## Surfboard Geometry and Design

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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.

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

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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

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.

The Funboard

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

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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.

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

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

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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

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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

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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

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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.

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

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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.

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

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.

Flex and Cant on Fins

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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.

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.

http://www.bulletfins.com

FCS are an Australian company who leads the world in surfboard fin technology.

Over 40 World Titles have been won on surfboards with FCS Fins, and world champion Kelly Slater has used them for several years.

This first four minute video discusses what has changed in Fins between 1993 and 2011, and how FCS was born.

The next video shows two new products that FCS have released that may look simple, but took up a lot of design and testing hours to get production ready and reliable.

The previous video showed a figure 8 type curved fin mounting insert, which is a break away from the traditional rectangular shaped fin box.

This new shape appears to give greater strength, as demonstrated in this next three minute video.

Fin Configurations

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Surfboard fins come in a variety of configurations, ranging from a single fin on a longboard, to three or four fins on a modern high performance short board.

The five fin box mounts setup is used to enable either a tri fin, or a quad fin, setup to be used.

The characteristics of the common fin configurations are as follows:

Single Fin: One long, wide fin that is usually set up at the back center of the surfboard. Almost always used on long boards.

A single fin rides great in directional surfing. They preform well if the take off is not too steep and can really hold a line in a tube.

Twin Fin: Two smaller fins mounted on either side of the rear. A twin fin works well in mushy, powerless surf.

There is usually very little or no entry rocker, and these are typically used on Fish type surfboards.

Thruster: Three fins of the same size, with two mounted slightly toed-in and canted outwards.

These two fins are on the sides, about 10 to 12 inches from the rear. The middle fin is straight, and mounted three to five inches from the rear.

This is standard of today’s pro-surfer and the recreational surfer as well.

Trifins can make tight turns, they are usually lightweight short boards, and require pumping down a line, to gain speed, and can be manoeuvered like skateboards.

There is also a Trifin “2 + 1 Setup” which is basically the same as the thruster set up, but with a larger middle fin and smaller side fins.

The middle fin is not the same size, as the twin pair of outer fins.

The “2+1″ setup is used in modern longboards. It gives boards of this size good down the line surfing, with a lot move manoeuverability.

Quad: Four fins, two on either side of the back of the board. This fin set up give the surfboard a looser feeling in the tail, which allows make tighter turn towards the curl (cutbacks) and other top to bottom manoeuvers.

This fin set up has gained in popularity over the past few years.

Note that for multifin boards, the fins are not always parallel to the direction the board goes in.

Outside fins have their angle offset from the centre line, and they point outwards. This angle is called “The Toe”. The front edge of the fins pointing towards the nose are angled in towards the center stringer and this alignment is called “toe in”.

In multifin setups, only the Trifin-Thruster centre fin is parallel to the centre line, and perpendicular to the surfboard bottom.

Bottom Contour

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

Bottom contours are associated with the shape of the bottom of the surfboard and influence how water travels under the surfboard.

Some common surfboard bottom contours are:

– Flat Bottom
– Concave Bottom
– Double Concave Bottom
– Vee Bottom
– Multiple Channel Bottom

Image Source: http://www.shapers.com.au

Flat surfboard bottom design is a fast bottom shape but one that can be difficult to control in larger/faster waves.

Flat bottom is good for small, mushy waves, where you need lots of speed.

Concave surfboard bottom design helps to prevent water being released under the rails giving the surfboard lift and speed.

A single Concave is often used in the front section of Longboards to aid noseriding.

On short surfboards a concave bottom will need increased rocker to allow the rider to retain good manoeuverability.

The Double Concave is seen on the majority of modern mainstream surfboards.

It is the most common contour of boards bought it straight off the rack in surf shops.

Generally the board will have a single concave from the nose which will gradually fade into a double concave towards the tail.

The single concave provides a good planing surface, giving the board drive and accelaration.

The double concave splits the water into two channels through the fins and creates a much looser ride, which allows great flowing manoeuvers.

Vee surfboard bottom gives a flat planing surface on each side of the surfboard that makes the surfboard fast through turns and easy to change direction.

A Vee surfboard bottom is not as fast as other boards when going in a straight line.

Multiple Channel bottom design can have up to 8 channels running along the bottom of a surfboard and there are a number of variations.

Their basic purpose is similar to a concave bottom i.e. To direct the water from nose to tail giving increased lift and speed.

