Passy's World of Mathematics

Mathematics of Ships at Sea

Posted on December 4, 2013 by Passy

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

This ship is the “Blue Marlin”, one of the most amazing vessels to ever to sail the seas.

The Marlin can carry an incredible 75,000 tonnes.

It reaches a top speed of a sedate 13 knots, powered by a gigantic 17,000 horse power diesel engines, with a crew of 24.

(Note 13 Knots = 13 x 1.852 = 24 km / hr).

Steering it is said to be like controlling a floating office block.

The barges pictured above – which weigh a total of 60,000 tonnes – were exported from Korea to be completed in Rotterdam.

No crane in the world is big enough to lift these types of cargoes onto the Marlin’s deck, and an ingenious method is used to load the cargo.

The deck is submersible and can disappear under 13 metres of water when the Marlin’s ballast tanks are pumped full of water.

Boats, oil rigs and ships can then be floated on before the deck is raised again by emptying the ballast tanks.

There is a great article about the Blue Marlin at the following link:

Click here for Daily Mail Blue Marlin Newspaper Article

The following three minute video shows a giant oil rig that was built in Korea being loaded onto the Blue Marlin for transport to the USA.

Obviously there is some very exact mathematics involved here, because ships are designed via Archimedes Mathematical Principle to float rather than sink!

In this lesson we look at some of the Mathematics of Ships.

This includes Archimedes Floating Principle, as well as the Geometry of Hulls and Stabilisers.

We also look at Ship Stability, the Plimsoll Line, and the Mathematics of Loading Ships with Cargo.

How Ships Float

Image Source: http://twistedsifter.wordpress.com

Today’s colossal size cruise ships, such as the “Oasis of the Seas” and the “Allure of the Seas”, remain afloat thanks to buoyancy and displacement.

Ships are designed to displace the amount of water equivalent to their own mass.

The ocean meanwhile pushes up and keeps the ship afloat, or buoyant.

In other words, the pressure of the water pushing up on the bottom (or hull) of a cruise ship counteracts the downward force of the vessel’s gravity, thereby making it float on the water’s surface.

(Basically Positive and Negative Integer values cancelling themselves out to zero).

This basic idea is often referred to as “Archimedes’ Principle”.

According to this principle, an item floats when it displaces more than its own weight of the liquid that it’s immersed in.

The surrounding fluid pushes back with a force equal to that of the amount displaced, so that the two opposing forces are equal and the object floats.

A Ship will float as long as it weighs less than the volume of water that its hull displaces.

The following one minute video explains Archimedes Principle

Tall Cruise Ships

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

However, there are some concerns about cruise liners becoming too tall.

In the past, there was a reasonable and safe ratio between a vessel’s draft (below the waterline) and air draft (above the waterline).

These days cruise liners seem to be getting designed primarily as multi-storey Hotels, and secondly as actual ships.

These cruise ships have lost the traditional proportions between what’s below and above the waterline, making the vessels dependent on stabilizers not only to battle through rough weather, but also to stay upright with only slight to moderate breezes.

“Stablisers” are discussed in the next section.

The following link is to an article about ships getting too top heavy and too tall.

Click here for Cruise Liners Article

The following 46 minute documentarty viceo is about the world’s largest cruise shipe “The Oasis of the Seas”.

Ship Stabilisers

Image Source: http://sonnyv.smugmug.com

The purpose of stabilizers is to reduce the roll (i.e., the sideways motion) of a ship.

Today, all modern cruise ships have stabilizers. Most ships have two stabilizers, one on each side of the ship. Larger ships like Cunard Line’s Queen Mary 2 and Royal Caribbean’s Voyager, Freedom and Oasis class ships, have four stabilizers, two on each side.

The stabilizers are shaped like airplane wings and extend out from the side of the hull in a perpendicular fashion when in use. They can pivot up and down like the ailerons on an airplane’s wings. Consequently, as the water flows over a stabilizer it can be turned upwards or downwards to exert dive or lift.

When a sensor detects that a wave is pushing the ship one way, the ship’s systems automatically pivot the stabilizers so as to exert pressure in the opposite direction. Captain William Wright of Royal Caribbean reports: “they eliminate about 85% of the roll of the vessel.”

When the ship is in port or when the seas are calm, the stabilizers can be folded back hydraulically into compartments along the side of the hull. Stabilizers can be deployed independently and so the ship’s officers have the option of deploying the stabilizers on one side of the ship or on both sides to suit the sea condition.

Stabilizers do create some drag and thus can in theory reduce speed and fuel efficiency. However, any such loss has to be balanced against the savings in speed and fuel efficiency resulting from reducing the ship’s motion.

The following diagram shows how the lift works for various stabiliser positions.

Image Source: http://www.uq.edu.au

Sea Sickness

Image Source: http://toohighortullo.wordpress.com

The purpose of cruise ship stabilizers is to reduce the side to side rocking motion of the ship.

They help a ship move more smoothly, which cuts down the chance of seasickness for passengers.

When there is a great deal of movement, it can cause a discrepancy between what a person sees and what her inner ear senses. This is what causes seasickness. The smoother the ride, the less chance for this to happen.

Stabilising Pool Tables

Image Source: http://images.dailydawdle.com

How do pool tables stay level on cruise ships?

Ships, and even some large yachts, have pool tables that are kept level when the ship has a slight roll by specially made gimbals.

These keep the pool table from rolling with the ship or yacht in mild conditions.

This makes the pool table self-levelling.

The pool table rests on gimbals and not just table feet on a floor or deck.

Gimbals are also often used on the stoves and cooking tops on ships and yachts.

A Gimbal consists of three concentric rings – the following picture shows what a basic pool table gimbal looks like:

Image Source: http://productions.andrewgentle.com

The following web page has a video of a pool table that uses gyroscopes and computers to create perfectly accurate self levelling.

Click the Link below to watch a Video of a Self-Levelling Pool Table.

Click Here for Pool Table Video

Swimming Pools on Ships

Image Source: http://i47.photobucket.com

What about swimming pools on ships ?

How do they endure stormy weather where the ship rolls side to side ?

Quite simple really – They have a Pool inside a Pool as shown in the following YouTube clip

However for passenger safety, they usually close the pool during storms on ships.

Geometry of Ship Hulls

Image Source: http://www.shipcruise.org

(Click to Enlarge the above Image on a new Screen)

A ship’s hull is the body of the ship, which sits below the main deck.

Through years of trial and error engineers have found that making the hull rounded, wide and deep, helps disperse the weight of the ship across the body of the ship.

Large cruise ship hulls are shaped like a squarish shaped letter “U”

This design allows water to flow away from the vessel, dissipates drag, and facilitates a smooth ride.

Hulls Inside Hulls

In addition to providing passengers with a fluid ride, the hull helps protect a cruise ship so that it doesn’t sink.

Icebergs, reefs, sandbars, and other large, sharp objects can rip apart a ship’s outer layers.

To prevent a major catastrophe shipbuilders typically use extra-strength steel and insert double hulls, or a hull inside a hull,
as an extra precaution.

Mega cruise ships also stay afloat thanks to bulkheads.

These vertical watertight dividers are installed throughout the interior of the hull and help keep damaged ships afloat by containing incoming water into compartments.

By containing the gushing water, the ship should not flood and sink.

Bulbous Bow

Most large ships have a bulb like protrusion under the water line which pushes out a bow wave to partially cancel the waves hitting the bow and provide “cutting through” ability.

However, recent design changes to save fuel consumption costs on large oil tankers has actually changed the traditional ship hull shape, and totally removed the Bulbous Bow.

The following video explains that the designers tried out over 40,000 different hull designs to come up with the final new design.

Prior to the use of computer modelling, it would have been impossible to manuallu do the necessary calculations for 40,000 different designs without it taking years and years of work!

Ballast Water

Image Source: http://www.netpeckers.co.in

Mathematically, there is no real limit to how big a ship can be, as long as it fits through the waterways it sails, or the harbors it enters.

The “Carnival Destiny”, for example, is too large to fit through the Panama Canal.

Of course, the larger a ship gets, the more passengers and cargo it can carry.

This additional weight does not put the ship in danger of sinking.

The extra weight pushes the ship deeper into the water, which displaces more water and increases the buoyant force.

Balast tanks are filled in order to add weight to the ship once cargo has been discharged, and improve its stability.

When a ship is sailing with only Ballast Water, and no cargo or passengers, it is called a “Light Ship”; as in it is light in terms of weight.

In some extreme conditions, ballast water may be introduced to dedicated cargo spaces in order to add extra weight during heavy weather or to pass under low bridges.

Effect of Ballast Water

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

Most of the time, extra weight actually helps a ship remain stable.

The weight gives the ship a lower center of gravity – the point around which the weight of an object is concentrated.

This means the pull of gravity is greater near the base than the top, so the ship is less likely to topple over.

To stabilize a ship, shipbuilders place the vessel’s heaviest machinery and equipment, like the engines and fuel, at the lowest level, below the waterline.

As the ship’s load lightens, from using up fuel, for example, the ship takes in ocean water for ballast, or added stability.

But the water needs to be in the right place on the ship.

Too much water in the wrong place can make a ship sink.

Water entering a ship’s lowest level can be disastrous; especially if the water fills up only one end of the ship. That’s what happened in the sinking of the Titanic in 1912.

Image Source: http://upload.wikimedia.org

The Normandie, was an ocean liner built in the early 1900s.

While the ship was docked in New York Harbor in 1942, a fire broke out and Firefighters sprayed so much water on the ship’s top deck that the ship became top-heavy.

The Normandie rolled over and sank, right at the dock.

The Safe Loading Level

Image Source: http://ebrd-stories.com

Ships have a red bottom painted onto them, and when a ship is sailing without cargo, the water level will be at the top of this red line boundary along the length of the ship.

A ship can be sailing out of the water partially exposing the red bottom without any risk of sinking, but might be less stable with more side to side roll occuring in big seas.

It is important to realise that not all water weighs the same, and this effects buoyancy.

Freshwater is less dense than sea water, and so a ship will sink down lower in the water, when it sails into a freshwater port.

