In Perspectives on the Sea, we look into how simulation can help us face societal challenges, particularly those related with the ocean.

Artificial reefs can help in many different contexts.

We have seen how artificial reefs can not only help the regeneration of endangered habitats in struggling coral reefs but also how they can protect the coastline from harsh sea conditions.

But artificial reefs are truly multifunctional structures.

Throughout the upcoming blog posts, we will show how artificial reefs can also be engineered to create the perfect surfing wave.

In this post, we start off by outlining the basic principles of surf and which surfability metrics can be defined to characterize surf waves.

The Science Behind the Perfect Surf Wave

There is something positively awe-striking and almost logic-defying going on when we see a surfer dominating a wave:

Video 1: A compilation of the best clips from the Red Bull Surfing sessions in 2020.

It’s hard to pin down exactly when activities reminiscent of modern-day surf became widespread. Evidence comes from Polynesia from 800 years ago [1] and as far back as 4000 years ago from Peru [2].

It wasn’t until the end of the XIX^th century that our contemporary idea of surf was born, mostly in Hawaii and California.

But surf — as a sport and leisure activity — has come a long way.

13 to 24 million surf boards are sold annually worldwide [3]. By 2021, the surf equipment industry was worth in excess of 4 billion dollars and one of the fastest-growing sport sectors — projected to hit 5.5 billion dollars by 2028 [4].

But what is behind such a sought-after unique phenomenon?

A wave is born

The principle behind the formation of waves on the sea is pretty straightforward.

By mechanically pushing against the water surface, wind currents transfer kinetic energy and generate pressure throughout the fluid. The resulting force causes the water on the surface to accelerate, displacing it from its equilibrium position.

With an uneven mass distribution, the pulling gravity force creates the up-and-down motion of the water surface. The disturbance is now both modulated but also with kinetic energy, propagating in the direction of the wind.

A sea wave is born.

But now, here comes the hard part — maneuvering the wave:

Video 2: “As a surfer, you are a master of complicated physics”. Credits: Ted-Ed

But this magic can only happen if some key ingredients are present.

Beaches with planar seabeds simply do not produce good waves for surfing. For instance, consider incoming waves propagating parallel to the coastline over a ramp. Either:

• the wave is too small or too slow for its crest to advance with respect to its trough and hence the wave won’t break at all.
  Without anything ahead which can obstruct any section of it, the wave will simply continue moving as a whole unaffected until inertia dominates, grinding it to a halt. These are known as surging waves.
• the wave is tall or fast enough, allowing the crest to tip over its own weight. Since this situation is satisfied along all points of the wavefront, the entire wave collapses at once. This situation is known as a closeout [5].

Fig. 3: Closeout waves are challenging to impossible to surf since the wave collapses at the same time along all points. Credits: [6].

Neither case is desirable for surf. In fact, no maneuver can be performed in these conditions.

Something else is needed.

Naturally suitable surf waves are found whenever the coast has an irregular seabed topography. Since both gravity and pressure depend on the depth, the potential energy stored at each point varies.

It’s the delicate balance between the potential energy water can utilize and the kinetic energy it has as it moves along that determines where and how waves break.

However, quantifying these conditions requires some surfability metrics to be established.

Quantifying the perfect surf wave

Unlike other tasks with more objective parameters, the quality of a wave entails subjective aspects beyond Physics.

The skill level of the surfer — tied to his or her fitness and mobility intuition — as well as the intended purpose of the maneuver ultimately dictate the suitability and quality of waves for surfing.

Nonetheless, some physical parameters help establish and distinguish wave profiles.

Let us outline the four most used metrics to characterize surf waves.

Wave height

Naturally, the vertical extent of the wave determines much of what can be executed. This is quantified through the wave height.

Measuring the height of a wave seems something seemingly trivial to accomplish.

However, it isn’t.

It took 18 months for a team of scientists to establish that the 26 meter (86 feet) tall wave that Sebastian Steudtner rode in Nazaré, Portugal in 2020 was indeed the tallest ridden wave ever recorded [7].

Video. 3: The 26-meter wave surfed by Steudtner in Nazaré, Portugal in 2020. Credits: Oliver Raatz

To further complicate matters, the ambiguity on how to measure the wave height leads to different methods to be used [6, 8].

The most used one is the scientific or Bascom method. By standing on the beach and aligning the wave crest with the horizon in sight, the wave height is taken as the distance from the aligned crest to the average sea level.

Fig. 4: A schematic of where measurements of the wave are taken according to the Bascom method and Hawaiian method. The latter tends to be roughly half of the former. Credits: Surfer Today.

However, a single wave is of little significance for surfing practices.

Waves come in sets and surfers ride the largest waves in a set. Also, atypical incoming wave conditions can produce waves with abnormal heights which do not represent frequent occurrences for surfers.

For this reason, assembles of waves allow for more appropriate heights to be calculated. A common such measure is given by the significant wave height, which is the average of the top 30% of the incoming waves in a particular area.

Wave peel angle

As the wavefront propagates towards the shore, the seabed profile can induce wave breaking in specific spots.

From there, the tumbling and rolling mass of water at the crest forces the neighboring parts of the wave to also break.

It’s like a domino effect.