If they are deep, long and have hard edges they may do this too well and make the surfboard prone to wanting to stay in a straight line “track”.

Information Source: http://www.shapers.com.au/

The Surfboard Deck

Image Source: http://vimeocdn.com

The deck is the top surface of the surfboard.

Because it is normally slippery fibreglass, it either needs to be thickly waxed, or have rubber grips put onto it.

The most common designs are domed, flat, and step decks.

A number of companies now offer soft top surfboards. These lower the chance of injury and are very durable without sacrificing too much performance.

Making Surfboards

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

The traditional way of making surboards is to start with a foam “blank”, which is a rough cutout of a surfboard shape, with a stringer down the centre.

A “Shaper” is a person who turns this blank into a proper surfboard, using cutting and sanding tools, along with fibreglass and resins.

Image Source: http://b.vimeocdn.com

The customer orders their board in a surf shop on an order form, and then the shaper works from blueprints to carefully measure and remove material from the blank to create the required surfboard.

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

Click to enlarge the above image to full size.

The following five minute American video goes through the process of hand making a surfboard, starting from a foam blank.

There is an in depth technical article on how to build your first surfboard at the following link:

http://www.surfersteve.com/design.htm

Shaper Measuring Tools

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

There are a number of specialised measuring tools used by shapers to check the geometry of the board, the fin placement, and the symmetry.

These include the “Fin Angle Finder”, the “Marking Gauge”, the “C-Caliper”, and the unique “E-Z Square Pro”.

The Fin Angle Finder is is a Plastic Protractor and Angle Finder with articulating arms that make it very simple to reproduce or mark out the appropriate cant angle required for the fin system.

The Marking Gauge is a tool for making marks on the blanks for various steps of the shaping process. It is also helpful in drawing the outline prior to masking tape decks and bottoms for air-spraying.

The “C-Caliper” is Used to measure the thickness of blanks and finished surfboards. Thickness is vitally important, as the board volume, expressed in total litres of foam for correct buoancy, is dependant on correct thickness.

The “E-Z Square Pro” is a clear plastic sheet overlay measuring / template tool. It can accurately measure the nose, width, and tail dimensions of a surfboard, and also has the ability to mark fin placement.

The following five minute video shows how to the “EZ Square Pro” foam measuring template tool can be used for marking out a blank that is going to be shaped into a surfboard.

For complete information about the measuring tools discussed here, check out the following webpage:

http://www.foamez.com/shaping-surfboards-surfboard-measuring-devices-c-2_28.html

If you want to know all about shaping Surfboards, then watch the “EZ Foam” videos collection at the following link:

http://vimeo.com/user6626751/videos

Computer Design and Manufacture

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

There are computerised shaping machines with the ability to reproduce existing “master key” board designs ten times faster than the average shaper.

These machines are able to successfully capture the integrity of any proven board up to ten feet long, which can then be manipulated by computer to any specified dimension.

“Borst” in Southern California, is one such company at the cutting edge of automated surfboard manufacture.

The “master key” foam blank is either made manually by a master shaper, or computer designed and cut, and then checked and finished by a master shaper.

The “master key” board is then scanned using a 3D image scanner so that all vital geometry and size measurements can be captured.

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

The surfboard’s shape is captured into sets of “Parametric Geometrical Equations” that use three dimensional X Y and Z Coordinates. Simple 2D parametric equations are the y = x^2 Parabola, and the x^2 + y^2 = r^2 Cirle Equation. Parametric Equations allow the board to be “morphed” into larger or smaller sizes. Eg. “Upsizing” a board is more than simply applying a common scale factor to all key measurements.

Once the surfboard’s shape has been captured, it can then be adjusted in a computer designing app to any standard length, and then manufactured the the surfboard maker machine.

This two minute Borst music video shows how a raw blank is machine shaped into a board that is then ready to be fibreglassed.

The finished computer manufactured products look like this:

Original Image Source: http://borstsurfboards.com

http://borstsurfboards.com/

Another site worth checking out is by a guy called Louis Robert who built his own shaping machine so he could increase his surfboard production capacity.

http://www.revolucionsurfboards.com/

Louis Robert’s machine automatically cuts out the blank from a block of foam with hot wires, as well as then being able to be tooled up to do auto shaping.

Here is a short two minute video showing how this is done.

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

Companies like “Firewire” Surfboards, have developed online web apps that allow surfers to design their own boards, and view them in rotational 3D.

