Holy Moly Batman! I used to live at 84-246 Makau St. Did you know the kid with the artificial leg, named Lucky. Nice friendly kid. Rell Sunn also lived down the street. Small world, eh?

Thanks to all of your input in this discussion! We are getting closer to resolving one of "mtb"'s questions about whether it is possible that a surfer on a wave is able to actually offset the loss of Total Energy due to Drag, and maybe even INCREASE his total energy somehow, enabling him to go faster. But...HOW?

Could it be that by "pumping" or climbing-and-dropping and powering through a series of turns, he IS actually able to increase his TOTAL Energy Budget? Fascinating question!

When I first theorized about the wave dynamics (as related to surfing) back in 1976, right after I bought my first scientific calculator, I assumed that the energy available to the surfer was obtained at the drop-in, and that it was thereafter a closed system. You take off on the highest part of the wave, and after turning, the wave gradually loses height in most cases.

In "Part 1" of my essay on "Surfer Speed vs. Wave Speed and Peel Angle", I used a right triangle, where the SHORT side represents the speed of a surfer moving WITH the wave, at the same speed as the wave moving toward the beach, so I called it "Vwave". It's the same as the speed of the wave over the bottom, so it is a GPS speed.

The component of the surfer speed that is occuring at a right angle to the wave motion toward the beach is represented by the LONGER side of the triangle (longer that is, if the Ride Angle, or Break Angle is greater than 45 degrees, or the Peel Angle is less than 45 degrees). I agonized over the question as to what I should call that speed. Should it be labled "Vsurfer,curl,boat" to describe the surfboard speed through the WATER (i.e., "boat speed"), or maybe just "Vsurfboard,curl"?

But because I wanted to reserve the use of "Vsurfer" to describe the resultant motion of the surfer moving 'across' AND 'with' the wave simultaneously (GPS speed), I decided to simplify the label for the speed of a surfboard that is JUST barely keeping up with the curl, without being passed up by the wave, (i.e., a close-out section). I decided to call it "Vcurl".

Note to "mtb": Vcurl is the maximum speed of the SURFBOARD, while staying just ahead of the curl. It's NOT the maximum speed of the curl itself. The maximum speed of the curl is INFINITY!

So, I ended up with a formula that said: (Vsurfer)^2 = (Vcurl)^2 + (Vwave)^2

Or, if you use the equivalent of the 'squared speeds' for the parts of the right triangle:

3.28gH = 2gH + 1.28gH

The ratio between Vcurl and Vwave is the Tangent of the Maximum Ride Angle (measured away from straight off), so

TAN B = Vcurl / Vwave, so, (TAN B)^2 = (Vcurl)^2 / (Vwave)^2

Angle A = 38.65980825 degrees (= Minimum Peel Angle)

Note that Angle B is the complement of A: B = 90 - A, so, B = 90 - 38.65980825 = 51.34019175 degrees.

I ride a paipo board, where every takeoff is a late takeoff. We make our turn while we are still high up on the face of the wave, where the slope is steep, greater than 45-50 degrees. Sometimes we make angling takeoffs, beginning our high-speed run as soon as we get moving on the wave. I believe we are no more than 20-25% down from the lip of the wave when we catch the wave. Yet we can go fast from the get-go...no bottom turn required, and no climb-and-drop or 'pumping' needed to reach high speeds. Up high is where we can really fly!

My question for "mtb": Is the speed potential on any given wave, then, only a function of wave slope steepness, or only of wave height, or a combination of BOTH? Any idea how we could use your graphs to answer this vexing question?

I always believed that for a 'steady-state' condition, (surfer maintaining a high track, just ahead of the curl, not down low, IN the tube), the maximum makeable speed would be dependent ONLY on the wave height. I guess it might also depend on how Steep the wave is where he's riding.

So, guys: maybe we haven't yet discovered "How Fast Can a Surfer GO?"

Thanks to all of your input in this discussion! We are getting
closer to resolving one of "mtb"'s questions about whether it is
possible that a surfer on a wave is able to actually offset the loss of Total
Energy due to Drag, and maybe even INCREASE his total energy somehow, enabling
him to go faster.

While that's a valid (and interesting) question, that's not my primary
interest, which is:

What is the maximum sustained speed that can be achieved when racing across
the face of a breaking wave?

By "sustained" I mean that the surfer maintains his position
relative to the curl (steady-state conditon). An example of a wave
approximately fulfilling this condition would be traversing across the wave
face at the maximum possible speed with the curl chasing (or leading, if in the
tube) at identically the same speed. The model I referenced in an earlier post
in this thread addresses this question in considerable detail and makes
predictions on speed, wetted area, location of the rider on the board, etc.
(and the effect of where one rides on the wave face and trims the board) that
are generally compatible with observation (see attached graphics) . In the
subsequent discussion. I'll refer to this model as the equilibrium or steady-state model.

An alternative model is your 'Total Energy Budget' model. I have a problem
with this model in that it predicts that the fastest speeds occur when riding
low on the face of a wave, while direct observation reveals that the maximum
speeds are associated with riding high on the face of the wave. So another of
my questions is how does one reconcile this discrepancy?

{Note: the
maximum sustained speed predicted by the equilibrium model generally increases as the slope of the wave increases. However, there is a limit to this trend at very large slopes as the hydrodynamic (planing) efficiency of
the surfboard begins to diminishes as the wave slope becomes very steep.}

But getting back to your question about the effect of pumping...

LarryG wrote:

Could it be that by "pumping" or climbing-and-dropping and
powering through a series of turns, he IS actually able to increase
his TOTAL Energy Budget? Fascinating question!

The input of new energy into the system via pumping increases the energy of
the system such that the loss of energy via drag, etc. no longer balances the input of energy into the system via "pumping". So the system will
respond with an increase in speed to bring the system back into balance. In my
opinion, estimating the magnitude of this increase is a more difficult question
to answer than calculating the maximum steady-state speed of a surfer racing
across the face of a peeling wave.