So when loading in a fresh water port, especially in the tropics (warm water is less dense than cold water), the ship will be lower in the water, (but will not sink).

Then when it sails out into tropical sea water it will rise up higher out of the water, because sea water is more dense (heavier than fresh).

Ships also rise up out of the water as they consume fuel, and their fuel weight consequently decreases.

The safe loadng level in a Port involves making use of markings on the ship’s hull which are called “The Plimsoll Line”.

The Plimsoll Line

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

The Plimsoll Line was originally a long horizontal line painted on the side of merchant ships.

When a ship was loaded, the water level was not supposed to go above the line.

However, the water could reach different parts of the line as its temperature and saltiness varied with season and location.

The basic symbol, of a circle with a horizontal line passing through its centre, is now recognised worldwide.

The Oxford Companion to Ships and the Sea defines the Plimsoll Line as:

A mark painted on the sides of British merchant ships which indicates the draught levels to which a ship may be loaded with cargo for varying conditions of season and location. The Plimsoll Mark shows six loading levels, those which may be used in tropical fresh water; fresh water; tropical sea water; summer, sea water; winter, sea water; and winter, North Atlantic, for vessels under 100 metres (330 ft) in length.

The maximum load capacity of a ship is determined by loading the ship up so that the Plimsoll Line styas just above the water.

Sinking is the most dramatic form of overloading. Before the ship sinks, it will sit so low in the water that even the smallest wave will wash over its deck, flooding the ship and possibly causing it to sink. To avoid this, ships are loaded so that their decks are well above any wave the ship is likely to encounter.

This safe height is identified by the Plimsoll line. The Plimsoll line usually is marked by a color change on the hull that differentiates the portion that is meant to be underwater from the portion that is meant to remain above the water. An empty ship floats high in the water. As cargo is loaded, the ship sits lower. Eventually only the portion that is supposed to remain above the water can be seen. Most prudent owners will not load cargo beyond this maximum.

The original “Plimsoll Mark” was a circle with a horizontal line through it to show the maximum draft of a ship. Additional marks have been added over the years, allowing for different water densities and expected sea conditions.

Letters may also appear to the sides of the mark indicating the classification society that has surveyed the vessel’s load line. The initials used include AB for the American Bureau of Shipping, LR for Lloyd’s Register, GL for Germanischer Lloyd, BV for Bureau Veritas, IR for the Indian Register of Shipping, RI for the Registro Italiano Navale, NK for Nippon Kaiji Kyokai, and NV for Det Norske Veritas. These letters should be approximately 115 millimetres in height and 75 millimetres in width.[7] The Load Line Length is referred to during and following load line calculations.

The letters on the load line marks have the following meanings:
TF – Tropical Fresh Water
F – Fresh Water
T – Tropical Seawater
S – Summer Temperate Seawater
W – Winter Temperate Seawater
WNA – Winter North Atlantic

The following five minute YouTube video which explains the Plimsoll line markings including summer winter tropical etc.

It also covers the the 1/48th of Draft Rule for Winter and Summer Markings.

Fresh water is considered to have a density of 1000 kg/m³ and sea water 1025 kg/m³.

Fresh water marks make allowance for the fact that the ship will float deeper in fresh water than salt water. A ship loaded to her Fresh Water mark in fresh water will float at her Summer Mark once she has passed into sea water. Similarly if loaded to her Tropical Fresh water mark she will float at her Tropical Mark once she passes into sea water.

The Summer load line is the primary load line and it is from this mark that all other marks are derived. The position of the summer load line is calculated from the Load Line Rules and depends on many factors such as length of ship, type of ship, type and number of superstructures, amount of sheer, bow height and so on. The horizontal line through the circle of the Plimsoll mark is at the same level as the summer load line.

The Winter load line is one forty-eighth of the summer load draft below the summer load line.

The Tropical load line is one forty-eighth of the summer load draft above the summer load line.

The Winter North Atlantic load line is used by vessels not exceeding 100 metres in length when in certain areas of the North Atlantic Ocean during the winter period. When assigned it is 50 millimetres below the winter mark.[8]

The calculation of the Fresh Water load line is quite complicated.

The Fresh Water load line is an amount equal to \tfrac{\triangle}{4T} centimetres above the summer load line where \triangle is the displacement in metric tonnes at the summer load draft and T is the metric tonnes per centimetre immersion at that draft.
In any case where \triangle cannot be ascertained the fresh water load line is at the same level as the tropical load line.
The position of the Tropical Fresh load line relative to the tropical load line is found in the same way as the fresh water load line is to the summer load line.

Information Source: http://en.wikipedia.org/wiki/Waterline

That’s it for our look at some of the Geometry and Mathematics associated with ships.

If you want to look further into some of the Hydrodynamics and Mathematics of ships at sea, then here are some weblinks:

Centre of Buoancy and Centre of Gravity

Ship Hull Measurements Diagram

Mathematics of Hull Shape

MAV Conference PowerPoint

To download the PowerPoint Presentation from the Mathematics Association of Victoria Conference, click the link below and save the file to your computer.

Mathematics of Ships at Sea

Click the link below to download this 4MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/ShipsMathPPTv1.pptx

AMC Teacher Resources

The Australian Maritime College has an excellent set of free resources for Mathematics Teachers.

Among these are some very good mathematical exercises related to Ships.

The AMC is planning to add more online interactive lessons, but as of late 2013 the current the interactive lessons are as follows:

A Study of Similar Vessels (an application of Curve Fitting) – 5mins

Wind Farm Feasibility Study (an application of Probability) – 12mins

Vessel Speeds in Waves (an application of Differentiation) – 5mins

Ship Hydrostatics (an application of Integration) – 12mins

Wave Refraction (an application of Trigonometry) – 12mins

Ocean Waves (an application of Superposition) – 5mins

Scaling Laws (an application of Algebra) – 8mins

Go to the following link to find their Resources page:

http://www.amc.edu.au/why-study-maths

To get the workbooks and other resources, it is necessary to register your school and complete an order form online.

These resources are free to Australian schools, and if you would like to see what some of the workbook exercises look like, then check out the following link:

Click here for AMC Online Maths Workbook PDF

The AMC can be contacted about this program at the following email address: whystudymaths@amc.edu.au

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Surfboard Geometry and Design

Posted on December 4, 2013 by Passy

Image Source: http://wikimedia.org

Surfboard Design has a number of different geometries associated with the overall size and shape of the surfboard and its underside fins.

For a high performance board capable of doing very tight turns, and spectacular skateboard like manoeuvers, there is a specific geometrical setup of the component parts of the board.

Here is a short two minute video of World Champion Kelly Slater showing some spectacular modern day manoeuvers on a very basic small wave at Bells Beach in Australia.

[youtube=http://www.youtube.com/watch?v=GwVnq_t-Wc4&w=550&h=309]

In this lesson we look at Modern Surfboard Design, and its associated mathematical geometry.

First we look at the main types of Surfboards, and then we examine the specific geometries of the component parts. We also look at computerised shaping and manufacture of surfboards.

Types of Surfboards

Image Source: http://dscuento.com

For a large Poster of the Main Surfboard types, click the poster image below to enlarge it to full size on a new screen.

Image Source: http://sandiegosurfingschool.com

The Surf Science website at http://www.surfscience.com/ has the following classification scheme for types of Surfboards:

Shortboard
Fish
Longboard
Fun Board
Gun
Hybrid
Other

In the sections which follow, we examine each of these surfboard types.

The Shortboard

Image Purchased by Passy World from Dreamstime.com

The shortboard surfboard design reinvented high performance surfing in the 1970s, and continues to allow surfers to push the boundaries of the sport.

The shortboard is usually found at a length of under seven feet long.

It has a greater amount of “rocker” curve on its top deck, which allows it to surf in critical sections of the wave.

It is most commonly seen with sharp noses, thins rails, and either three or four fins.

The Fish

Image Source: Google Images

The Fish design became popular in the 1970s and originated from the knee-board.

It is typically found shorter and wider than the shortboard, and because of this it works very well in small mushy surf.

A fish is a relatively flat surfboard with only a small amount of rocker deck curve, allowing it to paddle very well and carry speed through flat sections of a wave.

It also has an angular swallow tail which helps disperse bumpy water away efficiently.

The Longboard

Image Source: http://surfingaustralia.com

The Longboard is the oldest and most traditional surfboard design.

Longboards typically range from 8 feet to 12 feet long, at least 2.5 inches thick and twenty inches wide.

This extra volume allows them to paddle incredibly well and catch waves with ease.

A beginner should start on a longboard shape to learn wave selection, paddling technique, and turning basics.

Longboards are great boards for small 1 to 1.5 metre high waves, (3ft to 5 ft surf).

Expert longboarders are known for their smooth style, surfing in a very smooth and fluid manner on waves they can ride right into shore.

Another exciting aspect of longboards is running up and down the board and “nose riding”, or “hanging five” and “hanging ten”.

The following six minute video shows some brilliant longboarding by local surfers at Maresias in Brazil.

[youtube=http://www.youtube.com/watch?v=xNMLOCNo4BY&w=550&h=309]

The Funboard

Original Image Source: Google Images

A funboard is a larger surfboard with a lot of volume via its thickness and length, but not quite as much as a longboard.

They typically range from 6 feet to 8 feet long.

The funboard is a perfect for surfers who want to transition to a smaller board, while still maintaining additional paddling power and stability.

The funboard is similar to the “Malibu” and can come in a variety of tail shapes, nose shapes, and foils.

The Gun

Original Image Source: http://wikimedia.org

The Gun is the surfboard that needs to be used when the waves are huge.

A typical gun ranges in length from 6 ft 6 inches to 10 foot.

The extra length allows a surfer to gain enough speed while paddling to catch the huge and fast moving waves, (eg. 10 foot, 15 foot, and higher waves).

It is designed for big drops and handling very high speeds with good control.

A gun will have a great deal of rocker, (eg. the deck curves dramatically upwards), and will most likely have three of four fins.

There is a very good article about Gun boards at the following link:

http://www.grantmillersurfboards.com.au/boards/gun

The Hybrid

Image Source: http://surfingaustralia.com

A hybrid surfboard design mixes a few design elements from different board types.