The following video shows this application in action, as well as discussing the mathematics and geometry associated with computer design.

If you would like to try out the Free Firewire Design App, then click the link below:

http://custom.firewiresurfboards.com/custom/intro

Computers and People

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

What happens in a lot of cases these days is a “best of both worlds” combination of computerised manufacture, and traditional handshaping.

Australian companies such as “Burford Blanks”, supply blanks of surfboards which are precut to a basic shape by computerised manufacturing, and then a Shaper can finish the board off for you to your own custom design.

These blanks can even have the “rails” precut as a polygon shaped profile that then just needs to be rounded off by the Shaper.

The Shaper can even store your designs on computer, so that when you want your next board, there is your history available as a starting point.

Surf Aid Mathematics 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.

The Surfaid website can be found at this link:

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

A Typical example might be the following questions about travelling 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 24h
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 travelled?

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.

It was very surfing orientated and might alienate non-surfers and female students, but could easily be adapted to be a group of friends planning a Bali Holiday, and undertaking various tours and activites whilst in Bali.

We will be going through all of the Surfaid Materials, and seeing what could be incorporated into some middle school mathematics, even though the school Passy works at has a tiny minority of students who have ever been Surfing.

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

The following is a list of stand out resources which we found on the Internet while researching this lesson.

Surfboard Design

http://www.surfscience.com/topics/surfboard-design/

http://www.edsinnott.com.au/Surfboard%20Design/

http://www.hydroepic.com/technology.php

http://www.surfertoday.com/surfing/7124-the-effects-of-surfboard-design-in-wave-performance

CAD and Surface Meshing to Design Ultimate Surfboard Performance

http://www.sharc.co.uk/html/case_surf.htm

Surfboard Fins

http://blog.surfride.com/surfboard-fin-guide/

http://www.surfertoday.com/surfing/8993-what-are-the-best-fins-for-your-surfboard

http://www.surfertoday.com/surfing/8993-what-are-the-best-fins-for-your-surfboard

http://wavegrinder.com/surfboard-fin-science/

http://www.surffcs.com.au/community-story/community-blog/2013/03/01/a-guide-to-fcs-fins

http://www.surffcs.com.au

http://www.tactics.com/info/guide-to-surfboard-fins

http://surfcabarete.com/surfing-caribbean/different-types-of-surfboard-fins/

Surfboard Bottom Contours

http://www.shapers.com.au/pages/Contours.html

Manufacturing Surfboards

http://borstsurfboards.com/

http://surf.transworld.net/1000074167/features/surf-science-modern-mechanics-of-boardmaking/

http://www.edsinnott.com.au/Surfboard%20Design/

http://www.firewiresurfboards.com/

Image Copyright 2013 by Passy’s World of Mathematics

I have been a keen Surfer for over 20 years, and have surfed all along the East Coast of Australia, all around Indonesia, as well as visiting Fiji and Hawaii.

I was sidelined from serious surfboard Surfing around 6 years ago after a spine operation to repair cumulative damage caused by Surfing and Bike Riding accidents. However I still body surf and boogie board surf whenever I can.

My son is a keen body boarder, and that is him first learning to surf in the bottom right hand corner photo.

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.

Surfboard Design and Geometry

http://passyworldofmathematics.com/MAVconfPPTs/SurfGeomDesignPPTv3.pptx

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## Mathematics of Ocean Waves and Surfing

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

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:

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.

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.

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.

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.

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

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

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

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

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.

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.

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

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.

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.

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.

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.

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

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.

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:

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.

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

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.

It was very surfing orientated and might alienate non-surfers and female students, but could easily be adapted to be a group of friends planning a Bali Holiday, and undertaking various tours and activities whilst in Bali.

We will be going through all of the Surfaid Materials, and seeing what could be incorporated into some middle school mathematics, even though the school Passy works at has a tiny minority of students who have ever been Surfing.

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:

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 following is a list of stand out resources which we found on the Internet while researching this lesson.