However, one might be able to obtain a rough estimate (and an upper bound) of
the increase in speed by assuming that the transfer of energy from the rider
to the board via the contraction and extension of his leg muscles occurs without loss of energy. As
best I can recall, the rate of energy loss from the equilibrium model simulation shown in the graphics system is about 3-4 horsepower. The
maximum rate of power generation by an Olympic class athlete is (briefly) a bit
over a horsepower. Power associated with kinetic energy varies as the cube of
the speed. Hence assuming that the rate of loss of energy (due to drag) is 3.5
horsepower, and the rate of power input from the rider pumping is 1.0 horsepower, one
might expect that the (maximum) increase in speed resulting from the pumping
action is approximately 9 percent:

When I first theorized about the wave dynamics (as related to surfing) back
in 1976, right after I bought my first scientific calculator, I assumed that
the energy available to the surfer was obtained at the drop-in, and that it was
thereafter a closed system. You take off on the highest part of the wave, and
after turning, the wave gradually loses height in most cases.

In "Part 1" of my essay on "Surfer Speed vs. Wave Speed and
Peel Angle", I used a right triangle, where the SHORT side
represents the speed of a surfer moving WITH the wave, at the same speed as the
wave moving toward the beach, so I called it "Vwave". It's the same
as the speed of the wave over the bottom, so it is a GPS speed.

The component of the surfer speed that is occuring at a right angle to the
wave motion toward the beach is represented by the LONGER side of the
triangle (longer that is, if the Ride Angle, or Break Angle is greater
than 45 degrees, or the Peel Angle is less than 45
degrees). I agonized over the question as to what I should call that speed.
Should it be labled "Vsurfer,curl,boat" to describe the surfboard
speed through the WATER (i.e., "boat speed"), or maybe just
"Vsurfboard,curl"?

But because I wanted to reserve the use of "Vsurfer"
to describe the resultant motion of the surfer moving 'across' AND 'with'
the wave simultaneously (GPS speed), I decided to simplify the label for the
speed of a surfboard that is JUST barely keeping up with the
curl, without being passed up by the wave, (i.e., a close-out section). I
decided to call it "Vcurl".

Note to "mtb": Vcurl is the maximum speed of the SURFBOARD, while
staying just ahead of the curl. It's NOT the maximum speed of the curl itself.
The maximum speed of the curl is INFINITY!

Agreed...(and is the case in the equilibrium simulation model.

LarryG wrote:

I ride a paipo board, where every takeoff is a late takeoff. We make our turn
while we are still high up on the face of the wave, where the slope is steep,
greater than 45-50 degrees. Sometimes we make angling takeoffs, beginning our
high-speed run as soon as we get moving on the wave. I believe we are no
more than 20-25% down from the lip of the wave when we catch the wave. Yet we
can go fast from the get-go...no bottom turn required, and no climb-and-drop or
'pumping' needed to reach high speeds. Up high is where we can really fly!

Again...the hydrodynamic efficiency of the surfboard decreases with increasing wave steepness (via the changes in shape and size of the wetted area) and so the maximum sustained speed occurs before the slope approaches vertical. So it would seem that for both practical (read: "survival") and theoretical reasons one shouldn't go the last little bit to where the slope becomes vertical.

LarryG wrote:

My question for "mtb": Is the speed potential on any given wave,
then, only a function of wave slope steepness, or only of wave height, or a
combination of BOTH?Any idea how we could use your graphs to answer this
vexing question?

The speed potential is a function of wave slope steepness and wave height -- and more.

There's lots going on in the interaction between the board and the breaking wave. The model only incorporates first order modifications to adapt empirical observations and relationships for planar surfaces to the significant spatial curvatures present in breaking waves. So I don't put a lot of faith into the numerical accuracy of the model predictions. But I think the simulations at least help to indicate--and in some cases, quantify--how the surfboard responds to changes in the characteristics of the wave and changes in the board design. The effect of a sloping sea surface on the "lift" characteristics of the board is especially interesting.

One thing that I found interesting is that it is difficult to make changes in a board design that will result in really significant increases in board speed as nature fights back by decreasing the rate of energy input as the speed of the board across the face of the wave increases..

LarryG wrote:

I always believed that for a 'steady-state' condition, (surfer maintaining a
high track, just ahead of the curl, not down low, IN the tube), the maximum
makeable speed would be dependent ONLY on the wave height. I guess it might
also depend on how Steep the wave is where he's riding.

Yes, it directly depends on the slope of the wave face (along the pathline of the surfer). The latter, in turn, depends on the peel angle and the "straight-off" steepness of the wave at the location of the rider and board.

LarryG wrote:

So, guys: maybe we haven't yet discovered "How Fast Can a Surfer
GO?"

My original formula for finding Drop-In speed used the conversion of Potential Energy to Kinetic Energy to find that the surfer reaches a speed at the bottom that's about 1.25 times the Wave Speed, that's all.

But the surfer doesn't want to STAY there, obviously, so using his excess speed, he turns his board back up face of the wave, going up to where the wave is steeper, and the 'drive' of gravity is stronger, whereupon his board gains BACK the speed he lost climbing back up the wave face, and THEN SOME. There's more power up there, so he can pick up more speed, ending up maybe going 1.6 times as fast as the Wave Speed (GPS speed). That's what I called "Vsurfer" in the formula.

The equivalent speed through (or OVER) the water is what I called "Vcurl", and is the "boat speed". It is also 1.25 times the Wave Speed, same as the speed aquired at the Drop-In.

Bob Shepherd's speedometer experiments showed 27-28 MPH on "20-foot" surf at Waimea Bay, which breaks in at least 31 ft of water, so is actually at least 24 ft wave heights, including the Trough. The Wave Speed is at least 27.33 MPH. Pretty close...

Other guys shared their speed measurements in this forum, and they seemed to agree with my formula, so I was happy about that.

But, Paipo riders don't ever make bottom turns. We turn while still high on the wave, and we have no problem reaching high speed soon after catching the wave. So, it's evident that you really don't need to drive down very far on the takeoff to pick up the speed you need to make the wave. Stand-up surfers only do a bottom turn because they HAVE to. If they turn at the top, they'll probably spin out on the steep upper face of the wave. (Think Waimea or Pipeline: only bodyboarders can pull that off!)

