The waves that always arrive square to the beach

The shaking a uniform Earth could not carry found waves trapped in a layer because their speed changes with depth, and the same principle applies to sound in the ocean, as the channel with no walls describes. In both, a wave turns toward where it travels slowest. That rule is as old as optics, and the bend at the boundary gives it for a sudden change in medium. The most familiar example in the world, seen by anyone at a beach, generally goes unremarked.

Stand on a long straight beach on a day when the swell has come from a distant storm, and look at the lines of waves coming in. They are almost parallel to the shore. A storm in the middle of an ocean sends swell off in every direction, and that swell reaches each coast from a direction set by the geometry of the ocean. It may come from far to one side. On a chart of the wave’s direction at sea, the angle might be 50 degrees or more from the line straight offshore. Yet at the waterline, the same crests are close to parallel with the beach. Whatever way a coast happens to face, the waves appear to be heading straight for it. Beach-goers tend to assume that this is how waves are, or that wind has something to do with it. It is Snell’s law.

A wave that feels the bottom #

A wave on water has a phase speed depending on its wavelength and the depth. The speed that depends on the length derived the dispersion relation,

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

A swell’s period stays the same as it travels, because it is set by the source and any steady change in the medium conserves frequency. So at each depth, the wave has the wavelength required by that relation for its fixed $\omega$. The relation has two limits. In deep water, $\tanh(kh) \to 1$, and the phase speed is $g/\omega$, which is independent of depth. For a 10 s swell, this is 15.6 m/s, and the group speed is half that. In shallow water $\tanh(kh) \to kh$, and the speed is $\sqrt{gh}$, with no dependence on wavelength.

Between the limits, the speed depends on $h/\lambda$. A 10 s swell has a deep-water wavelength of 156 m, and it begins to feel the bottom at about half that, around 80 m. By then the water’s orbits near the bottom have flattened to a back-and-forth movement. Shoaling reduces the crest speed steadily, and by 5 m of depth the crests move at 6.8 m/s. The speed thus plays the role of a refractive index, with $n = c_0/c$, running from 1 in deep water to perhaps 5 or more near the beach. A chart of depth contours is a map of the medium, and a nautical chart shows at a glance how swell behaves.

A striking difference between light and water waves arises here. In glass the refractive index is fixed by the material. Here, it is set by depth and frequency. Long-period swell feels the bottom further out than short waves do, and so it starts refracting further offshore and turns more. That is why a long-period swell from a distant storm wraps more thoroughly around a headland than a short wind chop does. A surfer on a beach with a sheltered corner watches this happen: long waves reach right round into the bay, while short ones do not.

Waves that turn to face the beach #

Take a beach whose depth changes only with distance offshore. A crest coming in at an angle has one end in shallower water, and this end moves more slowly. The crest swings around, like a line of marchers wheeling at a muddy edge. The same thing can be stated exactly: because the medium is uniform along the shore, the component of $\mathbf{k}$ along the shore must be conserved, which gives Snell’s law,

$$\frac{\sin\alpha}{c} = \text{constant}.$$

The result is a ray that bends steadily toward the normal to the shore. For the plane beach in the figure, a slope of 1 in 50, a swell starting at 50° offshore is turned to only 5° from square-on when it is about to break. In this case, there is a further reason for the near-complete turn. Unlike an optical prism’s index, which is bounded by that of its material, the wave’s speed falls almost to zero at the water’s edge. So the sine of its angle must fall almost to zero too: the formula gives a final angle that depends on how shallow the water is at the point of interest. As the depth goes to zero, a wave’s direction must become exactly normal to the shoreline, whatever it was to start with.

A sailor thus does poorly in inferring the direction of an ocean swell from what is happening on the beach. The offshore angle is nearly erased: from 0 to 80 degrees offshore, the angle at 2 m ranges only from 0 to 16 degrees. This has consequences for sailors and for coastal engineering alike, since a beach’s response to a swell is much less dependent on the direction of the source than a naive reading would suggest.

What the remaining angle does #

The small angle that remains matters a great deal. A wave breaking at a slight angle to a beach delivers momentum both onshore and along the shore. The onshore component piles up water at the beach, in what is called setup. The alongshore component drives a current that runs parallel to the beach, called the longshore current, typically at up to a metre per second. Michael Longuet-Higgins explained this in 1970 in terms of the radiation stress, the momentum flux carried by a wave, in the same way that the drift a sound leaves behind describes a momentum flux in another wave. It is the same mechanism that transports the sand.

