A mathematical model of bodysurfing

Very few papers on bodysurfing had been published in peer-reviewed academic journals. These papers are mostly medical in nature, focused on injuries sustained while bodysurfing rather than the activity itself. Regardless of where in the world you are whomping, back and neck injuries appear to be most common. No-one who has ever been to Sandy Beach (Hawai'i) on a hairy day will seriously doubt that.

In this sea of dry academic earnestness, Nevill de Mestre's paper on the mathematics of bodysurfing stands out as a slight curiosity. De Mestre, an avid bodysurfer, research mathematician specialising in fluid dynamics and professor of Mathematics at Bond University in Australia, proposed a mathematical model to account for the forces involved in bodysurfing. Despite the obvious simplification in reducing a three-dimensional activity to a two-dimensional plane, the model gives some insight into the fundamental forces involved in catching and surfing waves.

The paper starts off with brief history of what is known about the origins of bodysurfing in the modern age. De Mestre then briefly describes swell generation in the open ocean based on three factors---action time of the storm, fetch, and wind intensity---and explains why waves break when they reach the shore. This is a well researched area, and equations governing wave behaviour up to the point of breaking have been available for some time.

Differences between spilling (rolling) and plunging (barrelling) waves are explained. De Mestre notes that slower spilling waves are ideal for bodysurfing, while faster plunging type waves are ideal for boardsurfing but generally too fast for bodysurfing. This seems to be written from the point of novice bodysurfers. Catching a wave is the most difficult part of bodysurfing it---the surfer has to match the wave's speed in order to catch it. Yet most rideable waves travel faster than humans can swim. To overcome this obvious hurdle, you can jump into the wave by pushing off the bottom, or use swim fins to increase swimming speed. Good positioning, good timing and quick acceleration are needed to catch unbroken waves in deep water.

Physical forces acting on the bodysurfer at the point of catching a wave are delineated: vertically, gravity is balanced out by buoyancy. Horizontally, drag is overcome by propulsive force from the surfer's swimming efforts. To improve swimming speed, propulsion can be increased with technique training and the use of swim fins. Drag can be decreased by minimising the surfer's cross-sectional area---presented in the direction of travel through the water--- through technique training, weight loss, and use of special low-drag swim suits. Water density also affects drag, with warmer water providing less resistance than colder water.

There is only a small window of opportunity for a surfer to catch a wave as it passes underneath. When the water depth decreases sufficiently for the passing wave to reach breaking point, some of the energy is converted to a shoreward force which reaches a maximum in the wave crest. This force, dropping off quickly to either side of the crest, enables a surfer to match the wave's speed and catch it.

Actual wave-riding is somewhat glossed over in the paper. The proposed wave model is used to explain why a surfer drops off the wave as it nears the shore and loses energy. Basically the wave force---the same force enabling the surfer to catch the wave---decreases as the wave expends its energy, and simultaneously the area in which this force acts upon the surfer diminishes. As the wave size decreases, the surfer's legs also start dragging behind the wave instead of being carried along by the wave force, increasing drag and causing the surfer to decelerate and drop off the wave.

The paper concludes with some remarks on various bodysurfing styles and skills. Interestingly, the popular style of using one or both outstretched hands as planing surfaces---with or without a hand plane---referred to as hydroplaning, is seen as a variation of bodysurfing. The practice of bending one leg at the knee in a 90 degree angle to aid stability when riding a turbulent wave is mentioned, although it seemed to have fallen out of use in recent years. Planing sideways across the wave face, rather than straight to shore, not only increases speed and length of ride, but is also the safest approach when surfing plunging waves. Riding the barrel of a close-out dumping wave is considerably safer than being flung shoreward by its lip.

You do not need to understand the mathematics to enjoy bodysurfing. Even so, comprehension of the forces at work might help improve your bodysurfing technique. Plus there is the obvious geek factor.

Source: Mathematics and bodysurfing, THE AUSTRALIAN MATHEMATICAL SOCIETY GAZETTE, Volume 31 Number 4, September 2004