Take another look at the "Fearless Phyliss" video clip. She makes a super-late take-off, turns high, then proceeds to cruise past the guys on their surfboards, who are trimming lower on the wave, and therefore going slower.

If you trim high on a 5 foot wave, you can't go as fast as a similar trim position on a 20 ft wave. So, it's obviously a combination of Wave Slope steepness and Wave Height that governs ultimate speed on the wave. That's assuming the SAME type of board design, of course.

So, now I have one more question for "mtb":

I love the information that is in your graphs, but I STILL don't know what Wave Height gives that Ideal speed of 22.7 MPH you quoted for one of your graphs, or the Maximum Tracking Angle of near 51 degrees for 11 degrees Angle of Attack and 48 degrees Wave Slope.

Also, I wanted to know if you are using the Logarithmic Curve, (plotted on a Polar Coordinate diagram), for the shape of the wave face. In other words, how far up the wave face is that optimum 48 degree slope to be found?

Three times in the last 40 years in Hawaii, I have had the thrilling experience of riding in front of a near-Vertical Wave Face, where the tradewinds were stiff enough to hold the wave (and ME) up and allow me to track just below the pitching lip of the wave. (Waimea Bay, 20 ft; Klausmeyers,15 ft, and "Papa Nui", half a mile out on the South Shore, about 15 ft, or triple overhead.) Only paipo riders and bodyboarders can get away with that.

"mtb"... which factor do you suppose might be the most important for going fast: Wave Height or Waveface Slope?

By the way, my center of gravity is positioned over the "hump" on my bellyboards, which is thickest at about 18 inches from the tail. Your graph shows the surfer CG at about 3 ft from the tail. I'm laying down on my board, which is about 4.25 to 4.5 feet long, or about half as long as a surfboard suitable for 15-20 ft surf. Hmmm... Interesting...!

The wetted length of my boards is about 45 to 48 inches, and the straight part of the bottom is about 2/3rds of the total board length. Lots of planing area there...about 3.2 square feet, same as the wetted area.

Thanks for your fascinating graphs! I'd love to see the underlying functions, but I realize that is your propietary intellectual property, so I won't ask.

My original formula for finding Drop-In speed used the conversion of Potential Energy to Kinetic Energy to find that the surfer reaches a speed at the bottom that's about 1.25 times the Wave Speed, that's all.

But the surfer doesn't want to STAY there, obviously, so using his excess speed, he turns his board back up face of the wave, going up to where the wave is steeper, and the 'drive' of gravity is stronger, whereupon his board gains BACK the speed he lost climbing back up the wave face, and THEN SOME. There's more power up there, so he can pick up more speed, ending up maybe going 1.6 times as fast as the Wave Speed (GPS speed). That's what I called "Vsurfer" in the formula.

The equivalent speed through (or OVER) the water is what I called "Vcurl", and is the "boat speed". It is also 1.25 times the Wave Speed, same as the speed aquired at the Drop-In.

Bob Shepherd's speedometer experiments showed 27-28 MPH on "20-foot" surf at Waimea Bay, which breaks in at least 31 ft of water, so is actually at least 24 ft wave heights, including the Trough. The Wave Speed is at least 27.33 MPH. Pretty close...

Other guys shared their speed measurements in this forum, and they seemed to agree with my formula, so I was happy about that.

But, Paipo riders don't ever make bottom turns. We turn while still high on the wave, and we have no problem reaching high speed soon after catching the wave. So, it's evident that you really don't need to drive down very far on the takeoff to pick up the speed you need to make the wave. Stand-up surfers only do a bottom turn because they HAVE to. If they turn at the top, they'll probably spin out on the steep upper face of the wave. (Think Waimea or Pipeline: only bodyboarders can pull that off!)

Take another look at the "Fearless Phyliss" video clip. She makes a super-late take-off, turns high, then proceeds to cruise past the guys on their surfboards, who are trimming lower on the wave, and therefore going slower.

If you trim high on a 5 foot wave, you can't go as fast as a similar trim position on a 20 ft wave. So, it's obviously a combination of Wave Slope steepness and Wave Height that governs ultimate speed on the wave. That's assuming the SAME type of board design, of course.

So, now I have one more question for "mtb":

I love the information that is in your graphs, but I STILL don't know what Wave Height gives that Ideal speed of 22.7 MPH you quoted for one of your graphs, or the Maximum Tracking Angle of near 51 degrees for 11 degrees Angle of Attack and 48 degrees Wave Slope.

Nor do I. The last time the model was used for a simulation was back around 1994 and unfortunately the file containing the input parameters were lost in a subsequent drive crash-- so I can't look them up. As best I recall most (but not all) of the simulations were executed assuming 6 to 8 foot waves.

LarryG wrote:

Also, I wanted to know if you are using the Logarithmic Curve, (plotted on a Polar Coordinate diagram), for the shape of the wave face. In other words, how far up the wave face is that optimum 48 degree slope to be found?

The model is a steady-state model, it only computes the steady-state situation, not the history as to how it arrived at that state. The steady-state results are determined by the balance of forces (that's why it's "steady-state"--there is no net force to cause a change in speed or direction. That also means that the result is independent of the initial state.

LarryG wrote:

Three times in the last 40 years in Hawaii, I have had the thrilling experience of riding in front of a near-Vertical Wave Face, where the tradewinds were stiff enough to hold the wave (and ME) up and allow me to track just below the pitching lip of the wave. (Waimea Bay, 20 ft; Klausmeyers,15 ft, and "Papa Nui", half a mile out on the South Shore, about 15 ft, or triple overhead.) Only paipo riders and bodyboarders can get away with that.

"mtb"... which factor do you suppose might be the most important for going fast: Wave Height or Waveface Slope?

Both are important. As to which is more important, let's get a rough estimate of how much of a doubling of Wave Height would increase the steady-state speed; then compute the speed increase if the slope were doubled instead--and then compare the two predicted increases.

First we examine the effect of doubling the wave height.