The current carries sediment along the coast, sometimes at hundreds of thousands of cubic metres per year. A groyne placed to stop it causes sand to pile up on one side and be eroded on the other, and a harbour’s breakwater interrupts this flow, so that a harbour on a sandy coast often has one beach growing and its neighbour disappearing. On the east coast of the United States, the Outer Banks and many other barrier islands are maintained by this drift. Santa Barbara’s breakwater, built in the 1920s, caused an enormous sand spit to form and famously starved the beaches to the south. The current’s driving force comes from the 5° that refraction left behind. Had refraction done its job entirely, the sand would not move along the coast at all.

Rip currents are also due to this. Water pushed shoreward by breaking waves has to return seaward somewhere, and it does so in narrow, fast channels, where the setup is lower. Those are the rip currents responsible for most surf drownings. They form at gaps in sandbars and near structures, and the sandbars that reflect the sea discusses a further effect of bars on waves. The bending described here sets the angle at which the momentum arrives, and that sets what the near-shore currents have to do.

A wave that shrinks before it grows #

In a steady state, the energy flux in a wave must be conserved, aside from any losses. The flux per unit length of crest is the energy density times the group speed, $\tfrac18\rho g H^2 c_g$. For waves coming straight in, the flux stays the same, and so

$$H = H_0\sqrt{\frac{c_{g0}}{c_g}}.$$

The group speed in deep water is half the phase speed. As the depth falls, it first rises, since the $2kh/\sinh 2kh$ term in its formula grows faster than the phase speed falls. It then falls with the phase speed. So the wave’s height first shrinks. It reaches 0.91 of its offshore value near a depth of 25 m for a 10 s swell, and it passes its offshore value again at around 6 m. This small early dip is hardly mentioned in popular accounts, but anyone with a wave buoy can see it, and it has a satisfying explanation: for a while, energy is moving faster into the shallows than it was before.

Near the shore, the height rises steeply as depth to the minus one-quarter, a result due to George Green in 1838. This comes directly from $c_g \approx \sqrt{gh}$. It is a slow rise as a function of depth, which is why a tsunami a metre high in deep water grows only to a few metres near shore on a gentle shelf, and any larger amplification is due to focusing or to harbour resonance. The taper that matches every note deals with the same gradual-change limit in a horn, where a similar fourth root arises from the same flux conservation. Refraction spreads crests, since a ray from a given offshore distance covers a larger shore distance, and thus decreases the height slightly, as the second curve shows.

The wave eventually breaks. A wave collapses when its height approaches a fixed fraction of the water’s depth, because the water at the crest starts to move faster than the wave itself. A common rule of thumb gives this at $H/h \approx 0.78$, a value that goes back to McCowan’s calculation for a solitary wave in 1894. In the example, a 1.5 m swell breaks in about 2.5 m of water. The front that steepens until it cannot gives a more general account of how nonlinear steepening works. At this point, the linear theory used in every figure here has stopped being applicable.

Why the waves break on the point #

So far the contours have been straight. On a real coast, depth contours follow the shoreline, with shoals running out from headlands and deeper water in the bays. A swell approaching such a coast undergoes refraction that depends on position, and this concentrates energy into some places and spreads it out of others.

The figure marches each ray through the local speed field, with the curvature of its path equal to the rate at which the speed changes perpendicular to it. Offshore, the 21 rays are evenly spaced, each representing an equal share of energy. Near the coast they are not. Over the shoal off the headland the water becomes shallow earlier, and the rays bend toward it from both sides. Those heading into the bay bend away from its centre and spread. The rays are focusing, in a sense no different from the rays through a lens. Energy travels along rays, and so in the middle of the bay the energy per unit length of shoreline falls well below its offshore value, while at the headland it rises above it, with heights that vary as the square root of this density.

That has long been known to mariners and engineers. In the 1940s, Walter Munk and Harald Sverdrup developed wave forecasting and refraction diagrams to plan the Allied amphibious landings in the Second World War, when landing at the wrong spot on a coast could mean a much bigger surf. Refraction diagrams of this type were produced by hand for the Normandy beaches, as well as for many Pacific landings, and their authors found a strong need for precise offshore charts in order to draw them.