One typical hybrid is a larger and wider version of either the Shortboard or the Fish.

Other hybrids mix characteristics and performance of a shortboard with the stability and tail design of a fish.

This is a great board for medium size days or even just a heavier surfer who wants to surf a smaller style board.

Other Types / SUP

Original Image Source: http://www.seabreeze.com.au

Other Types of surfboards mainly include the “Stand Up Paddle Board” or “SUP” design, as well as the “Tow In” board.

Some other less common surfboard designs are the “Bonzer”, “Mini Simmons”, and “Alaia”.

Each of these types excels in specific conditions.

The Stand Up Paddle Board can catch waves that normal surfers cannot, because using its big paddle produces a 4:1 ratio of paddling power compared to a surfer paddling with their arms.

Think of the types of Surfboards, as being like types of cars.

The Gun is like a big long cigar shaped high speed Drag Racer, the Longboard is like a big stable slow turning 4WD, the Hybrid is an SUV like the Ford Territory, the Shortboard is like a high performance sports car, and the Fish is like a rally car suited for good performance in bumpy conditions.

Choosing a Surfboard

The correct surfboard for your body is usually the one that is one hand palm above your head in length.

Choosing a board also depends on your weight, heavier people need a longer and wider board so that the foam volume is greater.

In addition, choosing a board depends on the type of waves and surfing you want to do, as well as your experience/commitment level.

For example, if you are a casual surfer who rides small-to-medium waves at your local beach break, you can choose to use a longboard or a funboard/malibu.

However, if you are an experienced serious surfer, who wants high manoeuverability, you can use a Fish or a Short Board.

Here are some sizing tables for a range of different board types.

Original Image Source: http://www.surfertoday.com

There is a great sizing table for all board types at the following link:

http://www.surfertoday.com/board-size-chart/surf

The following video talks about choosing a surfboard.

[youtube=http://www.youtube.com/watch?v=_SU5cL7w1tE&w=550&h=309]

This next video is a great video about the types of boards beginner surfers should choose.

[youtube=http://www.youtube.com/watch?v=fLSP-Qr01NI&w=550&h=413]

After carefully selecting the right size and design, it is important to know the main features of a surfboard.

A rounder template/outline will create wider turns with longer curves and more laid-back surfing.

Width and length are very important issues. Wider boards offer more fluctuation and are the perfect choice for very small fat waves.

Wider boards also have more stability, which is important for beginner surfers.

Rocker (the upward curve of the deck) is considered the most important feature in a surfboard, by many shapers who design and make boards.

For small mushy surf, people often use “Fish” type boards, which are a flat looking board with very little rocker.

In terms of tail shapes on surfboards, the rounded pin tail, round tail, and squash tail are very common because they hold the surfboard quite well in all-round surf conditions.

These component parts: Rocker curve, Tail shape, and so on are discussed in detail in the next section.

Parts of a Surfboard

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

The component parts of a Surboard are:

Nose, Rocker Rails, Stringer, Fins, Tail, Bottom Contour, and Deck.

In the sections which follow we investigate the Geometry associated with each of these component parts, and how this effects the board’s performance characteristics.

The Nose

Image Source: http://surfboardexchange.com.au

The nose is the very front of the surfboard.

It’s Geometric shape plays a significant role when dropping into waves.

Noses vary in width, curvature and thickness.

There are two basic types of surfboard nose: The “Pointed Nose” and the “Round Nose”.

The Pointed nose is found on most high performance boards, like short boards and fish.

The pointed shape adds more rail curve compared to a round nose longboard, which is handy for dropping into big waves, and paddling through chop.

It is also easier to duck dive under a wave with a pointed nose because the nose is lot smaller and less buoyant than a round nose.

The Round Nose allows for the front of the board to have more surface area, giving the board more stability.

Round nose boards make it easier to paddle, glide, and plane on the surface because it helps to keep more of the surfboard out of the water.

Longboards typically have a round nose for the purpose of allowing the surfer to nose-ride.

The Rocker

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

A Surfboard has a bow or upward bend in it; eg. the deck is not flat along the entire board.

This bow, or upward banana-shaped curving, is called “The Rocker”.

Actually there are usually two Rockers on a typical board – the Nose Rocker and the Tail Rocker.

The in between part in the middle of the board, which includes most of the board’s deck is essentially flat.

The “Fish” design has a very flat deck, and so do most longboards; but other boards have a larger amount of curving upwards, (or more “rocker”), on their decks.

The Stringer

Image Source: http://www.hydroflex-surfboards.com

Traditionally, all surfboards required a stringer (stiffener) placed down the middle of the board to strengthen the core and add rigidity.

Without a stringer, the core would be too flexible and too prone to breakage.

Stringers have usually been wood, but carbon fibre has been used for some boards in recent times.

The Stringer has the two sides of the board foam laminated onto it, and comes as part of the initial “blank” that a shaper builds a board from.

Parabolic Stringers

This is an additional stringer system where two stringers are placed along the perimeter of the rails.

They are typically made of wood or carbon fiber and create “memory flex”, which dictates how quickly the shape can bend and bounce back to its original shape.

“Memory Flex” from Parabolic Stringers increase the board’s handling, and enable faster and tighter turning.

The added flexibility on the perimeter of the board allows you to lean into the stringer in turns as opposed to the foam.

This bends the stringer and shoots you out of the turn as it flexes back.

This results in more speed, acceleration and torque through each turn.

Multiple Stringers

Image Source: http://findthebest.com

Longboards often have multiple stringers for the purpose of adding strength.

This is needed in a lot of cases because the larger surface area of a longboard can flex and wobble too much.

Additional stringers will minimize this effect.

Stringerless Boards

Some surfboards have no stringers.

Epoxy and Carbon Fiber boards often have a stringerless design.

Their materials and sandwich construction give the board added stiffness so a stringer is not needed.

Read more about epoxy boards at the following link:

http://www.tactics.com/info/epoxy-surfboard-construction

Surfboard Rails

Rails are a critical part of surfboard design and are the edge of a surfboard, where the deck and the bottom meet.

“Soft” rails are rounded with no hard edges. The transition from the rail to the bottom of the board is very smooth.

A “Hard” rail, (also called a down rail), is more squared off, sloping down in a more distinct manner and may form a corner or edge with the bottom.

The softer the curve of the rail, the more the surfboard will hold its “track” in the wave, but with reduced turning ability.

Softer rails are found on more traditional long-boards, but harder edges can be used in certain places for drive and manoeuverability.

Hard Rails produce boards which do a tighter turns and have explosive acceleration.

Modern short-boards have a combination of both soft and hard railing, and they are kept lower to disguise the thickness and give better turning.

Because the thickness of the board varies from nose to tail, forming the rails along the edge of the board can be quite complex, as shown in the following cross-sectional measurements diagram:

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

There is a great detailed article all about rails at the following link:

http://www.surfscience.com/topics/surfboard-anatomy/rail/ignore-the-rail-at-own-risk/

Tail Shapes

Original Image Source: http://www.surfertoday.com

The different tail shapes on surfboards are: Squash, Square, Thumb, Rounded Pin Tail, Pin Tail, Swallow Tail, Bat Tail, and Wing Tail.

Tail shape influences the hold and release on the surface of the wave.

Water is a “sticky” liquid, and follows the lines of the board when it flows off the back of the board.

The basic hydrodynamics are that rounded curves hold water flow, whereas corners allow water to break away.

A round or pin tail will hold the water longer, making it stable in bigger surf and it also allows rounder and smoother turns.

A square or angular tail releases water quickly, making it looser, skateier and snappy.

Angular shaped tails like the “Wing”, create more pivot, which allows for sharper turns.

The Square tail is typically found on “Longboards” and historically is the earliest surfboard tail design. The square tail is wide and helps ad stability to a surfboard.

The Squash Tail is a square tail with an edgier shape which allows for quick release. This makes the board very responsive and easier for sharper and looser turns.

The Squash shape allows for more width, which increases the surface area in the tail. More surface area means more lift, or easier to plane and maintain speed.

This makes the Squash Tail good for slower spots of a wave, and is the most common type of tail found on “Shortboards”.

The Swallow Tail is ideal for smaller waves, which is why “Fish” boards typically have a very pronounced swallow tail.

Pin tails are typically found on big wave “Gun” boads. This is because tracking and control in high speeds is far more important than manoeuverability when dropping down the face of huge waves.

There is an excellent article all about Surboard Tail Types at the following link:

http://www.surfscience.com/topics/surfboard-anatomy/tail/basic-tail-shapes/

Surfboard Fins

Image Source: http://mpora.com

Underneath every surfboard there is as least one “fin”.

(They are called a “Fins” because they look like Dolphin and Shark Fins).

Fins are required to turn the board, to stop sideways slipping, and to hold the board onto the face of the wave.

In the early days of surfing, boards just had one reasonably large fin to enable the board to follow the line you wanted (“tracking”), and not slide out sideways underneath you when you turned.

In today’s high manoeuver surfing of tight turns, 180’s, snap backs, and skateboard type tricks, fins and the number of them underneath a board has become a far more complex matter.

Today there is a lot of engineering and design that goes into making surfboard fins and configuring them onto different types of boards.

Fins have flex built into them to cushion their action, and are “foiled” or curved like an aeroplane wing to create pressure differences on each side of the fin.

The idea of the pressure difference, is to have the fin have a low pressure side next to the wave, so that the board is sucked onto the face of the wave and has “grip” as it tracks through the water.

Video About Surfboard Fins

The following seven minute video provides a good tutorial about the properties and behaviour of surfboard fins.

[youtube=http://www.youtube.com/watch?v=nm57d4C9rZs&w=550&h=413]

Fin Characteristics

Original Image Source: http://www.surffcs.com.au

The key characteristics of surfing fins are the following:

– Base Length
– Depth
– Surface Area
– Sweep or Rake
– Foil

Base is the length between the leading and trailing edge where the fin meets the board.

Base Length is primarily linked to drive. Fins with a longer base length offer substantially more drive and acceleration.

Depth is the distance the fin penetrates into the water. Depth directly relates to hold.