We begin by noting that the surfer's speed is roughly linearly proportional to wave speed, and that wave speed is proportional to the square-root of the effective water depth (i.e. adjusted for wave height):

Vwave = sq-root (g x (h + 3H/4)) h = water depth inj absence of waves, H = wave height (See: Van Dorn - Oceanography and Seamanship)

The change in wave height as the waves move into shoaling water and breaks depends on the ratio of the of the slope of the wave face of the deep water waves to the slope of the ocean bottom, but normally lies within the range of 0.78 to 1.2 (Van Dorn). As an approximation, let's assume a ratio of 1. Now let's see how much a doubling the wave height (H) will increase the wave speed:

Vwave1 = sq-root (g x (H +(3/4) x H) = sq-root ((7/4) x H)

Vwave2 = sq-root (g x (2H + (3/4) x (2H)) = sq-root ((7/2) x H)

So the ratio is:

Vwave2 / Vwave1 = sq-root (2)

...hence doubling the wave size can be expected to roughly increase the surfer's speed by roughly 40%.

Now let's estimate what change in speed will occur by doubling the slope of the wave face at the position of the surfer.

The steady-state speed is achieved when the thrust generated by the combination of the sloping face of the wave and gravity (via the weight of the surfer and board) is equal to the drag. For a 1st-order approximation, we assume that the surfboard is moving sufficiently fast that the primary source of drag is from skin friction, In that case, the drag will be proportional to the square of the speed. We also note that the force associated with the sloping sea surface is linearly proportional to the slope of the wave face. So doubling the slope of the wave approximately doubles the thrust associated with gravity. The system responds by increasing the speed of the surfer so as to double the associated drag. Since drag increases as the square of the surfer's speed, increasing the speed of the surfer by a factor = sq-root (2) will double the drag and thus offset the doubling of the thrust associated with gravity and the sloping wave face.

So the end result is that the two factors are roughly equal -- and to get a better estimate, one must go to a higher order of approximation (which greatly complicates the analysis).

LarryG wrote:

By the way, my center of gravity is positioned over the "hump" on my bellyboards, which is thickest at about 18 inches from the tail. Your graph shows the surfer CG at about 3 ft from the tail. I'm laying down on my board, which is about 4.25 to 4.5 feet long, or about half as long as a surfboard suitable for 15-20 ft surf. Hmmm... Interesting...!

The wetted length of my boards is about 45 to 48 inches, and the straight part of the bottom is about 2/3rds of the total board length. Lots of planing area there...about 3.2 square feet, same as the wetted area.

Thanks for your fascinating graphs! I'd love to see the underlying functions, but I realize that is your propietary intellectual property, so I won't ask.

There are many difficulties in computing an estimate of the speed of a board. Not the least of these is that changes in any one factor affects the magnitudes of many other factors. By way of an example, increasing the angle-of-attack of the board relative to the face of the wave not only changes the lift generated, but also the induced drag. Moreover, the altered lift and drag change the wetted area of the board. The change in wetted area changes the aspect ratio of the wetted surface, thus changing the lift coefficient for the bottom of the board. The change in lift coefficient changes the lift generated by the board, which in turn changes the aspect ratio, ...etc. And so it goes. Only nature has an easy time in accounting for all these changes.

The situation is much simpler for a hydrofoil board with fully submerged foils since the wetted area (and aspect ratio) is normally constant. For the interested reader, I've attached a MS Word file that contains an outline of a procedure to determine the steady-state speed of a hydrofoil-based board utilizing fully submerged foil(s) to illustrate the representations of the various forces involved in solving this simpler case. Unlike the surfboard model (which requires a computer to execute the simulations) this model can be (and has been) executed using a spread-sheet. Although less representative of surfboard hydrodynamics, it still can provide some insight as to the relative importance of various processes.

LarryG wrote:

Gotta go for now. Time for dinner...

UPDATE: I was unable to attach the MS Word file as it exceeds the file size limit by about 20% (and I don't have the time to rewrite it). Sorry!

Yesterday I attempted to attach a MS Word file outlining a simple model of a surfboard and rider traversing across the face of a wave if the rider maintains his position on the wave relative to the breaking point of the wave crest (i.e. the break point and the rider move along the face at the same speed).

However, I was unable to do so as the file exceeded the attachment limits for posting here. So I broke the original file into two parts and have attached them to this post. I hope you find it interesting.

Rod Rodgers is in California at the moment but I suspect he would be happy to post your MS Word file as is, to complement Larry's writings and the articles on the exisiting Terry Hendricks articles he has up. Could pm me or Rod if you are interested in this as an option.

If anybody asks the question: "Where's the Beef?", you have certainly provided it, in spades! Thanks! I hate the vagueness in most people's answers in the various forums that I frequent.

Now, I wonder how many surfers on this forum can follow your mathematical formulas with full understanding. You definitely need to have a solid background in math to be able to digest it all. At least, you didn't get into Vector math or Calculus! We thank you for that...

Let's put this to a test: How about a mind-experiment?

Which surfer, riding identical surfboards, would be able to go faster on a wave that is peeling fast enough, where:

1): Surfer #1 is ONE-THIRD of the way up on the face of a 24-ft wave, where it's not too steep

and

2): Surfer #2 is HALF-WAY way up on a 16-ft wave, where it is steeper, but not TOO steep to handle

AND...

3): Hey! How about that bodyboarder (or paipo rider) tracking way up high at 3/4ths of the way up the face of the 16-ft wave, where it's really steep!

I'm curious to know what kind of speeds the different riders would be capable of reaching, IF the wave wasn't peeling too radically. What IS the maximum makeable peel angle in each case above? The faster rider will be able to make it across a faster-peeling wave, right?

So, which of the two stand-up surfers would be able to go faster: the one on the steeper part of the smaller wave, but at about the same height above sea level as the other guy, who's tracking lower on the face of a bigger wave, (which is breaking in deeper water, so is moving faster)?

I think maybe the surfer speed could be simply related to: HOW HIGH he is above sea level (or the bottom of the wave, say) and HOW STEEP the wave face is where he's tracking. The results would probably be different from what we have already theorized so far.

This is maddening! Mind-boggling! Ha!

I guess we will have to await more experimental results from guys who tow-in, or who enter speed contests. But, they need to report wave heights and wave speeds more reliably and consistently, otherwise the data is almost useless to a statistician. (Garbage IN, Garbage OUT)

Let's put this to a test: How about a mind-experiment?