The same arithmetic also shapes coasts over time. Headlands receive a focused wave attack and erode quickly, while bays receive reduced energy and accumulate sediment. The result is a slow tendency for coasts to straighten: that is a general conclusion of coastal geomorphology and is visible on any large-scale map of a coastline with mixed rock types. The waterfall at every river mouth concerns a related interface at river mouths. Surfers look for offshore reefs or canyons that focus long swell into exceptional surf. Nazaré in Portugal has an undersea canyon that points at its beach, so waves in the canyon’s deep water are not slowed or refracted while those on either side are, and the two wave trains converge to give very large waves, the largest ever ridden.

Where a ray bends strongly enough, neighbouring rays can cross, forming a caustic. Linear ray theory then predicts infinite height, which of course does not occur, because diffraction takes over at the caustic. A real focus has a finite size, a few wavelengths across, and this is what the fringes below the rainbow describes in the optical case with the same Airy function.

Long waves turn first #

Refraction’s dependence on period sorts a mixed sea. A storm far away sends out waves with a range of periods, and the whistle that arrives sorted shows how dispersion sorts a similar spread of frequencies with distance. On reaching a coast, the long periods meet the bottom first. A 20 s swell has a deep-water wavelength of about 620 m and feels the seabed at 300 m or so; a 6 s wind sea has a wavelength of 56 m and notices nothing until about 28 m. On the same shelf, the 20 s swell has had tens of kilometres in which to refract, so the swell seen at the shore has come round further than the local chop arriving with it. Standing on a beach with both, an observer can see the two systems arriving from noticeably different directions, the long swell almost square and the chop still at an angle.

The longest waves of all take this to an extreme. A tsunami, with a period of many minutes, travels at the shallow-water speed everywhere in the ocean, because even the deep sea is shallow compared with its wavelength. So it refracts over ocean ridges and across continental shelves throughout its whole journey, and maps of its travel time are computed from depth charts alone, with $\sqrt{gh}$ as the only input. A chain of seamounts can act as a lens: the 2011 Tohoku tsunami’s energy was steered by ridges in the Pacific, and arrived with unusual strength in some distant places. The same chart and the same law determine these, though the wave is a hundred times longer than swell.

What the ray picture leaves out #

Everything here is a ray-based, linear calculation for a single frequency. Actual swell has a spread of frequencies and directions, which smooths out focusing. Diffraction is not included. A ray model fails wherever the depth changes significantly over a wavelength, and at the coast, a steep slope reflects a part of the wave, an effect that the sandbars that reflect the sea considers. Bottom friction and white-capping take energy out, and shoreward of the first break, the linear flux conservation is meaningless. Wind is left out, and it can change wave heights over short distances. Currents in the sea also refract waves, sometimes strongly, as with the wave-current interaction in tidal entrances.

The plane-beach slope and the wavy-coast shape are simple choices and not any real location. The breaking criterion is a rule of thumb, and real values vary with the slope of the bed and the wave’s steepness. Finally, these are water waves in which particle motion is large compared with what ray theory assumes; that said, the wave heights in the figures are only taken as small compared with depth, and the calculation is not meant to apply in the surf zone.

Still open: how a wave decides where to break #

Linear refraction and shoaling are now standard in forecasting models, and they work impressively well offshore. The surf zone, where waves break, is far more difficult. A breaking wave generates turbulence, entrains air and pumps sediment into suspension, and these interact on scales from millimetres to kilometres, and no computational model can represent them all. Phase-resolving calculations can follow individual waves in a limited region, while models for whole coasts have to rely on bulk parameters based on field calibration, and the numbers involved in sediment transport are often uncertain by as much as a factor of several. Predicting how a given beach will respond to a storm a week ahead, or how a coast will change with rising sea levels, remains a difficult problem.

Rogue waves in the open ocean are a related issue. Linear superposition produces occasional large waves by chance, and some studies suggest refraction by currents as a further focusing mechanism. How far nonlinear effects add to this has been argued for decades, since the even wave train that cannot stay even shows an instability that can concentrate energy, and yet real ocean spectra are broad, which reduces its importance. Long-term buoy records and targeted experiments are slowly narrowing what can be said about how often extreme waves actually occur.

A wave’s speed falls with depth, and the part of a crest nearest the shore lags, so the crest turns towards the beach. Snell’s law takes a swell from 50° offshore to about 5° at the break, and it reaches exactly zero where the speed does. Energy flux takes the height down to 0.91 near 25 m before it grows, and over a wavy coast the same turning gathers rays on the headland and spreads them across the bay. The waves arrive square because they are slowest at the shore.