The greater the depth the more hold, the shorter the depth the more a board will slide and release.

The Depth of the fin cannot be excessively large, or else the fin will create excessive drag and slow the board down.

Area is the total surface area of the fin.

Longboards have the option of installing a large rounded rectangular fin which has a large surface area, and provides stability for nose riding.

Sweep and Rake

Surfboard fins typically are heavily raked or swept back from the vertical, but there is a raking limit.

Sweeping back encourages downwash, the situation in which water flows from one side of the fin to the other.

Heavily raked fins tend to stall during hard turns, cause the fin tip downwash creates a large vortex behind the fin as it travels though the water.

Fins with a large rake give the surfboard a tighter turning radius, but don’t offer as much stability.

Fins with a small rake, like a longboard fin for nose riding give the surfboard more stability but a larger turning arc.

A balance needs to be made, determined by the type of waves the surfer usually rides and the style of board they have.

Rake is discussed later in this lesson in the Fins section.

Foil is associated with the curve of the exterior surface of the fin along its sides.

Foil is discussed in detail in the next section.

Foil on Fins

Original Image Source: http://magicseaweed.com

Foil refers to the shape and geometry of the inside and outside faces of the fin.

Foils directly affect the flow of water over the surface of the fin.

The above photo diagram shows the manoeuverability effect of using fins with different foils.

There are four main types of foils: “Flat”, “Inside”, “50/50”, and “70/30 or 80/20” foil.

Information from the FCS Fins webpage describes these foils as follows:

FLAT FOIL: A flat inside face combined with a convex outside face. The traditional flat sided foil offers an even combination of drive, pivot and hold and provides a very consistent, reliable feel over a wide variety of conditions.

INSIDE FOIL: A sophisticated hydrodynamic foil consisting of a convex outside face, a rounded leading edge and a concaved inside face. Inside foil increases the efficiency of water flow over the surface of the fin adding lift and reducing drag. The result is a fin with more options through increased hold and speed.

50/50 FOIL: A symmetrical foil used on all centre fins where both sides are convex. Even water flow on both sides creates stability and control.

70/30 or 80/20 FOIL: Combines the performance of a centre and side fin offering increased speed, smooth rail-to-rail transitions and a consistent feel in a variety of conditions. (Ideal for all board types and rear fin placements)

The following diagrams compare the manoeuverability of a surfboard across the wave face for a three fin board, and a four fin board, when different foil profiles are present on the fins.

Original Image Source: www.fcs.com.au

Click the above Image to view full size

The following thirty second video explains FCS Fin Ratings in terms of Drive Speed, Turning Pivot, and Wave Hold/Release.

[youtube=http://www.youtube.com/watch?v=mWEpNAgJpZQ&w=550&h=309]

Flex and Cant on Fins

Original Image Source: Google Images

Click to enlarge the above image to full size

Flex refers to the distortion of the fin from its original shape, caused by lateral pressure during a manoeuver.

This is effectively how much side to side movement the fin makes when the tip of the fin is pushed or pulled.

Eg. “Flex” is the amount a fin flexes from the straight line position.

Fins with little flex are more responsive and will have more speed and direct drive.

Fins with flex are more forgiving, easier to use, and offer a “whipping” sensation.

Ideally a fin should have a stiffer base for drive and a flexible tip for release

Cant refers to the angle of the side fins measured from a vertical line perpendicular to the flat bottom surface of the board.

Cant has a direct effect on acceleration and manoeuverability. Less cant produces faster acceleration and a stiffer feel.

More cant will increase manoeuverability and gives the board a loose feel.

The Angle the fins make with the centreline of the horizontal stringer is called the “toe”.

Detailed Fin Measurements

Specifying the Geometrical Design of a Fin requires marking in quite a few measurements:

Image Source: http://wavegrinder.com

Click the above image to go full size.

The above diagram compares a standard surfboard fin with a radical “Wave Grinder” winged keel type fin.

Read more about this specialised fin with winglets at the following link:

http://wavegrinder.com/

Bulbous Bullet Fin

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

The design of this fin draws on lessons learned from the marine engineering of ship bows.

On ships they build a spherical “bulbous” lump onto the ship’s bow below the water line, this makes the ship cut through the water easier by creating its own out of phase wave to negate bow waves.

Just as with the bow of a non-bulbous ship, the traditional surfboard fin creates a wave as it displaces the water in its path.

The resulting turbulence places drag on the surfboard, which slows the board down.

The bulb of the Bullet Fin design greatly reduces this drag by creating a new (primary) fin wave in front of the original (secondary) wave.

This new bulb wave is designed to be nearly 180 degrees out of phase with the original fin wave to subtract its turbulence thus reducing fin drag.

The result is a fin that gets more efficient as you approach the surfboard’s optimal hull planing speed.

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Posted in Math in the Real World, Ocean Mathematics | Tagged computer made surfboards, fcs fin mathematics, geometry of surfing, how fins work, how to make a surfboard, math for surfers, mathematics for surfers, mathematics of fin foils, mathematics of making a surfboard, mathematics of surfboard fins, mav conference ocean mathematics, ocean mathematics, passy world ocean mathematics, surf board math, surf science, surfbaord design math, surfboard design, surfboard geometry, surfboard math, surfboard mathematics, surfboard science, surfboard shapes science, surfboard technology, surfing resources for teachers, surfing science, surfing technology, web design surfboard, what makes a surfboard turn | 3 Comments

Mathematics of Ocean Waves and Surfing

Posted on December 4, 2013 by Passy

In this lesson we look at some of the Mathematics associated with Surfing.

We cover the Mathematics of Ocean Waves, as well as the physics associated with catching and riding a surfable wave.

A knowledge of Mathematics is not required to be a surfer; however the mathematics of waves is used for designing artificial surfing reefs, computer wave modelling for testing ship designs, bridges, and coastal construction, and for converting wave energy into electricity.

In later lessons we look at the Geometry of Surfboards, as well as the Generation of Electricity from Sea 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.

Three factors influence how big these waves are:

– the speed of the wind

– the distance the wind travels over the water, which is called the “fetch”

– the length of time the waves travel for

The amount of energy imparted from Wind to Water is highly efficient, being proportional to the fourth power of wind speed!

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

The biggest surf waves are created by storms a long way from shore, out in the open ocean.

These storms are caused by low pressure to high pressure gradient differences, like the gradients that exist between cold air in Alaska and warm Pacific Ocean Air.

Storm waves start out as huge, choppy waves, and then gradually join together into strong, smooth separated lines of peaks called “swell”.

Large waves with the most energy are formed by strong wind blowing over a long period of time over a long distance (or “fetch”).

Eg. Wave Energy = Wind Speed x Wind Duration x Fetch Distance

These waves will evolve into lines of long waves (called “swell”), which has a long “Period” time between successive waves, so that breaking waves will not interfere with each other.

These longer period waves also travel faster, resulting in them producing greater kinetic energy when they eventually reach a shoreline and break.

These waves also have their energy extending deeper below the surface of the water, resulting in a bigger wave pushing up at the shoreline which has greater refraction bending which further focuses its energy.

Predicting Large Waves

There is an extensive system of open Ocean Buoys which continuosly collect data about how high waves are, (the “Amplitude”), and how long it is between successive waves (called “The Period”).

The buoy system is used for Tsunami monitoring, and also by maritime agencies to alert rough storm sea conditions to ships at sea.

However Surfers also use the publicly available data to predict when powerful waves will reach big wave surf spots such as “Mavericks” in Northern California.

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

Click the above image to view full size.

Here is a graph produced from ocean buoy measurements of one of the largest storm swells to ever reach the “Mavericks” surf break in California.

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

Click the above image to view full size.

Breaking Waves

Image Source: Bells Beach from magicseaweed.com

The shape of the ocean floor and the direction of the wind are the two main factors that cause how a wave breaks or crashes against the shoreline.

The best surfing waves are usually caused by underwater features like sand banks, rocky points or reefs.

To get the hollow tubes that surfers love, the ocean floor needs to have a steep underwater slope.

Waves tend to break more gently and farther out if the slope of the ocean floor is gradual.

When the wind blows from the beach out to the sea, it is called “Offshore” and helps to maintain clean glassy waves, which are better to surf.

When the wind blows from the ocean to the beach (“Onshore”), or across the beach (“Cross-Shore”), it makes the waves messy and choppy.

The key mathematical measurements made on Surf Waves are the following:

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

In water waves, the energy travels but the water does not.

The water particles move in small circular motions as each wave passes by.

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

The size of the circular motion decreases as we get deeper below the wave, and dies out at a depth that is equal to half the wavelength.

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

As shown in the previous diagram, the Energy in the wave is stored between the top of the wave and a depth which is about one half the wavelength.

When the water depth decreases to be come about half the wavelength, then the wave becomes what is known as a “Shallow Water Wave”.

As the water becomes shallower, the wave rises up, it becomes higher, and eventually its potential energy is converted into kinetic energy and we get a breaking wave.

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

Note that in our diagram, we have referred to the speed of the wave as “v”.

However, in most diagrams the speed variable is labelled as “c” for “Celerity” which is the term oceanographers use to refer to wave speed.

Water Wave Equations

As per our previous diagram, there are three main types of incoming wave:

– Deep Water where Depth > L / 2

– Transitional where L / 20 < Depth < L / 2

– Shallow where Depth < L / 20

The shape of water waves is not actually Sinusoidal, it is Trochoidal.

A "Trochoid" can be defined as the curve traced out by a point on a circle as the circle is rolled along a line.

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

The Trochoid shape does approach the sine curve in shape for small amplitudes, found in Deep Water Waves.

However at Transitional water depth the shape is different, with a narrowing of the peaks of the trochoid compared to the sinusoid.

This narrowing or steepening of the peak becomes more pronounced as the wave amplitude increases, as the wave starts to rear up and break.

The Trochoidal shape can be approximated to the shape of the Hyperbolic Tan Function graph, tanh(x) which is shown below.

Image Source: http://functions.wolfram.com

As a result of the wave shape change from approximately sinusoidal, to trochoidal as a wave approaches the shoreline, we end up with three different Wave Speed Formulas, for out three differing water depth wave types.