Which surfer, riding identical surfboards, would be able to go faster on a wave that is peeling fast enough, where:

1): Surfer #1 is ONE-THIRD of the way up on the face of a 24-ft wave, where it's not too steep

and

2): Surfer #2 is HALF-WAY way up on a 16-ft wave, where it is steeper, but not TOO steep to handle

AND...

3): Hey! How about that bodyboarder (or paipo rider) tracking way up high at 3/4ths of the way up the face of the 16-ft wave, where it's really steep!

I'm curious to know what kind of speeds the different riders would be capable of reaching, IF the wave wasn't peeling too radically. What IS the maximum makeable peel angle in each case above? The faster rider will be able to make it across a faster-peeling wave, right?

So, which of the two stand-up surfers would be able to go faster: the one on the steeper part of the smaller wave, but at about the same height above sea level as the other guy, who's tracking lower on the face of a bigger wave, (which is breaking in deeper water, so is moving faster)?

I think maybe the surfer speed could be simply related to: HOW HIGH he is above sea level (or the bottom of the wave, say) and HOW STEEP the wave face is where he's tracking. The results would probably be different from what we have already theorized so far.

Good questions! -- but virtually impossible for me to estimate without generating simulations for each case. Moreover, those simulations would need to be carried out not with the simple model that I described in my previous post, but rather with a more complex model (e.g. the model that generated the graphic output predictions that I presented two postings ago to this thread).

LarryG wrote:

This is maddening! Mind-boggling! Ha!

Agreed!

LarryG wrote:

I guess we will have to await more experimental results from guys who tow-in, or who enter speed contests. But, they need to report wave heights and wave speeds more reliably and consistently, otherwise the data is almost useless to a statistician. (Garbage IN, Garbage OUT)

I agree that it would be a good beginning. But I think more information than just wave height and speed (combined with surfer speed and direction (e.g. peel angle)) would be required to construct a model with reasonable predictive capabilities. For example, a measurement of the slope of the wave face along the surfer's pathline at the location of the surfboard on the wave would be very useful as (it is the slope at the position of the surfboard along his pathline at the position of the surfer on the face of the wave rather than his elevation above the sea level that is important.

I'd like to thank "Strychnine" for his comments regarding the ability of skateborders and snowboarders to "pump" at the bottom of the Half-pipe or the Transition and thereby gain more speed in preparation for the next maneuver. I guess that's really what ice-skaters and roller-bladers have been doing for a long time. OK...not really rocket science, after all.

Well, "mtb", how do we quantify THAT? I'm about ready to give up on this...It's getting too complex for my puny brain do handle.

If the shape of the wave face of a good (i.e., "Plunging" or curling) wave is that of a Logarithmic Spiral (a Logarithmic function when plotted on a Polar Coordinate system), then we could use calculus to find the Slope of the curve (Derivative) at any point on the wave face.

But, I think a surfer tracking at a point on the face of a 16-foot wave where the slope is, say, the Optimum 48 degrees, would be going FASTER than a surfer on an 8-foot wave who is ALSO riding where the slope is 48 degrees. We need to find out...

I'm talking about "Steady-State" conditions, here; No climbing and dropping, and NO pumping!

Including the surfer's speed-increasing maneuvers complicates the situation significantly, so let's try to deal with that separately, after we have accumulated more experimental data in the surf.

Oh well...I guess I need to brush up on my calculus. This is going to be interesting!

Are any of you guys using the "S-Box" to measure and plot your rides on the waves?

We need to "Shake the Bushes" some more, trying to find more people interested in determining just "How Fast CAN a Surfer Go on a Wave?" Hopefully, some day we'll have a better idea...

Holy Moly Batman! I used to live at 84-246 Makau St. Did you know the kid with the artificial leg, named Lucky. Nice friendly kid. Rell Sunn also lived down the street. Small world, eh?

Bill ThrailkillSHAPER SINCE 1958Howzit, guys!

Thanks to all of your input in this discussion! We are getting closer to resolving one of "mtb"'s questions about whether it is possible that a surfer on a wave is able to actually offset the loss of Total Energy due to Drag, and maybe even INCREASE his total energy somehow, enabling him to go faster. But...HOW?

Could it be that by "pumping" or climbing-and-dropping and powering through a series of turns, he IS actually able to increase his TOTAL Energy Budget? Fascinating question!

When I first theorized about the wave dynamics (as related to surfing) back in 1976, right after I bought my first scientific calculator, I assumed that the energy available to the surfer was obtained at the drop-in, and that it was thereafter a closed system. You take off on the highest part of the wave, and after turning, the wave gradually loses height in most cases.

In "Part 1" of my essay on "Surfer Speed vs. Wave Speed and Peel Angle", I used a right triangle, where the SHORT side represents the speed of a surfer moving WITH the wave, at the same speed as the wave moving toward the beach, so I called it "Vwave". It's the same as the speed of the wave over the bottom, so it is a GPS speed.

The component of the surfer speed that is occuring at a right angle to the wave motion toward the beach is represented by the LONGER side of the triangle (longer that is, if the Ride Angle, or Break Angle is greater than 45 degrees, or the Peel Angle is less than 45 degrees). I agonized over the question as to what I should call that speed. Should it be labled "Vsurfer,curl,boat" to describe the surfboard speed through the WATER (i.e., "boat speed"), or maybe just "Vsurfboard,curl"?

But because I wanted to reserve the use of "Vsurfer" to describe the resultant motion of the surfer moving 'across' AND 'with' the wave simultaneously (GPS speed), I decided to simplify the label for the speed of a surfboard that is JUST barely keeping up with the curl, without being passed up by the wave, (i.e., a close-out section). I decided to call it "Vcurl".

Note to "mtb": Vcurl is the maximum speed of the SURFBOARD, while staying just ahead of the curl. It's NOT the maximum speed of the curl itself. The maximum speed of the curl is INFINITY!

