Cross Sea – where ‘two seas meet’ | Physics, Mathematics and Chemistry

01 · The myth – and what you actually see

A collision between two seas? Or two wave trains that geometry forces to cross each other?

‘Grenen is a unique natural phenomenon where the two seas, the Skagerrak and the Kattegat, meet and the waves culminate with each other.’ (translated) Tourist information, enjoynordjylland.dk

What you see from the tip of Grenen is a cross sea: two wave trains that cut across each other at an angle, so that the surface rises in short, pyramid-shaped peaks exactly where the crest lines cross. But the trains do not meet because two seas ‘collide’ – they meet because Skagen Reef and the tip of the spit bend the waves towards each other: refraction over the shallow water, diffraction round the tip, and finally ordinary linear superposition.

The proof that you don't need two seas lies closer than Skagen: at a sandbar on Gjerrild Nordstrand, on Djursland, exactly the same pattern appears in miniature – in one and the same sea, the Kattegat, with one and the same wind.

The author standing at the tip of Grenen while waves arrive obliquely from both sides and cross at the water's edge — **The tip of Grenen.** Wave trains arrive obliquely from each flank of the spit and cut across each other at the water's edge. The refraction follows the geometry of the reef – not a boundary between bodies of water.

Sunset over the Kattegat at Gjerrild Nordstrand, where a narrow sandbar stretches out into the sea and small waves meet over it — **Gjerrild Nordstrand, 20 April 2021.** A sandbar stretches out into the Kattegat; small waves are refracted in towards the shallow water from both sides and meet over the ridge. Grenen in miniature – in one sea.

Two families of small wave crests crossing each other over the sandbar in the evening light — **A minute later.** Two families of crest lines cut across each other over the sandbar – where they cross, the surface momentarily culminates in short peaks.

02 · The wave tank: two trains cross

Turn the angle between the trains and watch the chequered pattern appear – turn one train right down and watch it disappear.

Two identical wave trains with the same wavelength cross at the angle 2α. In linear theory they simply add:

η = a cos[k(x cos α + y sin α) − ωt] + a cos[k(x cos α − y sin α) − ωt] = 2a cos(ky sin α) · cos(kx cos α − ωt)

The result is a standing modulation across the mean direction and a progressive wave along it. The bright ‘whitecaps’ in the simulation show where two crests cross and momentarily culminate together – the trains pass through each other undisturbed and never collide.

Angle between the trains 2α 60°

Wavelength λ 44 px

Train 2 – amplitude 100 %

Distance between neighbouring crests across: λ / sin α = 62 px

Set train 2 to 0% and watch the crossing pattern disappear: without the second train there is no cross sea. At 90° the famous ‘square waves’ appear, as at Île de Ré.

03 · One wave meets the sandbar

Here only one wave train is sent in – and the sandbar makes the cross sea all by itself.

The simulation solves the wave equation η_tt = ∇·(c²∇η) with the depth-dependent speed c = √(gh). The model is linear (with a little extra damping in very shallow water as a substitute for breaking): it captures the geometry and the linear shoaling, but not the steepening and breaking of the crests. Waves arrive from the deep water at the top and meet a submerged sandbar sticking out from the coast at the bottom – as at Gjerrild, and like Skagen Reef at Grenen. Watch for three things: the crest lines slow down and turn in over the flanks of the sandbar (refraction), they bend round the tip (diffraction), and behind it they cross themselves – one wave train has become a cross sea.

Wavelength λ 28

Wave direction 0°

Water depth over the sandbar 10 %

Try ‘Remove the sandbar’: without shallow water the wave train travels undisturbed towards the coast, and the crossing pattern disappears. Put the sandbar back, and geometry does the rest – with no help from a second sea. Turn the depth over the sandbar up and down, and watch the refraction get weaker and stronger. And try oblique incidence: far from the sandbar the crest is still one unbroken line all the way across — from one side to the other.

04 · The physics behind it – and the rest of the story

Three mechanisms, a grain of truth and a global genre of where-the-seas-meet myths.

Refraction

The dispersion relation ω² = gk·tanh(kh) gives, in shallow water, the speed c ≈ √(gh): the shallower the water, the slower the wave. The crest lines turn towards the shallows according to Snell's law in its normalisation-free form sin θ/c = constant – water waves have no vacuum from which to take a refractive index, and the seabed is one continuous medium, not two with an interface. The invariant is the component of the wavenumber along the depth contours, k sin θ. If you want an index, it is n_eff = √(h_deep/h): the reef acts as a lens of ‘optically dense’ shallow water that gathers the rays round the tip.

Diffraction

The ray picture requires the depth to vary slowly on the scale of a wavelength – at the tip it breaks down, and the waves bend into the shadow region behind the spit: diffraction, Huygens' principle at full scale. That is why the sandbar simulator solves the full wave equation instead of following rays.

Superposition

Where the crest lines cross, the displacements add, and the surface momentarily culminates in short peaks. Amusingly, the tourist sentence's ‘culminate’ is closer to the physics than ‘collide’ would have been – the trains pass through each other undisturbed.

Does linearity hold? Yes, where the argument uses it: in the propagation, where the steepness ka and the ratio a/h are small, and where any real exchange of energy between free wave trains requires slow four-wave resonances – the trains really do pass through each other undisturbed. Over the ridge of the sandbar and at the water's edge it is different: shoaling makes the amplitude grow (as h^−1/4), the Ursell parameter Hλ²/h³ grows out of the linear regime, the crests steepen and finally break (H/h ≈ 0.8). The geometry of the pattern is linear kinematics – the spray when two nearly breaking crests culminate together is fully non-linear hydrodynamics, and the extreme waves in cross seas are a weakly non-linear effect on top of the linear pattern.

There is a grain of truth in the story: the two sides of the spit face different wave climates, so the crossing trains at Grenen often have different periods and directions. And ‘two seas meet’ has a real hydrographic content: saltier Skagerrak water meets brackish surface water from the Baltic outflow, and the front can occasionally be seen as a line of foam or colour. But that is a density phenomenon in the bodies of water – something other than the wave pattern above. The boundary between the waters itself is a cartographic convention: a line from Grenen across to the Swedish coast.

The genre is global, by the way: Cape Reinga at the northern tip of New Zealand is sold as the meeting of the Tasman Sea and the Pacific, and at Île de Ré the famous ‘square waves’ are photographed – pure cross sea, in an ocean with no neighbour to collide with. Both places have exactly the geometry this page is about: a point with a submerged reef.

Cross sea with a chequered pattern seen from the lighthouse on Île de Ré — **Île de Ré.** The ‘square waves’ at the Vieux Phare des Baleines – one sea, two wave trains, one chequered pattern. If the cells are square, the trains cross at 90°. Photo: Michel Griffon, CC BY 3.0, via Wikimedia Commons.

Columbia Bank off Cape Reinga: a submerged reef with surf along its edge — **Cape Reinga.** Columbia Bank / Te Nuku-o-Mourea: the submerged reef off the point, where the waves break and where the change of colour reveals the shallow water. Skagen Reef again – on the other side of the globe. Photo: Wikimedia Commons.

And the distinction is more than pedantry: cross seas are over-represented in the statistics of shipping accidents and can raise the likelihood of extreme single waves. The bathing ban at the tip of Grenen rests on the same reality. The physics is both more beautiful and more serious than the romance of colliding seas.