The Speeds of these three wave types have the formulas shown below, where the Speed is assigned the oceanographer’s name of “c” for “Celerity”.

Original Image Source: http://scubageek.com

Click the above figure to view as enlarged on a separate screen.

Some things to note about these equations are the following:

1) In Deep Water the celerity, (or speed), is independent of water depth, because deep water waves do not interact with the sea bottom. Hence there is no “d” in the Deep Water equations. Their speed is a function of Wavelength, and their wavelength remains constant in the open ocean due to the dispersive nature of water.

2) Shallow water surface waves, on the other hand, do feel the bottom, and slow down as the square root of the water depth. Their speed is not a function of the wave length, but a function of water depth, and the earth’s gravitational force constant.

3) The Wave Period “T” is independent of the water depth. The time between successive crests of waves (The Period “T”) remains constant, irrespective of depth. (See “Constancy of Wave Period” in references below, for more information).

As a result of this constant “T”, in deep water, the wave length is constant, but as waves approach a beach the wave length decreases as the square root of the depth.

4) Tsunami waves behave like shallow water waves, after they break they surge forth (often for long distances), and then recede again back out into the ocean.

Information from the “Scuba Geek” site relating to Tsunamis is as follows:

“Wind-generated waves typically have periods from 1 to 25 seconds, wave lengths from 1 to 1000 meters, speeds from 1 to 40 m/s, and heights less than 3 meters. Seismic waves, or tsunamis, have periods typically from 10 minutes to one hour, wave lengths of several hundreds of kilometers, and mid-ocean heights usually less than half a meter. Because of their long wavelengths, tsunamis often satisfy the criterion for shallow-water waves. For example, when a tsunami with a wave length of 200 km passes over a depth of 4 km (the average depth of the oceans) the relative depth is d/L=.02. Since this is less than .05, this tsunami is a “shallow-water wave”, and its celerity depends only on the water depth.”

5) The Three Zone Water Wave Equations as presented above, represent a simplified overview of real world wave behaviour, compared with the water wave equations used in computer modelling of waves. (We look at Computer Modelling in the next section).

If you need to find out more details and derivations of these equations, click the links below:

Click here for Constancy of Wave Period Proof

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/watwav2.html

http://scubageek.com/articles/wwwceler.html

http://functions.wolfram.com/ElementaryFunctions/Tanh/introductions/Tanh/ShowAll.html

Computer Modelling Waves

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

Computer Modelling of Waves can be used as part of designing breakwaters, marinas, light houses, oil rigs, ships, tourist resorts, water fun parks, and artificial surfing reefs.

If you are in need of a massive amount of water wave Mathematics and Equations, then work your way through the US Army Corps of Engineers Coastal Engineering Manual.

This manual goes into water wave equations in full depth.

The full set of equations for water waves are quite complex, but the good news is that the equations can be programmed into a computer, and then combined with statistics and grid meshing methods to create an interactive water wave model.

The equations only need to be put into the program once, and from there onwards it is simply using the Computer App to adjust variable values and perform investigations.

The type of equations which can be programmed into a computer to do very realistic simulation exercises for Water Waves involves the the RCPWAVE Model:

Here are the definitions of the Terms and Variables found in the RCPWAVE Model Equations:

Image Source: US Army Coastal Engineering Manual

Some of the mathematical equations used in the RCPWAVE model are shown below.

Image Source: US Army Coastal Engineering Manual

The above material is from Chapter 3 of the US Army Coastal Engineering Manual.

If you would like to examine a full copy of the entire manual, then click the link below:

http://140.194.76.129/publications/eng-manuals/

Effect of Sea Floor

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

The shape of the Ocean floor (called “bathymetry”) plays a very big part in whether or not breaking surf waves will form.

Eg. There can be large surf waves at one beach, and then around the point at the next cove, there are not any surfable waves of any reasonable size.

The ocean floor needs to rise up suddenly in the form of a rocky or coral reef, or a sandbar, for surf waves to form.

These reefs and bars create a focus point where waves are produced.

A few places around the world have perfect reef conditions for very big waves to form.

One such place is “Mavericks” in Northern California, where breaking waves can be up to four building storeys high!

Here is a short two and a half minute video showing the huge waves at Mavericks.

[youtube=http://www.youtube.com/watch?v=KMlZM9kDpMc&w=550&h=309]

Multibeam sonar can be used to map the sea floor, and at places such as Mavericks in California, it shows that we also get refraction which further focuses the wave energy, resulting in a very large wave forming at a certain location.

The refraction or bending happens because the part of the wave which hits the reef first slows down a bit, while the other part of the wave still in deeper water keeps going at a faster speed, which causes bending of the wave.

The sea floor at Mavericks has grooves cut into the steep reef, made by icebergs grinding the surface during the last ice age, as well as a fault line which runs through the reef. These grooves also channel the water and add to the power of the wave.

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

There is a great slideshow presentation on Mavericks to watch at the following link.

Note that an advertisement runs on the page while the slideshow loads.

http://www.surfline.com/surf-news/mechanics-of-mavericks_62313/

The slideshow includes information about ocean buoys to measure sea and storm swells for shipping, and the data from these is used by surfers to figure out when a big swell is going to hit Mavericks.

There is a great slide about this in Part 2 of the “Mechanics of Mavericks” slideshow at the following link:

http://www.surfline.com/surf-news/-mavericks-part-two-epic-swells_62537/

The following YouTube video discusses the mathematics and science of the big waves found at Mavericks in California.

[youtube=http://www.youtube.com/watch?v=fP1QK0XCt1U&w=550&h=413]

Surfing and Tides

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

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

The sea floor shape may be perfect at high tide and produces fabulous waves, but at low tide the waves are breaking on a different part of the sea bed which produces unsurfable dumper waves.

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

Tides, which occur from varying gravitational forces between the sun, moon, and the earth, are easy to predict far in advance.

Surfers use Tide Charts to know when various tide conditions producing the best waves are going to occur at their favorite breaks.

Here is a sample Tide Chart for guests at the “Blue Oasis Club” Resort in Bali

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

Click the above image to view full size.

The effect of tides on wave quality differs between surf breaks.

Some breaks can be excellent on a low tide, but can suffer from a drastic drop in wave quality during a high tide, during which the water depth is too great, causing the wave face to break more slowly and with less power.

Other surf breaks may experience the opposite effect and have better wave shape during high tide.

Bathymetry, coastal geography, full moons, and man-made coastal features such as seawalls, harbors, piers, and dredging all impact how a surf break will respond to tides.

Parts of a Breaking Wave

Original Image Purchased from Dreamstime.com

The ideal place to catch a wave is in the “Impact Zone”.

The best place to surf along the wave is in the Shoulder’s clean and smooth face; however it is sometimes necessary to turn into the whitewater occasionally to pick up some more power.

Moving up and down the wave also allows gravitational potential energy conversion to kinetic energy of movement.

The surfer also needs to have a standing position which maintains their centre of gravity in balance.

Catching and staying on a wave is a delicate balance of a number of mathematical factors.

In the next sections we look at the Mathematics of paddling a board to “catch” a Wave, and then the potential to kinetic energy exchange as the surfer commences riding the wave.

Catching a Wave

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

The surfer needs to catch the wave just as it is breaking, which is when it has maximum velocity (speed).

In order to catch the wave, the surfers’ momentum must be equal to the wave’s momentum.

The surfer has to start paddling as the wave approaches, and most people agree that the required number of paddle strokes to catch a wave is five strokes.

The fundamental mathematics of Surfing depends on Newton’s equations where mass times acceleration of the surfer, must be equal to the propulsive force produced by paddling, plus the “drag force” due to wave particle motion.

When you Paddle, the Forces involved are as follows:

Image Source: http://www.abc.net.au

You must produce enough acceleration to get your
speed as close as possible to the wave’s speed.

The Drag component of this equation expands out to something quite mathematically complicated, but basically the drag is an assistant force due to the circular movement of water particles in the wave, and actually gives the swimmer/surfer some extra speed.

Think of the wave drag as being like the current that helps you when you swim downstream in a river.

The end result of the Mathematics underlying the ma = P + D equation, is that the surfer must paddle fast enough to attain equal to, or a little bit less than, the actual speed the wave has.

If the swimmer or surfer does not achieve this required speed, the wave will pass by without them catching it.

There is a great five minute video featuring Professor Neville de Mestre which discusses this at the following link:

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

A much fuller detailed paper by Professor Neville de Mestre published by Austrlalian Mathematics Association is available at the following link:

http://www.austms.org.au/Publ/Gazette/2004/Sep04/demestre.pdf

Paddling Speed Equation

The Speed a Surfer needs to paddle to catch a wave is dependent on the Height of the Wave.

This is the case because larger waves have more energy and travel at higher speeds.

The following formula has been developed for the required Paddling Speed by Professor David Sandwell.

(Note that the diagram is an aerial view of waves breaking along a headland, or what surfers often refer to as a “Point Break”).

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

Videos About Catching Waves

The following is a great practical video about the steps to catching a wave:

[youtube=http://www.youtube.com/watch?v=hVyatMUYwZw&w=550&h=309]

This next video is all about the importance of catching a wave at the peak.

[youtube=http://www.youtube.com/watch?v=hcHWWBWdzzc&w=550&h=309]

Types of Surfing Waves

There are four main types of Wave which are associated with Surfing.

In order from smallest to largest, these are the Longboard Wave, the Fun Wave, the Tube Wave, and the Tow-In Wave.

Image Source: Google Images

Geometrical Shape of Tube Waves

One of the four types of Surfing waves is the “Tube Wave”. This is the classic curling wave which surfers love the best. Tube waves contain plenty of power and provide an exciting ride, especially if you can get yourself inside the tube of water.

Being inside a tube is awesome!

It is like being transported temporarily into a calm and peaceful cylindrical crystal palace.

Image Source: http://ocean.si.edu

The only drawback is that if the “Crystal Palace” unexpectedly caves in, the weight of the seawater (at around 1kg per cubic meter), can pack a decent punch to your head!)