So, I ended up with a formula that said: (Vsurfer)^2 = (Vcurl)^2 + (Vwave)^2

Or, if you use the equivalent of the 'squared speeds' for the parts of the right triangle:

3.28gH = 2gH + 1.28gH

The ratio between Vcurl and Vwave is the Tangent of the Maximum Ride Angle (measured away from straight off), so

TAN B = Vcurl / Vwave, so, (TAN B)^2 = (Vcurl)^2 / (Vwave)^2

(Tan B)^2 = (2gH) / (1.28gH), = 2 / 1.28, = 1.5625

Tan B = SQRT(1.5625) = 1.25

Angle B = 51.34019175 degrees ( = Maximum Ride Angle)

In other words, Vcurl is 1.25 times Vwave. That's the same as your drop-in speed, at the bottom turn. That's what you start out with.

If you prefer to use the Peel Angle, A (as measured away from the crest of the wave), then The Tangent of A is Vwave / Vcurl, so...

(TAN A)^2 = (Vwave)^2 / (Vcurl)^2, =(1.28gH) / (2gH), = 1.28 / 2, = 0.64

TAN A = SQRT(0.64), = 0.8

Angle A = 38.65980825 degrees (= Minimum Peel Angle)

Note that Angle B is the complement of A: B = 90 - A, so, B = 90 - 38.65980825 = 51.34019175 degrees.

I ride a paipo board, where every takeoff is a late takeoff. We make our turn while we are still high up on the face of the wave, where the slope is steep, greater than 45-50 degrees. Sometimes we make angling takeoffs, beginning our high-speed run as soon as we get moving on the wave. I believe we are no more than 20-25% down from the lip of the wave when we catch the wave. Yet we can go fast from the get-go...no bottom turn required, and no climb-and-drop or 'pumping' needed to reach high speeds. Up high is where we can really fly!

My question for "mtb": Is the speed potential on any given wave, then, only a function of wave slope steepness, or only of wave height, or a combination of BOTH? Any idea how we could use your graphs to answer this vexing question?

I always believed that for a 'steady-state' condition, (surfer maintaining a high track, just ahead of the curl, not down low, IN the tube), the maximum makeable speed would be dependent ONLY on the wave height. I guess it might also depend on how Steep the wave is where he's riding.

So, guys: maybe we haven't yet discovered "How Fast Can a Surfer GO?"

But it's been fun trying to find the answer!

Any more ideas?

While that's a valid (and interesting) question, that's not my primary interest, which is:

What is the maximum sustained speed that can be achieved when racing across the face of a breaking wave?By "sustained" I mean that the surfer maintains his position relative to the curl (steady-state conditon). An example of a wave approximately fulfilling this condition would be traversing across the wave face at the maximum possible speed with the curl chasing (or leading, if in the tube) at identically the same speed. The model I referenced in an earlier post in this thread addresses this question in considerable detail and makes predictions on speed, wetted area, location of the rider on the board, etc. (and the effect of where one rides on the wave face and trims the board) that are generally compatible with observation (see attached graphics) . In the subsequent discussion. I'll refer to this model as the equilibrium or steady-state model.

An alternative model is your 'Total Energy Budget' model. I have a problem with this model in that it predicts that the fastest speeds occur when riding low on the face of a wave, while direct observation reveals that the maximum speeds are associated with riding high on the face of the wave. So another of my questions is

how does one reconcile this discrepancy?{Note: the maximum sustained speed predicted by the equilibrium model generally increases as the slope of the wave increases. However, there is a limit to this trend at very large slopes as the hydrodynamic (planing) efficiency of the surfboard begins to diminishes as the wave slope becomes very steep.}

But getting back to your question about the effect of pumping...

The input of new energy into the system via pumping increases the energy of the system such that the loss of energy via drag, etc. no longer balances the input of energy into the system via "pumping". So the system will respond with an increase in speed to bring the system back into balance. In my opinion, estimating the magnitude of this increase is a more difficult question to answer than calculating the maximum steady-state speed of a surfer racing across the face of a peeling wave.

However, one might be able to obtain a rough estimate (and an upper bound) of the increase in speed by assuming that the transfer of energy from the rider to the board via the contraction and extension of his leg muscles occurs without loss of energy. As best I can recall, the rate of energy loss from the equilibrium model simulation shown in the graphics system is about 3-4 horsepower. The maximum rate of power generation by an Olympic class athlete is (briefly) a bit over a horsepower. Power associated with kinetic energy varies as the cube of the speed. Hence assuming that the rate of loss of energy (due to drag) is 3.5 horsepower, and the rate of power input from the rider pumping is 1.0 horsepower, one might expect that the (maximum) increase in speed resulting from the pumping action is approximately 9 percent:

Vpump / Vno_pump = cube root (POWERpump/POWERno_pump) = (4.5/3.5)^0.33 = 1.09

Agreed...(and is the case in the equilibrium simulation model.

Again...the hydrodynamic efficiency of the surfboard decreases with increasing wave steepness (via the changes in shape and size of the wetted area) and so the maximum sustained speed occurs before the slope approaches vertical. So it would seem that for both practical (read: "survival") and theoretical reasons one shouldn't go the last little bit to where the slope becomes vertical.

The speed potential is a function of wave slope steepness and wave height -- and more.

There's lots going on in the interaction between the board and the breaking wave. The model only incorporates first order modifications to adapt empirical observations and relationships for planar surfaces to the significant spatial curvatures present in breaking waves. So I don't put a lot of faith into the numerical accuracy of the model predictions. But I think the simulations at least help to indicate--and in some cases, quantify--how the surfboard responds to changes in the characteristics of the wave and changes in the board design. The effect of a sloping sea surface on the "lift" characteristics of the board is especially interesting.

One thing that I found interesting is that it is difficult to make changes in a board design that will result in really significant increases in board speed as nature fights back by decreasing the rate of energy input as the speed of the board across the face of the wave increases..

Yes, it directly depends on the slope of the wave face (along the pathline of the surfer). The latter, in turn, depends on the peel angle and the "straight-off" steepness of the wave at the location of the rider and board.

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## BOARDCOM_0.GIF

Hi, guys!

My original formula for finding Drop-In speed used the conversion of Potential Energy to Kinetic Energy to find that the surfer reaches a speed at the bottom that's about 1.25 times the Wave Speed, that's all.

