · SPRAY IS A DISPLAY EFFECT
04 / THE SWASH · STILL SHORELINE AT x = — m
What determines how far the water runs up the sand?
Water runs up the sand.
A thin sheet of water runs uphill, slows, and drains back under gravity and bed friction. Its highest reach is called runup: one measure used to understand coastal flooding.
TRY IT · DRAG A NUMBER
Set bed friction to 0.0060c_f. Compare the model’s runup with Hunt’s rule, an empirical prediction from wave-tank data.
Explore the mathematics
Quadratic bed friction and Hunt's runup rule: τ_b/ρ is bed friction per unit mass, c_f its coefficient, u water speed, R runup, ξ₀ the Iribarren number, and H₀ offshore height.τb/ρ = cf u|u| · R ≈ ξ₀ H₀
R RUNUP, MODEL
—m
R RUNUP, HUNT
—m
SHORELINE NOW
+0m
FRONT SPEED
—m/s
ξ₀
—
DERIVATION + SOURCES
The shoreline is wherever depth reaches zero; the finite-volume scheme carries a wet-dry front without special treatment, and hydrostatic reconstruction keeps still water still on a slope. Hunt's rule R/H₀ = ξ₀ holds for ξ₀ up to about 2.3 and is applied with that cap. Hunt 1959 · Audusse et al. 2004 · Stockdon et al. 2006 for the field version.
COMPUTED · WET-DRY FRONT · HUNT 1959 IS A FIT FOR REGULAR WAVES ON PLANE BEACHES · COMPARE THE MODEL WITH THAT FIT
05 / THE HANDOFF
Where the models meet.
The time between crests stayed the same. Height, wavelength and speed changed as the wave approached shore. Each model describes part of the journey. Where they meet, the simulation passes the wave’s motion from one description to the next.
MODEL · AIRY THEORY OFFSHORE, ANALYTIC · NONLINEAR SHALLOW WATER FROM THE SEAM, COMPUTED · ONE PROFILE, REPEATED ALONGSHORE · VERTICAL ×6
THE MODEL
Both models share a clock.
Offshore: Airy linear theory, evaluated analytically at every point, with energy-flux shoaling and a second-order Stokes shape. From the seam: the nonlinear shallow-water equations, integrated with a shock-capturing finite-volume scheme on a one-metre grid, blended in through a relaxation zone. McCowan's breaking index sets the seam's position.
THE LIMITS
What the model leaves out.
The shallow-water operator has no dispersion, so it breaks the wave later than the sea would, and it has no roller, so the bore is sharper than a real one. Spray is a particle effect. The alongshore direction repeats one computed profile. Vertical scale is exaggerated six times unless you switch it off.
NEXT
Now let’s track light.
Water waves and light can both be described using wavelength, frequency, and interference. What carries the wave is different. Follow light from its source to an observer in a schematic study, and change the settings to see how its path and the resulting image respond.
Linear wave theory. Water particles move in small closed orbits; only the wave shape travels. Wavelength, crest speed and energy speed follow from the period through the dispersion relation.
The shelf
Linear shoaling raises the wave height as depth falls; a second-order Stokes term sharpens the crests. The period is conserved. The Ursell number reports when the shallow-water balance takes over.
The break
The nonlinear shallow-water solver forms a bore where the crest outruns its base. The energy lost across the bore equals a hydraulic jump's loss. The Iribarren number classifies the breaker.
The swash
The same solver carries a thin sheet up the beach with bed friction and a moving shoreline. Modelled runup is shown beside Hunt's empirical rule.
