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Waves & Surfing Surfboard Design and Geometry Power Generation from Waves Tsunamis Sharks Ships. Paul Pascoe. Mathematics of Surfing . Image Purchased by Passy’s World from Dreamstime.com. Surfing Miscalculations and Random Events. Intro Video .

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Waves surfing surfboard design and geometry power generation from waves tsunamis sharks ships

Waves & SurfingSurfboard Design and GeometryPower Generation from WavesTsunamisSharksShips



Mathematics of surfing

Mathematics of Surfing

Image Purchased by Passy’s World from Dreamstime.com


Intro video

Surfing Miscalculations and Random Events

Intro Video

Source: http://www.youtube.com/watch?v=Fp5Ds_6ck5s


Mathematics of surfing1

  • What Causes Water Waves

  • Predicting Large Waves

  • Breaking Waves

  • Wave Speed Equations

  • Effect of Sea Floor

  • Parts of a Breaking Wave

  • Catching and Riding a Wave

  • Artificial Surf Breaks

Mathematics of Surfing


What causes water waves

What Causes Water Waves

Image Source: http://www.culut.com

When wind blows over the vast expanses of open

water, it transfers energy to the water surface and creates water waves. Surf Waves come from Ocean Storms.


What causes water waves1

Wave Energy = Wind Speed x Wind Duration x Fetch Distance

What Causes Water Waves

Image Source: http://www.seafriends.org.nz


Predicting large waves

Predicting Large Waves Distance

What Causes Water Waves

Image Source: Mechanics of Mavericks at http://www.surfline.com


Predicting large waves1

Predicting Large Waves Distance

What Causes Water Waves

Image Source: Mechanics of Mavericks at http://www.surfline.com


Breaking waves

Breaking Waves Distance

What Causes Water Waves

Bells Beach : http://magicseaweed.com


Deep water waves

Deep Water Waves Distance

What Causes Water Waves

Original Image Source: http://science.kennesaw.edu


Water wave motion

In water waves, (in open water) : Distance

The energy travels but the water does not

Water Wave Motion

Original Image Source: http://bc.outcrop.org


Water wave motion1

Water Particles subjected to wave energy, move in Distance elliptical motions, which decrease to zero with depth.

Water Wave Motion

Original Image Source: science.kennesaw.edu


The three wave zones

Deep Water Waves “BREAK” into Shallow Whitewater Distance

The Three Wave Zones

Original Image Source: science.kennesaw.edu


Water wave equations shape

The shape of water waves is not Sinusoidal, it is Distance actually “Trochoidal” (like a Hyperbolic Tan Graph)

Water Wave Equations - Shape

BUT - Deep Water waves are approximately Sinusoidal

Original Image Source: http://hyperphysics.phy-astr.gsu.edu


Three zone waves speed equations

“Celerity” - c - for Deep, Transitional, and Shallow Distance

Three Zone Waves - Speed Equations

Original Image Source: http://scubageek.com


Three zone wave s speed equations

Notes about the three equations Distance

- Wave Period is always constant : T is independent of d.

As a result, in Deep Water the wavelength “L” is constant

and T is constant, so the Speed is also constant.

- In shallow water L decreases as the square root of

Depth, but “T” remains the same; so the wave speed

decreases as the square root of gravity x water depth.

- If we substitute the values of Wavelength, Depth, and

T = 10 mins, d=4000m, L =200000m for Tsunami Waves,

we find that Relative Depth is d/L < 0.05 or d/L < 1/20

which Mathematically classifies them shallow water waves.

Three Zone Waves - Speed Equations


Computer modeling waves

Computer Modeling of Waves can be used as part of Distance designing breakwaters, marinas, light houses, oil rigs, ships,

tourist resorts, water fun parks, and artificial surf reefs.

Computer Modeling Waves

Original Images Source: Google Images


Computer modeling variables

Computer Modeling - Variables Distance

RCPWAVE Computer Model Variables:

Source: US ArmyCoastal Engineering Manual


Computer modeling equations

Computer Modeling - Equations Distance

RCPWAVE Computer Model Equations

Source: US ArmyCoastal Engineering Manual


Computer modeling waves1

Real Wave Equations like REF/DIF1 are programmed into Distance

Computer Apps, where we can add bathymetry data, and then adjust Equation Parameters, and view resultant effects.

Computer Modeling Waves

Original Images Source: Google Images


Effect of sea floor

The shape of the Sea Floor, (called “Bathymetry”), Distance

plays a big part in forming surfable breaking waves

Effect of Sea Floor

Source: Mechanics of Mavericksat http://www.surfline.com


Effect of sea floor1

Mavericks Surf Break in Northern California Distance

Effect of Sea Floor

Source: http://www.youtube.com/watch?v=KMlZM9kDpMc


Effect of sea floor2

Mavericks – Power, Steep Reef, Parabolic Refraction, Grooves

Effect of Sea Floor

Source: Mechanics of Mavericksat http://www.surfline.com


Effect of tides

Tides change the water depth, and so the bathymetry Groovesat a particular surf break varies over the tidal period.