But the surfer doesn't want to STAY there, obviously, so using his excess speed, he turns his board back up face of the wave, going up to where the wave is steeper, and the 'drive' of gravity is stronger, whereupon his board gains BACK the speed he lost climbing back up the wave face, and THEN SOME. There's more power up there, so he can pick up more speed, ending up maybe going 1.6 times as fast as the Wave Speed (GPS speed). That's what I called "Vsurfer" in the formula.

The equivalent speed through (or OVER) the water is what I called "Vcurl", and is the "boat speed". It is also 1.25 times the Wave Speed, same as the speed aquired at the Drop-In.

Bob Shepherd's speedometer experiments showed 27-28 MPH on "20-foot" surf at Waimea Bay, which breaks in at least 31 ft of water, so is actually at least 24 ft wave heights, including the Trough. The Wave Speed is at least 27.33 MPH. Pretty close...

Other guys shared their speed measurements in this forum, and they seemed to agree with my formula, so I was happy about that.

But, Paipo riders don't ever make bottom turns. We turn while still high on the wave, and we have no problem reaching high speed soon after catching the wave. So, it's evident that you really don't need to drive down very far on the takeoff to pick up the speed you need to make the wave. Stand-up surfers only do a bottom turn because they HAVE to. If they turn at the top, they'll probably spin out on the steep upper face of the wave. (Think Waimea or Pipeline: only bodyboarders can pull that off!)

Take another look at the "Fearless Phyliss" video clip. She makes a super-late take-off, turns high, then proceeds to cruise past the guys on their surfboards, who are trimming lower on the wave, and therefore going slower.

If you trim high on a 5 foot wave, you can't go as fast as a similar trim position on a 20 ft wave. So, it's obviously a combination of Wave Slope steepness and Wave Height that governs ultimate speed on the wave. That's assuming the SAME type of board design, of course.

So, now I have one more question for "mtb":

I love the information that is in your graphs, but I STILL don't know what Wave Height gives that Ideal speed of 22.7 MPH you quoted for one of your graphs, or the Maximum Tracking Angle of near 51 degrees for 11 degrees Angle of Attack and 48 degrees Wave Slope.

Also, I wanted to know if you are using the Logarithmic Curve, (plotted on a Polar Coordinate diagram), for the shape of the wave face. In other words, how far up the wave face is that optimum 48 degree slope to be found?

Three times in the last 40 years in Hawaii, I have had the thrilling experience of riding in front of a near-Vertical Wave Face, where the tradewinds were stiff enough to hold the wave (and ME) up and allow me to track just below the pitching lip of the wave. (Waimea Bay, 20 ft; Klausmeyers,15 ft, and "Papa Nui", half a mile out on the South Shore, about 15 ft, or triple overhead.) Only paipo riders and bodyboarders can get away with that.

"mtb"... which factor do you suppose might be the most important for going fast: Wave Height or Waveface Slope?

By the way, my center of gravity is positioned over the "hump" on my bellyboards, which is thickest at about 18 inches from the tail. Your graph shows the surfer CG at about 3 ft from the tail. I'm laying down on my board, which is about 4.25 to 4.5 feet long, or about half as long as a surfboard suitable for 15-20 ft surf. Hmmm... Interesting...!

The wetted length of my boards is about 45 to 48 inches, and the straight part of the bottom is about 2/3rds of the total board length. Lots of planing area there...about 3.2 square feet, same as the wetted area.

Thanks for your fascinating graphs! I'd love to see the underlying functions, but I realize that is your propietary intellectual property, so I won't ask.

Gotta go for now. Time for dinner...

Nor do I. The last time the model was used for a simulation was back around 1994 and unfortunately the file containing the input parameters were lost in a subsequent drive crash-- so I can't look them up. As best I recall most (but not all) of the simulations were executed assuming 6 to 8 foot waves.

The model is a steady-state model, it only computes the steady-state situation, not the history as to how it arrived at that state. The steady-state results are determined by the balance of forces (that's why it's "steady-state"--there is no net force to cause a change in speed or direction. That also means that the result is independent of the initial state.

Both are important. As to which is more important, let's get a rough estimate of how much of a doubling of Wave Height would increase the steady-state speed; then compute the speed increase if the slope were doubled instead--and then compare the two predicted increases.

First we examine the effect of doubling the wave height.

We begin by noting that the surfer's speed is roughly linearly proportional to wave speed, and that wave speed is proportional to the square-root of the effective water depth (i.e. adjusted for wave height):

Vwave = sq-root (g x (h + 3H/4)) h = water depth inj absence of waves, H = wave height (See: Van Dorn - Oceanography and Seamanship)

The change in wave height as the waves move into shoaling water and breaks depends on the ratio of the of the slope of the wave face of the deep water waves to the slope of the ocean bottom, but normally lies within the range of 0.78 to 1.2 (Van Dorn). As an approximation, let's assume a ratio of 1. Now let's see how much a doubling the wave height (H) will increase the wave speed:

Vwave1 = sq-root (g x (H +(3/4) x H) = sq-root ((7/4) x H)

Vwave2 = sq-root (g x (2H + (3/4) x (2H)) = sq-root ((7/2) x H)

So the ratio is:

Vwave2 / Vwave1 = sq-root (2)

...hence doubling the wave size can be expected to roughly increase the surfer's speed by roughly 40%.

Now let's estimate what change in speed will occur by doubling the slope of the wave face at the position of the surfer.

The steady-state speed is achieved when the thrust generated by the combination of the sloping face of the wave and gravity (via the weight of the surfer and board) is equal to the drag. For a 1st-order approximation, we assume that the surfboard is moving sufficiently fast that the primary source of drag is from skin friction, In that case, the drag will be proportional to the square of the speed. We also note that the force associated with the sloping sea surface is linearly proportional to the slope of the wave face. So doubling the slope of the wave approximately doubles the thrust associated with gravity. The system responds by increasing the speed of the surfer so as to double the associated drag. Since drag increases as the square of the surfer's speed, increasing the speed of the surfer by a factor = sq-root (2) will double the drag and thus offset the doubling of the thrust associated with gravity and the sloping wave face.

So the end result is that the two factors are roughly equal -- and to get a better estimate, one must go to a higher order of approximation (which greatly complicates the analysis).