The wave keeps breaking, leaving a foamy trail behind — this is known as peeling.
Surfers must surf at least as fast as this peeling rate in order to stay in front of the wave break point.

Fig. 5: The peel angle measures the dephasing between the wavefront and breaking trails. Credits: [9].

By the time that segment of the wave broke, the unbroken wave got further ahead.
The wave trail and front are no longer aligned and the angle between these two traces is the peel angle \(\alpha\).

The peel angle is a very good indicator on what the surfer can accomplish. Fast and therefore more challenging waves have \(\alpha \approx 0º\), while \(\alpha \approx 90º\) creates extremely slow waves.

In practice, surfable waves have peeling angles greater than \(30º\). Optimal surf waves are found anywhere between \(45º\) and \(66º\). The latter is in fact a theoretical upper bound — we will get to why in an upcoming post.

Breaking wave intensity

The wave shape, width, mass distribution and surface profile are also very important in determining which technical maneuvers can be tried.

In general, these features tend to be controlled mostly by both seabed gradients orthogonal to the wave propagation and wind profiles close to the shore [9].

Surfers have long classified some of these wave profiles. The most common scheme splits waves as spilling, plunging, surging, or collapsing.

Conventional surf waves are of the plunging type, providing a reasonable slope for motion stability to be reached. The dynamics of the remaining ones makes them more challenging or downright impossible to tame.

Fig. 6: A summary of breaker types and the ranges for their associated Iribarren parameters. Credits: [10].

An effective indicator of these types is given by the Iribarren parameter \(\xi_0\):

Here, \(\theta\) is the angle of the seabed slope beneath the wave, \(H\) the wave height and \(L_0\) the offshore wavelength of the incoming wave.

In deep water, and for a period \(T\), \(L_0 = \frac{g}{2 \pi} T^2\), where \(g\) is the gravitational acceleration.

This descriptor is continuous but different ranges separate the breaker types qualitatively.

Conventionally surfable waves are collapsing and have \(\xi_0\) between \(0.5\) to \(2.5\).

Anything lower will not induce a tunnel-like break, and the crest become positioned in between oscillations. It can collapse down in foam without a clear pattern being formed — a spilling wave is formed.

On the other side of the spectrum, the wave can get broader and the crest does not quite break. However, the bottom can, making the whole wave to slowly disintegrate into foam down to the average sea level. In this instance, it is a collapsing wave.

In the extreme case, no breaking happens at all and the wave moves unaffected by the seabed — these are surging waves.

Both the variables used for the Iribarren parameter and the characterizing ranges of the breaker type can all be slightly tweaked given the specific situation being studied.

Despite its reasonable theoretical foundation in idealized situations (namely by considering a perfect seabed ramp), the use of the Iribarren parameter and other breaking wave parameters as general descriptors of the breaking wave intensity is still a controversial matter [11].

Notwithstanding these shortcomings, the Iribarren is amply used in the literature for classifying the breaker type.

Wave section length

All the metrics so far assume the wave properties remain fairly constant in type.

However, as the wave propagates toward the shoreline, different sea, wind and seabed profiles can render it qualitatively different.

Once that happens, a new section has been reached.

Sections are propagation segments with specific wave heights, peel angles and breaker intensities.

For this reason, a single wave is suitable for many different maneuvers, depending on which section a surfer is located.

A length can be naturally associated to each section, quantifying how long a surfer can theoretically ride along the wave with guaranteed features.

Fig. 7: A long wave with 5 different sections. Each one corresponds to a combination of peel angle \(\alpha\), breaker intensity \(B_I\) and section length \(S_L\) and allows for particular surf maneuvers. Credits: [11].

Coming up

Surf is a sport with huge potential to grow.

But with an ever increasing demand for good spots to ride the perfect wave, this situation presents a challenge.

Many more surf spots are needed and nearby locations accessible from where surfers live. Also, due to changing coastline morphology and other human interventions, surf beaches can become degraded and unsuitable for surf more often throughout the year.

Artificial reefs can help, by providing a way to engineer the water propagation with given incoming wave and sea conditions so that a particular range of waves can be created.

In this way, otherwise unsuitable or unsatisfactory beaches can be reinvented as excellent places to practise surf and other water sports.

That’s what we will focus on in the next blog post.

🌊 Hang loose and sea you soon! 🌊

References

[1] - The History of Surfing and Its Origin

[2] - Were the first surfers Peruvians? The history of surf in Peru

[3] - Surfing Statistics: 2022 Round-up - House of Surf

[4] - Surfing Equipment: Report Overview

[5] - What is a closeout wave?

[6] - How to Measure Waves in Surfing

[7] - Washington Post - Surfing a record 86-foot wave took guts. Measuring it took 18 months

[8] - How is Wave Height Measured?

[9] - Scarfe, B. E., et al. “The science of surfing waves and surfing breaks-a review.” (2003)

[10] -Barceló, Fernando Roca. Effect of porosity on hydrodynamic performance of Porous Bonded Revetments–numerical modelling and application for shore protection at Valencia, Spain. Diss. Master Thesis, 123 sayfa, 2014

[11] - Robertson, B., and K. Hall. “Wave vortex parameters as an indicator of breaking intensity.” Proceedings of World Academy of Science, Engineering and Technology. No. 73. World Academy of Science, Engineering and Technology (WASET)