Effect of Tides

Image Source: http://photo.stellav.ru


Effect of tides1

The sea floor shape may be perfect at high tide and produce Groovesfabulous waves, but at low tide the waves are breaking on a different part of the sea bed resulting in unsurfable waves.

Effect of Tides

Image Source: http://www.ozcoasts.gov.au


Effect of tides2

Surfers use Data arranged in Tables called “Tide Charts”.

Effect of Tides

Image Source: http://www.blueoasisbeachclub.com


Parts of a breaking wave

Catch in the Impact Zone, Ride along the Shoulder Charts”.

Parts of a Breaking Wave

Original Image Purchased by Passy’s World from Dreamstime.com


Catching a wave

Surfer Momentum must Charts”.≈ Wave Momentum

Catching a Wave

Image Source: http://i3.mirrror.co.uk


Catching a wave the equation

When you Paddle, the Forces involved are as follows: Charts”.

Catching a Wave - The Equation

( of surfer )

D = Assistive Drag force of the wave.

Original Image Source: www.abc.net.au

You must produce enough acceleration to get your

speed as close as possible to the wave’s speed.


Professor neville de mestre video

The “ma = P + D” equation is from the following Charts”.video

featuring Professor Neville De Mestre.

Professor Neville de Mestre VIDEO

Source: http://www.abc.net.au/catalyst/stories/2377157.htm


Paddle speed equation

Professor David Charts”.Sandwell’s Equation

Catching the Wave on a Surfboard (Aerial View)

Paddle Speed Equation

Original Image Source: http://topex.ucsd.edu


Types of surfing waves

Based on Size, there are four main types of Charts”.Breaking Wave associated with Surfing.

Types of Surfing Waves

Original Images Source: Google Images


Geometry of the tube wave

A Geometrical Ratio is used to Describe Tube Waves Charts”.

Geometry of The Tube Wave

Original Images Source: Google Images


Ocean depth breaker height and wave speed

Ocean Depth, Breaker Height, and Wave Speed Charts”.

David Sandwell – http://topex.uscd.edu


Dropping in

It is impossible to paddle at the speed of big waves Charts”.

to catch them directly; and so “Dropping In” is used.

Dropping In

Image Source: http:govisitcostarica.com


The other dropping in

“Dropping In” also means stealing another surfer’s Charts”. wave by not giving way via the “Inside Rule”.

The other “Dropping In”

Source: http://www.youtube.com/watch?v=S8f9HVvezMQ



Speed of standard wave types

“Dropping In” produces these bottom of wave Charts”.

speeds for the four standard wave types.

Speed of Standard Wave Types



Surfing the wave

Riding the Wave - up and down and turn around Charts”.

Surfing the Wave

Image Source: http://picasaweb.google.com


Surfing the wave1

Examples of Riding the Wave and Charts”.Manoeuvres

Surfing the Wave

Source: http://www.youtube.com/watch?v=O_oADJ961vo


Surfing giant waves

Very high speed only allows basic Charts”.manoeuvres.

Surfing Giant Waves

Source: http://www.youtube.com/watch?v=O9crPOB_9tE


Tow in surfing for huge waves

A Jet Ski Tow-In gives enough speed to “safely” Charts”.catch gigantic size fast waves. (35mph / 66km/hr)

Tow-In Surfing for Huge Waves

Source: http://www.youtube.com/watch?v=3oS_28utt2Y


Biggest wave ride 100 ft 30m

Nazarre Charts”. Portugal – Surfers : Brazilian and American

Biggest Wave Ride – 100 ft / 30m

Image Source: http://i.telegraph.co.uk


Nazzarre portugal bathymetry

Huge Underwater Canyon that the water is channeled Charts”.

Along, but then suddenly bottoms out near the shore.

Nazzarre Portugal – Bathymetry

Image Source: http://i.ytimg.com


Other surfing statistics

Other Surfing Statistics Charts”.

Original Image Source: science.kennesaw.edu


Artificial surfbreaks

Narrowneck Charts”., Queensland, AustraliaCables, Western Australia, AustraliaPratte’s Reef, El Segundo, California, Los AngelesBagarra, Queensland, Australia

Artificial Surfbreaks

Mt Manganui, NZBornemouth, UKKovalam, India

Image Source: http://surfspotsmap.com


Artificial surfbreaks1

Artificial Reef in Charts”.Kovalam, India

Artificial Surfbreaks

Source: http://www.youtube.com/watch?v=mGYuj-Ow1rk


Continuous waves

Can occur in Rivers and Water Charts”.Parks

Continuous Waves

Source: http://www.youtube.com/watch?v=ljVbieeFn14


Surf aid mathematics resorces

Surf Aid Mathematics Charts”.Resorces

http://www.surfaidinternational.org/schoolsprogram


Mathematics of surfing2

Mathematics of Surfing Charts”.

Complete details of all material covered

in this presentation can be found in the

“Mathematics of Oceans” lesson on the

Passy’s World of Mathematics Website:

www.passyworldofmathematics.com


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