Mechanics of Mavericks

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

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

Physics and Energy of a Surfer – David T Sandwell

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

Speed of Ocean Waves

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

Near-shore swell Estimation from a Global Wind-Wave Model

http://www98.griffith.edu.au/dspace/bitstream/handle/10072/17987/49454_1.pdf

US Army Corps of Engineers Coastal Engineering Manual

http://chl.erdc.usace.army.mil/cem

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

http://www.marine.tmd.go.th/Part-II-Chap1.pdf

Artificial Surf Reefs

http://www.coastalmanagement.com.au/artificial-surfing-reefs/

http://www.surfingramps.com.au/CablesArtificialSurfingReef.htm

http://www.dailymail.co.uk/news/article-1372879/Britains-3-2m-artificial-surf-reef-closed-2-years-unsafe.html

http://en.wikipedia.org/wiki/Multi-purpose_reef

http://www.surfscience.com/topics/waves-and-weather/wind-and-weather/artificial-surfing-reefs/

Other Surfing Topics

http://www.surfscience.com/

http://science.kennesaw.edu/~jdirnber/BioOceanography/Lectures/LecHighEnergyEcosystems/HighEnergyMarineEcosystems.html

http://www.kidzworld.com/article/6839-the-science-of-surfing#ixzz2krLEbzMg

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

http://www.gogeometry.com/wonder_world/surfing_tube_ride_shape_golden_rectangle.htm

Maths and Science Teacher Resources

http://www.surfaidschools.org/

http://www.surfaidinternational.org/schoolsprogram

Image Copyright 2013 by Passy’s World of Mathematics

I have been a keen Surfer for over 20 years, and have surfed all along the East Coast of Australia, all around Indonesia, as well as visiting Fiji and Hawaii.

I was sidelined from serious surfboard Surfing around 6 years ago after a spine operation to repair cumulative damage caused by Surfing and Bike Riding accidents. However I still body surf and boogie board surf whenever I can.

My son is a keen body boarder, and that is him first learning to surf in the bottom right hand corner photo.

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 Ocean Waves and Surfing

http://passyworldofmathematics.com/MAVconfPPTs/IntroWavesSurfPPTv4.pptx

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## Wave Power Mathematics

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

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:

Oscillating Water Column “OWC” Technology

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.

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.

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.

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

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.

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.

Here is a longer four minute video about CETO.

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

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.

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.

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

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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:

– 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.

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.

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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

Image Copyright 2013 by Passy’s World of Mathematics

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

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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.

Image Copyright 2013 by Passy’s World of Mathematics

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).

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.

Image Copyright 2013 by Passy’s World of Mathematics

Click the following link to use the Calculator:

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.

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Please state in your email that you wish to obtain the free subscriber copy of the “Gradient Intercept Form”

Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

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Help us to maintain this free service and keep it growing.

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## Coordinates Bingo Game

This fun game teaches students to get very good at plotting (x,y) points, as well as reading points on the Cartesian Plane, and it is lots of fun.

The game was developed here at Passy’s World by Passy and Jack, and has been trialed in classrooms with great results. (Thanks to Jack for coming up with most of the ideas on this one!).

There is a PPT Presentation for this Game which will be very useful for instructing the students on how to play the game.

Overview of Game

Students plot 20 points of their choice on a Cartesian X-Y Grid, which ranges from -6 to 6.

A special Positive / Negative Virtual Dice is rolled, to generate a “Bingo Coordinate”.

If a student finds they have that coordinate marked down, they circle it in red pen on their grid.

(Repeat drawing Bingo Coordinates by rolling the special dice)

Once a student has FIVE Coordinates circled anywhere on their Grid, they call out BINGO, and get their grid checked and win a prize.

Image Copyright 2013 by Passy’s World of Mathematics

Equipment Required for Bingo Coordinates

Grid Paper / Graph Paper (one per student)

PPT Presentation (See end of this lesson for Details)

Laptop PC and Data Projector (for Virtual Dice and PPT)

Lollipops or individually wrapped candy / lollies for Prizes.

Instructional PowerPoint

This is available for free to any Subscriber to Passy’s World of Mathematics.

Becoming a Subscriber is Free.

You will be able to project the PowerPoint Instructional Slides in your classroom to show the students how the game works.

Once you are a Subscriber, Simply email us at the address shown below, and ask for a copy of the “Coordinates Bingo” PowerPoint.

Becoming a subscriber to Passy’s World is free, and to find out what happens when you subscribe, click the link below:

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Setting Up the Grid Paper

Students Rule up an X-Y Axis on their Grid Paper that forms a 6×6 Grid like the one shown below.

We need a 6×6 Grid because we will be rolling Dice to generate the Bingo Coordinates.