A Geometrical Ratio used to describe the shape of Tube Waves, that involves the measurement of the maximum width and length, as per the following diagram:

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

The geometry of tube shape is represented as the ratio between Length and Width.

A perfectly cylindrical vortex has a ratio of 1:1, while the classic almond-shaped tube is nearer 3:1.

When width exceeds length, the tube is described as “square”.

Information Source: http://www.gogeometry.com

Wave Speeds of the Four Wave Types

Professor David Sandwell has determined the following relationship between Ocean Depth and Breaker Height, as well as determining Average Wave Speeds.

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

From the speeds of the bigger waves, we can see that a surfer cannot paddle fast enough to “catch” these waves, and a technique called “Dropping In” is used to generate the required additional speed.

“Dropping In” involves the conversion of Gravitational Potential Energy into Kinetic Energy by the surfer dropping down the front of the wave. This is discussed in the next section.

Dropping In

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

“Dropping in” is a method of gaining enough speed to catch and stay on a wave that is moving too fast to catch merely by paddling as hard as possible. The drop down the wave provides the needed speed for the surfer to be able to stay with the wave.

This is the same idea as riding a bike down a really steep hill, or a skateboard down a steep ramp.

Going down a steep incline increases your speed, because you convert earth gravitational energy into moving kinetic energy.

“Dropping in” is also a term used for stealing another surfer’s wave by “Dropping In” on him.

Eg. The rules are that the surfer on the inside, closest to the “Impact Zone” and whitewater has the right of way.

In the following video two surfers drop into the same wave, but the one on the left of screen should have given way to the one on the right of screen. (via the who is on the Inside Rule).

The left guy illegally gets the wave, but then the Karma Gods catch up with him and he gets totally smashed and eaten by the wave.

[youtube=http://www.youtube.com/watch?v=S8f9HVvezMQ&w=550&h=413]

Dropping down the wave converts Potential Energy into Kinetic Energy. This Kinetic Energy can supplement paddling speed and enable catching the wave to occur.

The following calculation produces a Formula which calculates Surfer Speed at the bottom of a Wave.

Original Image Source: http://topex.ucsd.edu/ps/energy.pdf

Substituting the typical wave heights of our four standard wave types produces the following Results:

Original Image Source: http://topex.ucsd.edu/ps/energy.pdf

When a surfer “drops in” to accelerate down the face of the wave, this needs to be done at an angle across the “face” of the wave.

Anyone who has ever dropped straight down the face of a wave knows the subsequent “nose dive wipe out” crash which usually results.

The Drop In Angle can easily be calculated using our previous values of :

Initial Paddle speed “c” (which needs to match the wave speed to catch the wave),

and

Surfer’s Speed “v” at the bottom of the wave at the end of the Drop In.

As shown below, this “Cutting Across” angle ends up being a value of 50 degrees.

The wave moves forward breaking across horizontally at its moving forward speed, and we drop down the wave increasing our speed.

To stay out of the white water, and move across the wave face, we need to point at an angle of 50 degrees.

Original Image Source: http://topex.ucsd.edu/ps/energy.pdf

Note that the bulk of the diagrams in this section are modified from material supplied in David T Sandwell’s presentation titled:

“Physics of Surfing Energetics of a Surfer”. Click the link below to view his complete presentation.

http://topex.ucsd.edu/ps/energy.pdf

Surfing the Wave

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

Once we have paddled at sufficient speed and / or “Dropped In” to catch the wave, we continue to use the conversion of Gravitational Potential Energy to Kinetic Energy to move along, and go up and down the wave as we do this.

Remember that our speed has to stay close to the speed of the wave, or we will not be able to keep up with it, and it will pass by underneath us.

The following three minute video shows the mathematics of Surfing the Wave in action.

[youtube=http://www.youtube.com/watch?v=O_oADJ961vo&w=550&h=413]

For the remainder of the Surfer’s ride across the wave, (after the initial “Drop In”, there are usually repeated up and down the wave travel, combined with turns and acrobatic manoeuvres, until the wave has dissipated or the surfer elects to end the ride.

For really big fast moving waves there is not enough time for a lot of up and down the wave manoeuvres, as taking the drop down the face and surviving is the surfer’s main concern.

To demonstrate this, here is a video of Shane Dorian Surfing a really big wave at “Jaws” in Hawaii.

[youtube=http://www.youtube.com/watch?v=O9crPOB_9tE&w=550&h=309]

Shane Dorian was incredible to take the Drop and catch a wave like this at Jaws. Large waves like these can be moving typically at 35 mph (60km/hr), and often it is not possible to catch them by paddling a surfboard up to your best paddling speed.

Usually Jet Skis are used to tow a surfer up to speed and into a wave of this size, especially at huge breaks like Mavericks in California, and Jaws in Hawaii.

Tow In Surfing

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

“Tow-In Surfing” or “Tow Surfing” is a relatively new form of Surfing that was developed by the creative efforts of Laird Hamilton and his friends in Hawaii in the early 90’s.

Tow Surfing combines Surfing with wakeboarding and waterskiing, where a person is towed behind a vessel utilizing an extension rope with a grab handle, and standing on a special Tow-In Surfboard.

Tow Surfing derived from the desire of Big Wave Surfers to be able to power surf larger waves that were breaking on outer reefs and typically incapable of catching due to their size and speed.

The size of these waves generally tracked towards land at faster rates than could be paddled into by “Guns”, surfboards designed for larger waves.

Laird and his partner’s utilized technology and background products from windsurfing, and surfing which created specialized tow boards with foot straps, and designed for stability and high speed.

A 155hp Yamaha Waverunner with a top speed of 65 mph is a popular Jet-Ski used for tow in surfing.

Speeds of up to 40 mph (60 km/hr) are required for Tow Surfing, because the huge waves that are surfed are usually traveling through the water at around 35 mph.

Here is a video showing an awesome demonstration of Tow-In Surfing.

[youtube=http://www.youtube.com/watch?v=3oS_28utt2Y&w=550&h=413]

There is a great website dedicated to Tow-In Surfing at the following link:

http://www.towsurfer.com/

Surfing Statistics

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

Shown above is what is currently the biggest wave believed to have ever been Surfed: a 100 ft / 31m storm wave on the coast of Portugal.

Brazilian surfer Carlos Burle rode the monster wave recently in October 2013.

The wave was created by the St Jude storm at Praia do Norte, near the fishing village of Nazare.

Estimated at possibly over 100ft, it is believed to be the biggest wave ever ridden.

Forty Five year old Hawaiian surfer Garrett McNamara also surfed a 100 foot wave at the same location in January 2013. It will not be known until around April 2014 when they decide the Surfing Year’s biggest wave winner as to who exactly is the world record holder.

The following videos show these epic wave rides.

[youtube=http://www.youtube.com/watch?v=7tAp5WFhMwM&w=550&h=309]

[youtube=http://www.youtube.com/watch?v=csothimiWjc&w=550&h=309]

[youtube=http://www.youtube.com/watch?v=k5Gb0ceZibg&w=550&h=309]

The unique bathymetry of the Nazarre Portugal region sea floor which causes these monster waves.

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

As per the above image there is a Huge Underwater Canyon that the water is channelled along which suddenly bottoms out near the shore. This creates monstrous waves from incoming rogue waves from big storms in the Atalantic Ocean.

Here are some other interesting Surfing Statistics

Original Image Source: http://www.statisticbrain.com

Click to Enlarge the above Image.

Note that the Biggest Wave ever recorded (1740ft / 535m) was a giant Tsunami wave which hit the coast of Alaska in 1958.

How Heavy is a Big Wave

The following video has all the answers

[youtube=http://www.youtube.com/watch?v=M9-CaewvynA&w=550&h=309]

Artifical Surf Breaks

Image Source: http://surfspotsmap.com

Australia is currently leading the way in building coastal engineering structures which prevent beach erosion, as well as supply improved surfing conditions.

Places around the world with artificial Surf Breaks include:

Narrowneck, Queensland, Australia
Cables, Western Australia, Australia
Pratte’s Reef, El Segundo, California, Los Angeles
Bagarra, Queensland, Australia
Mount Manganui, NZ
Bornemouth, UK
Kovalam, India

The current cost of building an artificial reef is between 1 and 10 million dollars.

The quest to build artificial reefs has had mixed results.

Cables in Perth used granite boulders and has withstood the test of storms, and is still surfable. It provides a reasonable surf spot in suburban Perth; however there has been controvery that it does not supply enough surfable days througout the year, and works best in winter.

In all fairness to the builders of the reef, Perth is hampered by choppy water in summer (due to Rotnest Island absorbing a lot of wave power), and strong cross shore south westerly winds.

Unlike Perth’s granite boulders, the sand bags reef in Bournemouth UK suffered substantial movement, resulting in unsafe under currents and had to be closed in 2011.

Narrowneck in Queensland has also been closed for periods of time after storm damage.

The Indian artificial reef appears to have been quite successful; however its longevity is yet to be proven.

Working in conjunction with the Indian government, ASR Ltd completed construction of India’s first Multi-Purpose Reef in February, 2010.

The primary purpose of the reef is to mitigate coastal erosion from monsoon waves in the small town of Kovalam.

The Aritifical Reef also creates an excellent habitat for marine species.

Finally, as this video shows, the reef has helped create a nice surf spot.

[youtube=http://www.youtube.com/watch?v=mGYuj-Ow1rk&w=550&h=309]

If you would like to read more about Artificial Reefs, there is a good article within this web page:

http://www.seafriends.org.nz/oceano/beacheng.htm

Continuous Wave at Waimea Bay in Hawaii

Not all Surfable waves are found in the Ocean!

Kelly Slater & locals were able to surf a river wave at Waimea Bay in January 2011 after the river flooded and broke the beach in two and created a continuous wave.

Here is a video showing their exploits:

[youtube=http://www.youtube.com/watch?v=ljVbieeFn14&w=550&h=413]

Surf Aid Mathematics Teaching Resources

Image Screen Captured from Surf Aid Website

Surf Aid is a not for profit organisation sponsored by Billabong with a kean interest in the preservation of Mentawai in Indonesia.

The Mentawai Islands are a chain of about seventy islands and islets off the western coast of Sumatra in Indonesia.