There are many difficulties in computing an estimate of the speed of a board. Not the least of these is that changes in any one factor affects the magnitudes of many other factors. By way of an example, increasing the angle-of-attack of the board relative to the face of the wave not only changes the lift generated, but also the induced drag. Moreover, the altered lift and drag change the wetted area of the board. The change in wetted area changes the aspect ratio of the wetted surface, thus changing the lift coefficient for the bottom of the board. The change in lift coefficient changes the lift generated by the board, which in turn changes the aspect ratio, ...etc. And so it goes. Only nature has an easy time in accounting for all these changes.

The situation is much simpler for a hydrofoil board with fully submerged foils since the wetted area (and aspect ratio) is normally constant. For the interested reader, I've attached a MS Word file that contains an outline of a procedure to determine the steady-state speed of a hydrofoil-based board utilizing fully submerged foil(s) to illustrate the representations of the various forces involved in solving this simpler case. Unlike the surfboard model (which requires a computer to execute the simulations) this model can be (and has been) executed using a spread-sheet. Although less representative of surfboard hydrodynamics, it still can provide some insight as to the relative importance of various processes.

UPDATE: I was unable to attach the MS Word file as it exceeds the file size limit by about 20% (and I don't have the time to rewrite it). Sorry!

Yesterday I attempted to attach a MS Word file outlining a simple model of a surfboard and rider traversing across the face of a wave if the rider maintains his position on the wave relative to the breaking point of the wave crest (i.e. the break point and the rider move along the face at the same speed).

However, I was unable to do so as the file exceeded the attachment limits for posting here. So I broke the original file into two parts and have attached them to this post. I hope you find it interesting.

(Please let me know if you find any mistakes)

## BrdSimP0.doc

## BrdSimP2.doc

MTB,

Rod Rodgers is in California at the moment but I suspect he would be happy to post your MS Word file as is, to complement Larry's writings and the articles on the exisiting Terry Hendricks articles he has up. Could pm me or Rod if you are interested in this as an option.

Bob

Hi, guys!

This is for "mtb":

If anybody asks the question: "Where's the Beef?", you have certainly provided it, in spades! Thanks! I hate the vagueness in most people's answers in the various forums that I frequent.

Now, I wonder how many surfers on this forum can follow your mathematical formulas with full understanding. You definitely need to have a solid background in math to be able to digest it all. At least, you didn't get into Vector math or Calculus! We thank you for that...

Let's put this to a test: How about a mind-experiment?

Which surfer, riding identical surfboards, would be able to go faster on a wave that is peeling fast enough, where:

1): Surfer #1 is ONE-THIRD of the way up on the face of a 24-ft wave, where it's not too steep

and

2): Surfer #2 is HALF-WAY way up on a 16-ft wave, where it is steeper, but not TOO steep to handle

AND...

3): Hey! How about that bodyboarder (or paipo rider) tracking way up high at 3/4ths of the way up the face of the 16-ft wave, where it's really steep!

I'm curious to know what kind of speeds the different riders would be capable of reaching, IF the wave wasn't peeling too radically. What IS the maximum makeable peel angle in each case above? The faster rider will be able to make it across a faster-peeling wave, right?

So, which of the two stand-up surfers would be able to go faster: the one on the steeper part of the smaller wave, but at about the same height above sea level as the other guy, who's tracking lower on the face of a bigger wave, (which is breaking in deeper water, so is moving faster)?

I think maybe the surfer speed could be simply related to: HOW HIGH he is above sea level (or the bottom of the wave, say) and HOW STEEP the wave face is where he's tracking. The results would probably be different from what we have already theorized so far.

This is maddening! Mind-boggling! Ha!

I guess we will have to await more experimental results from guys who tow-in, or who enter speed contests. But, they need to report wave heights and wave speeds more reliably and consistently, otherwise the data is almost useless to a statistician. (Garbage IN, Garbage OUT)

Thanks again for the formulas!

Good questions! -- but virtually impossible for me to estimate without generating simulations for each case. Moreover, those simulations would need to be carried out not with the simple model that I described in my previous post, but rather with a more complex model (e.g. the model that generated the graphic output predictions that I presented two postings ago to this thread).

Agreed!

I agree that it would be a good beginning. But I think more information than just wave height and speed (combined with surfer speed and direction (e.g. peel angle)) would be required to construct a model with reasonable predictive capabilities. For example, a measurement of the slope of the wave face along the surfer's pathline at the location of the surfboard on the wave would be very useful as (it is the slope at the position of the surfboard along his pathline at the position of the surfer on the face of the wave rather than his elevation above the sea level that is important.

Hi, guys!

I'd like to thank "Strychnine" for his comments regarding the ability of skateborders and snowboarders to "pump" at the bottom of the Half-pipe or the Transition and thereby gain more speed in preparation for the next maneuver. I guess that's really what ice-skaters and roller-bladers have been doing for a long time. OK...not really rocket science, after all.

Well, "mtb", how do we quantify THAT? I'm about ready to give up on this...It's getting too complex for my puny brain do handle.

If the shape of the wave face of a good (i.e., "Plunging" or curling) wave is that of a Logarithmic Spiral (a Logarithmic function when plotted on a Polar Coordinate system), then we could use calculus to find the Slope of the curve (Derivative) at any point on the wave face.

But, I think a surfer tracking at a point on the face of a 16-foot wave where the slope is, say, the Optimum 48 degrees, would be going FASTER than a surfer on an 8-foot wave who is ALSO riding where the slope is 48 degrees. We need to find out...

I'm talking about "Steady-State" conditions, here; No climbing and dropping, and NO pumping!

Including the surfer's speed-increasing maneuvers complicates the situation significantly, so let's try to deal with that separately, after we have accumulated more experimental data in the surf.

Oh well...I guess I need to brush up on my calculus. This is going to be interesting!

Are any of you guys using the "S-Box" to measure and plot your rides on the waves?

We need to "Shake the Bushes" some more, trying to find more people interested in determining just "How Fast CAN a Surfer Go on a Wave?" Hopefully, some day we'll have a better idea...

Thanks again, guys!

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