Image Copyright 2013 by Passy’s World of Mathematics

If you need some printable Grid Paper, then try the following links:

Paper Snake Free Graph Paper

Custom Graph Free Graph Paper

Four to a Page Cartesian Grids

Picking Points – Pre-Game Discussion

Students will need to plot 20 different random points of their choice onto their Grid.

We will be rolling dice to generate the Bingo Game Coordinates, and they will have to hope they get lucky, and some of the Dice Coordinates match their coordinates.

(We have found that 20 points makes for a game that goes for about 20 minutes and shopuld have at least 6 winners).

(You can also award some tasty treats for good answers during this pre-game discussion).

BEFORE THEY PICK THEIR POINTS – GO THROUGH THE FOLLOWING:

See if the students can figure out how many different points there are on the Grid.

ANSWER: The Grid is 12 x 12 = 144 possible points.

But what if we allowed Fraction and Decimal Coordinates ?

For our game we are only using whole number integer coordinates – you can plot fractions or decimals if you like, but they will never come up on the dice!

There are certain whole number points on the Grid which should not be used – What are they ?

ANSWER: X-Axis points like (1,0) (-3,0) etc and Y-Axis Points like (0,2) (0,-5) etc as well as the Origin – because these coordinates have a zero in them, and zero can never come up on a Dice!

So how many usable points are there on our Grid for the Game ?

ANSWER: 144 – 6 – 6 -6 – 6 -1 (for the origin) = 144 – 25 = 119

So what are the chances of one of your coordinates being called as a Bingo Coordinate ?

ANSWER: 20 / 119 which is roughly 1/6 .

Note that the Game was not deliberately designed to make the chances 1/6 on the Grid, be the same as the Dice Chance; and if we played with only 10 coordinates, the chance for each one coming up would be 1/12.

What are the chances of a Bingo Coordinate coming up; AND then on the very next dice roll, the exact same coordinate coming up again ?

ANSWER: 1/6 x 1/6 = 1/36

Note from classroom trials, the game usually ends with no repeat Bingo coordinates happening during the whole 20 minute game.

Now get the students to mark on their grid their 20 chosen coordinates.

Their finished competition entry should look something like this:

Image Copyright 2013 by Passy’s World of Mathematics

Note the anime girl in the above image is upset because she just noticed that her friend plotted lots of her entry’s points on the X and Y Axis!

Positive and Negative Virtual Dice

The virtual dice which are needed to play the game can be found at the link shown below.

Two rolls need to be done for each coordinate, the first roll is the X-coordinate, and the second roll is the Y-coordinate.

The teacher, or a reliable student, needs to note down each coordinate which is rolled during the game, and also plot the Dice Rolled Coordinates on a Grid, for later answer checking.

Click the following link to access the virtual dice

Positive and Negative Virtual Dice App

Note that an alternative would be to use a Coin with a real Dice (Head = Positive, Tails = Negative)

Playing The Game

Project the Virtual Dice App onto a Screen at the front of class.

Roll the Virtual Dice once for the X-Coordinate, and one again for the Y-Coordinate,

Call out the resulting (x,y) Coordinate

(Teacher or an appointed student needs to note down the coordinate, and also plot it on the answer grid)

If a student has that coordinate they circle it in red pen

Keep rolling the dice until a student has 5 coordinates and calls out Bingo

Check the Student’s entry and award a prize

Keep playing for about 20 minutes until about 10 prizes have been awarded.

Positive Coordinates Game

Our Bingo Game can easily be adapted for beginners doing only Positive Coordinates.

Make a 12 x 12 Y-X Grid, which has (0,0) in the bottom left hand corner, and (12,12) in the top right hand corner.

Then use Two Dice to generate the coordinates.

There is a virtual Dice for this at the following link:

Note that the game will be a little biased towards Coordinates containing 7’s, because 7 is the highest probability result for rolling two dice.

(Eg. 2 and 5, 6 and 1, 3 and 4, 4 and 3, 5 and 2, 6 and 1 = 6/36 = 1/6; whereas 12 is only from 6 and 6 which is a 1/36 chance).

See if this becomes apparent at all, as the game progresses.

Play the rest of the game as normal – Roll the two dice and add the result to get the X Coordinate, then roll them again to get the Y-Coordinate.

Note that Coordinates containing Zero or One will never come up in the Bingo Coordinates, because the lowest possible result from two dice is 1+1 = 2

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