It is totally free to join the “Surfaid Schools Program”, and all that is required is submission of your email address.

http://www.surfaidinternational.org/schoolsprogram

Downloadable Free Units which are in ZIP and PDF format which could be of use to Mathematics Teachers are the following:

AUS Maths 12-14

Mathematics – Connections through Surfing

Technology

Destination Mentawai Islands

The Economics Of Aid

Crossing The Divide – Primary

These are all free PDF and ZIP file downloads

A Typical example might be the following questions about traveling to surfing locations on various Islands:

“If the motor launch can manage an average speed of 15knots, calculate the journey time between each
location. 1knot is 1 nautical mile per hour. You can assume there is sufficient sailing staff to keep moving 24 hours per day. Round up your answers to the nearest day and complete worksheet 5.3.

If you move 10º around the equator how many nautical miles have you traveled?

What angle around the equator (change in longitude) corresponds to 1 nautical mile?”

Another activity we looked at was all about “Planning an Overseas Surfing Trip”.

Eg. Costs, Savings Plan, Items needed and their Cost, and so on.

We suggest you could easily do the same for your classes.

AMC Mathematics Teaching Resources

The Australian Maritime College has an excellent set of free resources for Mathematics Teachers.

Among these is a very good mathematical exercise about Wave Refraction.

Go to the following link to find their Resources page:

http://www.amc.edu.au/why-study-maths

To get the workbooks and other resources, it is necessary to register your school and complete an order form online.

These resources are free to Australian schools, and if you would like to see what some of the workbook exercises look like, then check out the following link:

Click here for AMC Online Maths Workbook PDF

The AMC is planning to add more online interactive lessons, but as of late 2013 the current the interactive lessons are as follows:

A Study of Similar Vessels (an application of Curve Fitting) – 5mins

Wind Farm Feasibility Study (an application of Probability) – 12mins

Vessel Speeds in Waves (an application of Differentiation) – 5mins

Ship Hydrostatics (an application of Integration) – 12mins

Wave Refraction (an application of Trigonometry) – 12mins

Ocean Waves (an application of Superposition) – 5mins

Scaling Laws (an application of Algebra) – 8mins

The AMC can be contacted about this program at the following email address: whystudymaths@amc.edu.au

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Posted in Math in the Real World, Ocean Mathematics | Tagged body surfing math, catch a wave mathematics, computer modelling ocean waves, computer modelling water waves, drop in mathematics, geometry of surfing, how to catch a wave, how to ride a wave, math for surfers, mathematics for surfers, mathematics of body surfing, mathematics of nose riding, Mathematics of Ocean Waves and Surfing, mathematics of surfing, mathematics of tow in surfing, mav conference ocean mathematics, ocean mathematics, ocean wave equations, ocean wave speed equations, passy world ocean mathematics, physics of surfing, riding waves mathematics, science of surfing, sea wave mathematics, sea wave speed equations, shapes of sea waves, surf math, surf math resources, surf maths, surf science, surf wave energy, surf wave speeds, surfing a wave, surfing drop in, surfing math, surfing resources for teachers, surfing wave speeds, trochoidal waves, types of ocean waves, types of sea waves, water wave mathematics, water wave speed equations, wave angle for surfing | 6 Comments

Wave Power Mathematics

Posted on December 4, 2013 by Passy

Image Source: http://resources2.news.com.au

According to The World Energy Council, our planet currently uses approximately 15 trillion kWh of Electricity per year, representing just 0.02% of the energy contained throughout the world’s oceans.

Much of this electricity is generated by coal powered plants which are environmentally unfriendly, and contribute significantly to green house emissions.

As a result, people have started using “Green Energy” in the form of Wind Power and Solar Power.

So far, not much Wave Power from the Ocean has been harnessed and utilised.

Wave Power is called “Blue Power” and provides clean renewable energy for both Generating Electricity and for the Desalination of Water.

Image Source: http://2.bp.blogspot.com

In this lesson we look at Wave Energy from the Oceans, and some of the Mathematics associated with “Blue Energy”.

Mathematics associated with Blue Energy includes the following:

1) The pattern of movement of particles in water waves

2) Up and Down Simple Harmonic Motion resulting from the sinusoidal nature of swell waves

3) The use of Parabolic shapes to capture waves and concentrate their energy into a narrow region

4) A Variety of Buoy Shapes for positioning on and below the water surface

5) Clever Geometry of Turbine Blades so that they can generate power when going both forwards and backwards

6) A Variety of Geometrical Arrangements of Energy Capturing blades, buoys, turbines, and paddles designed to interact with ocean waves, and freshwater currents

What Causes Waves

Due to the earth’s rotation on a tilted axis, the planet is heated unevenly, which in turn causes winds to blow to try and re-establish temperature equilibrium.

As wind blows across the ocean, it transfers energy into the water, resulting in ocean waves which we see breaking onto the shoreline.

As waves move towards the coast, they are referred to as “swell”.

When they get close enough to shore to be affected by the bottom of the sea, they rear up into “breakers”, which is what surfers utilise.

It is these ocean swell waves which are the best type of wave to use for Blue Energy technologies.

Wave energy is different to Tidal energy.

Tides are caused by the gravitational pull of the moon and sun, and have a period of around 12 hours, whereas ocean waves have a period of around 10 seconds.

The “Period” is the time for the wave to rise and fall and complete its cycle, before it is followed by the next wave.

Due to these vast differences in Waves and Tides, the technology required to extract this energy is extremely different.

Although some scientists are looking at using tides to generate Blue Energy, most of the work is being done on obtaining energy from oceam swell waves.

Water Wave Motion

Image Source: http://upload.wikimedia.org

Waves cause the water particles to travel up and down in elliptical paths, with the maximum kinetic energy happening as the height of the wave passes through the average sea level.

You can feel this constantly varying effect when floating in the surf zone, as it pulls you back and forward, and up and down.

The wave is a disturbance which moves through the water heading toward the shoreline, but the water particles do not move into the shore; they merely bob up and down in the one spot.

This difference and the mathematics associated with it are illustrated in the following diagram:

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

Ways of Capturing Blue Energy

Currently, there are three main ways of capturing Blue Energy :

1) Oscillating Water Column or “OCW” Technology

2) Buoys Bobbing Up and Down

3) A Sea Snake like structure of joined cylinders, which move with the waves in a cantilever fashion

The following five minute Video demonstrates the workings of these three main types of Wave based Power Generators:

[youtube=http://www.youtube.com/watch?v=bEfrtAOMuvk&w=550&h=413]

Oscillating Water Column “OWC” Technology

Photo Source: Google Images

The above photo is of the Blow Hole at Kiama in New South Wales Australia.

Waves wash up to a cliff face into a Parabolic shaped opening, which concentrates their energy into a tunnel which links up with a vertical shaft in the rock.

Air and Water are forced up this shaft, resulting in a spectacular plume of spray being release up through an elliptical opening in the rock.

As the wave recedes, air is also sucked back down the shaft.

Engineers at Oceanlinx Australia have replicated the blowhole effect into machinery which has a two-way turbine attached to the top of the air shaft.

This turbine spins continuously generates electricity.

The wave capturing intakes are also Parabolic shaped and are claimed to be able to concentrate a 2m swell to produce the input effect of a 6m swell.

Here are what the Oceanlinx “OCW” production units look like in the Ocean.

Image Source: http://static.offshorewind.biz

The equipment has been trialled at Port Kembla on the Australian Coast, and is going into production trials supplying electricity to the city of Adelaide.

The following is a five minute Beyond Tommorrow YouTube Video about the Ocenalinx OCW Technology.

It shows the Kiama Blow Hole in action, and near the end it mentions the Parabolic Wave Trapping unit concentrating the energy by a factor of 3, eg. a 2m swell becomes a 6m swell.

[youtube=http://www.youtube.com/watch?v=z5oOMmcqT8o&w=550&h=413]

The oceanlinx website is well worth checking out, and can be found at the following link:

http://www.oceanlinx.com/

OWC Breakwater Generator

OWC Technology is also being used in Scotland.

The application of the generator unit is in the form of a breakwater.

Photo Source: Google Images

The Geometry of the breakwater structure has been optimised to trap the power from the waves and force it to push air back and forth to continually drive the power generating turbine.

The following three minute YouTube video explains fully the workings of the Scottish OCW Generator.

[youtube=http://www.youtube.com/watch?v=gcStpg3i5V8&w=550&h=309]

Fluid Piston Pump Technology

Photo Source: http://www.lamtengchoy.com

This Blue Energy technology has been developed by Henry Lam Teng Choy in Singapore.

Website: http://www.lamtengchoy.com/main/

This method of harnessing wave power uses “Fluid Piston Pumps” driven by energy from ocean waves, or from flowing river currents, and converts this energy into electricity.

Each piston pump is driven by a float; the pumped fluid (water and/or air) is supplied to drive a fluid motor or a turbine, which in turn operates a generator to produce electric power.

Henry owns international patent PCT/SG2011/000232 filed on July1, 2011.

He is starting to receive increased attention in the renewable energy industry and research community, but is yet to receive sufficient capital funding to bring his generators to the market place.

What is particularly amazing is the number of different trapping mechanisms and geometry which has been proposed for the piston pump technology.

Here are just a few of them:

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

The following five minute video details and explains all of the different geometric configurations which utilise wave driven piston pumps pumping water to drive an elctricity generating master turbine.

[youtube=http://www.youtube.com/watch?v=fYfs-qYGzvs&w=550&h=309]

CETO Underwater Buoys System

Photo Source: http://www.carnegiewave.com

Carnegie Wave Energy in Western Australia has a non-visual impact underwater system, which pumps water ashore into a combined Desalination Plant / Electricity Generator.

Image Source: http://www.energy.edu.au

Click the above Image to view it full size.

The following one minute video provides an overview of the CETO system.

[youtube=http://www.youtube.com/watch?v=mhDl5HG6vOQ&w=550&h=309]

Here is a longer four minute video about CETO.

[youtube=http://www.youtube.com/watch?v=q0tNneszP5w&w=550&h=309]

There is a full set of 35 videos on CETO at the following You Tube Channel Link:

http://www.youtube.com/user/carnwave/videos

There is also a company website at the following link:

http://www.carnegiewave.com

Carnegie are currently commercially trialling the CETO Buoy system by powering the Naval Base on Garden Island at Perth, as well as supplying desalinated fresh water to the island.

Advantages of Wave Energy

Wave energy can be predicted several days in advance, meaning it can be used for base-load power when utilised in an appropriate mix of energy generation sources.

The supply of wave energy and demand for power are well matched, with most of the world’s population living near to the coast.

There are wave energy converters on the market that have no negative impact on its environment and in fact can be beneficial to its environment by acting as artificial reefs and encouraging growth.

Power is provided by Blue Energy Wave Technology night and day, with waves being a far more reliable and regular source compared to Wind and Solar Energy.

When compared to other power generating resources Wave energy is a denser energy medium with higher predictability than wind and solar energy.

There are a wide-ranging number of potential sites to utilise wave energy, while many other resources are limited when it comes to large scale energy production.

Wave energy converters can have less visual impact on its environment during energy production or recovery in comparison to wind and fossil fuels.

Information Source: http://www.oceanlinx.com

Issues and Challenges With Wave Power

Wave power, like other early energy sources, is currently expensive.

While coal might be priced at five to 10 cents per kilowatt hour of energy, wave energy costs reach 20 to 30 cents.

The question is, “How quickly does that come down?”

In the United States, wave power companies need more research and development funds and greater subsidies.

Wave energy companies need more time to develop technologies that can withstand the harsh ocean environment.

Another tricky issue is property. Unlike building a power plant on land a company owns, the ocean is common space.

And although some believe the environmental issues related to wave power are less so than with other technologies, there are still Environmental Unknowns, such as how ecological systems will be affected by wave technologies and how shipping and fishing activities will be affected by device placement.

Information Source: http://www.smartplanet.com

Picture Credits

Electricity Coal Power Station – http://resources2.news.com.au

Oceanlinx Offshore Power Plant – http://2.bp.blogspot.com

Piston Pump Generartors – http://www.lamtengchoy.com

Kiama Blow Hole – http://www.visitnsw.com

Kiama Blow Hole Entrance – http://warragambadamspill.blogspot.com.au

OWC UNit in Scotland – http://www.voith.com

Scottish OWC Diagram – http://news.bbc.co.uk

CETO Buoys – http://www.carnegiewave.com

CETO Diagram – http://www.energy.edu.au

MAV Conference PowerPoint

To download the PowerPoint Presentation from the Mathematics Association of Victoria Conference, click the link below and save the file to your computer.

Click the link below to download this 3MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/WavePowerPPTv2.pptx

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Posted in Math in the Real World, Ocean Mathematics | Tagged advantages of wave power, alternative energy mathematics, blue energy, blue energy mathematics, blue energy power, cost of wave power, how blow holes work, how to make electricity from the ocean, how to make electricity from the sea, how to make electricity from waves, how wave power works, issues with wave power, mathematics of blow holes, mathematics of wave power, mav conference ocean mathematics, ocean mathematics, oscilating water column, oscillating water column, owc wave power, parabola wave power, passy world ocean mathematics, wave power, wave power from buoys, wave power from oceans, wave power from the sea, wave power generation, wave power geometry, wave power in australia, wave power in scotland, wave power mathematics, wave power maths, wave power on garden island, what is blue energy | 4 Comments

Gradient Slope Intercept Form

Posted on November 9, 2013 by Passy

Image Source: Google Images

Babies usually follow a straight line of increasing body length as they start growing.

This baby was born 20 inches long (y-intercept), and has been growing at a rate of a 1/4 inch per week.

There are many other important Straight Line Relationships in Real Life, as shown in our previous lessons at the links below:

Real World Line Graphs 1

Real World Line Graphs 2

In this lesson we cover the following:

– Gradient-Intercept Equation Form

– Identifying Gradient and Y-Intercept

– Rearranging Into y = mx + b Form

– Finding the Line Equation from a Graph

– Graphing y = mx + b

– Graphs with Y-Intercept at Zero

– Horizontal and Vertical Lines

As you can see, it is a big lesson containing several related items, and so work through everything very carefully, and make sure you watch the Videos.

Gradient-Intercept Equation Form

Let’s begin by looking at what is Gradient-Intercept Equation Form.

Depending on what country you are in, the equation may vary, but in both formats it says that the value of every Y-Coordinate can be found by taking the X-Coordinate multiplied by the Gradient Slope and then adding on the Y-Intercept value.

Identifying Gradient and Y-Intercept

If we are given a Line Equation in y = mx + b form, we can find the Slope and Y-Intercept, without drawing any graphs.

We simply take the number or fraction value in front of “x”, and this is the Gradient Slope.

The number or fraction value on the end of the equation is the Y-Intercept.

Rearranging Into y = mx + b Form

Often we have equations such as 5x + 3y = 15, we need to rearrange the equation into y = mx + b format.

We do this by solving the equation for “y”

Here is a video which shows how to rearrange equations into y = mx + c form, by applying opposite operations to both sides

We can then find the Gradient Slope, and the Y-Intercept, without having to draw any graphs.

Here are a couple of examples of Rearranging into y = mx + b form.

Finding the Line Equation from a Graph

If we are given a Line Graph, we can read off the Y-Intercept and Calculate the Gradient Slope.

We then substitute these values for “m” and “b” in the y = mx + b equation.

Eg. We replace the “m” and the “b” in y = mx + b with our values taken from the graph.

Graphing y = mx + b

There are four steps to making the Straight Line Graph:

1. Rule up and Label an X-Y Cartesian Grid

2. Line up the equation with y = mx + b and read off the Y-Intercept as the number matching up with “b”.
Dot in this y intercept value as a point on the Y-axis.

3. Take the value in front of “x” as the Gradient, and write it as an UP/Across fraction. (Eg. If we have 3x, write the Gradient as 3/1).

From the Y-intercept, move UP (or DOWN if negative), and then Across, and make another dot on the Grid.

4. Join the two dots with a line including arrow ends,and then write the equation of the line next to the line.

Here is a two minute video showing how to do the above four steps to graph a line which is in y = mx + b form.

Keep in mind the four steps, whenever graphing y = mx + b equations.

Example Graph for y = 2x + 2

Our first steps are to rule up a Cartesian Grid, and then mark a dot on it for the y-Intercept.

Basically the number or fraction at the end of the equation gets marked onto the Y-axis.

The next step is to use the Gradient Slope value to plot a second point onto the Grid.

Now all we need to do is dot to dot join our two points with a straight line.

Video on Graphing Straight Lines

This next video is a longer video and shows how to do graphs for a number of equations which are in y = mx + b form.

Practice Grid

Here is a blank Practice Grid to plot some y = mx + b Straight Lines.

Graphs with Y-Intercept at Zero

Some equations do not have a “b” number or fraction value on the end of them.

Eg. y = 2x, y= -4x, y=3x, y= -3x, y = 2/3 x, y = -3/4 x, y = 7x, etc

This simply means that the Y-Intercept is zero.

We therefore do a dot in the middle of the X-Y Cartesian Grid to mark in Y-intercept = 0.

Then all we have to do is use the Gradient Slope value to move from zero up or down to a second position and make a dot there.

Finally we join these two points to make the line.

Here is an example showing how we have plotted the graph of y = 3x.

In this second example y = -3/4x we have a NEGATIVE gradient, and so we need to move DOWN and then across.

Vertical and Horizontal Lines

Two types of lines which do not have a Gradient Intercept form equation are Vertical Lines and Horizontal Lines.

Eg. Their Line Equations cannot be written in y = mx + b form.

Vertical Lines cannot be written in y = mx + b form, because their Gradient Slope if undefined, and the lines go infinitely straight up.

Vertical Lines have equations such as x = 2, x = -3, x=7, x = -4, etc.

Vertical Lines have an X-Intercept, but they do not have a Y-Intercept.

Horizontal Lines have Zero Gradient Slope, because they are not going uphill, and they are not going downhill, they are going along flat.

Horizontal Lines have line equations such as y = 1, y = 6, y = -3, y = -5 etc.

Because Horizontal Flat Lines have a Gradient of zero, y = 0x + b simplifies to y = b

Horizontal Lines have a Y-Intercept, but do not have an X-Intercept.

The following video shows how to graph Horizontal and Vertical Lines on an X-Y Cartesian Grid, and then shows how to do these same lines on a Texas Instruments graphing calculator.

This next video shows how to determine the equations of Horizontal and Vertical Lines.

Finally, here is a video which shows just how quick and easy it is to graph Vertical and Horizontal Straight Lines.

Slope Intercept Form Worksheets

The following worksheet involves drawing the graph for a number of y = mx + b Equations.

(There are answers further down on the worksheet).

Drawing the Graph for y = mx + b Worksheet 1

The following worksheet involves drawing the graph for a number of y = mx + b Equations.

However, the equations have to be rearranged into y = mx + b form first.

(There are answers further down on the worksheet).

Drawing the Graph for y = mx + b Worksheet 2

Do Questions 1 to 8 on the following Worksheet which are questions where we rearrange standard equations into y = mx + b form.

(There are answers further down on the worksheet).

Rearrange standard equations into y = mx + b form Worksheet

The following worksheet is on finding the Gradient Slope “m” directly from an equation without drawing the graph.

Some equations need to be rearranged into y = mx + b form first.

(There are answers further down on the worksheet).

Finding the Gradient Slope Worksheet

This final worksheet is on Horizontal and Vertical Lines but does not have an Answer Key on it.

Horizontal and Vertical Lines Worksheet

Online Graphing Calculator

The following Online Calculator Graphs the input Equation, and also supplies full working out with explanations of each step.

The working out is by the X and Y Intercepts method, but it will give you an accurate picture of what your graph should look like.

Click the following link to use the Calculator:

Click here to use the Online Graphing Tool

Online Quiz for y = mx + b

There is a great Online Quiz at the “Maths is Fun” website which you can do by clicking the link below.

The quiz also gives a complete fully worked explanation for any question you get wrong.

Click here to do the Online Quiz

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