Dynamics - The Trim Equation

The Trim Equation

To truly make sense of this, you will have to had to have read my essays in Dynamics – Surfing the Force. Perhaps its not completely necessary to have read them all, but to have at least looked at some of the illustrations is bound to help.

I was hoping to pop this essay at the end of the Dynamics – Surfing the Force, but the posts in the thread where straying off topic, as post often do, so I felt it best to start a Part II, so to speak.

So here’s the trim equation,

Rtrim = W [sin(roll angle)/cos(pitch angle)] i + W[sin(roll angle)/cos(pitch angle)] [ 1/ tan(yaw anlge)] j

Lets surf. The example is a surfer in trim, perhaps in the tube.

For those who have been exposed to vectors, this should all be pretty straightforward. (Please check my work!) For those who haven’t had the pleasure, this can still be understood, but it will require that you at least understand how I define pitch, roll and yaw, see figure 1. Remember that the bottom plane of the board is perpendicular to the propulsion vector. (If you can remember your HS geometry you should be fine.)

By the way, I figure that the way you find out if you are right, at least in the math, is by putting ‘it’ out there and letting those sufficiently interested, go over what you’ve done and point how wrong you actually are, or in general how much of a mess you’ve made. Since you need a forum (a bunch of potentially interested people) for such a thing, Swaylock’s seemed like the place, given the topic (and my resources.)

I’ve made some assumptions in order to keep the confusion to a minimum.

First, the amount of surface area that produces the propulsion remains constant. Second, the surfer is in trim, neither dropping nor rising on the face –i.e. he’s holding a horizontal line. Third, drag is not considered, nor are fins. But they’re obviously there. These are admittedly important things to leave out. But I feel that, by not including them at this point, it will not take away from the point to be made, which should be apparent soon enough. They will likely be other assumptions that I have made and have forgotten to state, and perhaps you’ll be kind enough to point them out, but my strategy has been to simplify in order to convey the bigger picture.

If you make the assumption the surfer and surfboard weight is balanced by the vertical component of propulsion, then what’s left is basically the sum of the propulsion vector components in the x and y directions (see diagram.) That is the surfer will be moving both towards-the-beach and down-the-line. Under these conditions, the orientation of the surfboard that of pitch, roll and yaw is indicated in the diagram (again, you’ll have to use a little geometry to get there.)

Now with the vertical component of P, that is Pzk being equal to W (which is, as you may have guessed, mass times the acceleration due to gravity), and with some substitution, you get the following for the propulsive force left after you balance its vertical component with the weight,

Here the simplified vector equation is

R = P + W

we have trim so,

Pz = P cos(pitch angle) = W

so, we now have

P = W/ cos(pitch angle),

which eventually gives,

Rtrim = W [sin(roll angle)/cos(pitch angle)] i + W[sin(roll angle)/cos(pitch angle)] [ 1/ tan(yaw angle)] j


force in the towards-the-beach direction = W [sin(roll angle)/cos(pitch angle)]


force in the down-the-line direction = W[sin(roll angle)/cos(pitch angle)] [ 1/ tan(yaw angle)]

By the way, if you’re wondering why yaw wouldn’t impact the force towards-the-beach, it does, see diagram, the effect is just sort of buried.

So, you’re in the tube, but you sense you need a little more down-the-line speed, what do you do? From the above relationships, it would appear you’ve got a few options.

You might consider decreasing you yaw, or trying to somehow pull the nose towards the face of the wave. This would reduce tan(yaw angle), which amounts to increasing 1/ tan(yaw angle), which would give you a little speed in the down-line-direction. (By the way to see how the cosine, sine and tangent functions change with the angle see figure 3, its crude the actually relationships are curves, but its enough.)

Or you might consider increasing your roll. Now the problem with that strategy is that roll will also send you towards the beach a lot faster, something which is likely to get you in trouble. You, being the computer you are, decide the roll option is not the best one and grab your rail pulling up to prevent it.

What’s left, increase your pitch, thereby decreasing cos(pitch angle)? Maybe, but possible a little dangerous, for your current pitch is just about right to balance your weight against the propulsive force. You decide not to go for too much more pitch, but it’s a nice option to have.

Your solution? Grab the rail and try and swing the nose towards the wave face (decrease yaw.) You accomplish this by ever so gently trying to torque the board about its center of mass by placing a little more weight on you rear foot, and given you’ve grabbed the rail in front of your center of mass, pull a little towards the wave.

But you know that by screwing with the yaw which, given that you’re in trim, will likely impact both your towards-the-beach and down-the-line velocity (see diagram for relationship), you pull extra hard on that rail, and kept the pressure up on that rear foot. You want to keep roll to a minimum.

Are these the only options, in my simplified example probably. But obviously there are more.

Cameras click.

Does this prove all of wild and crazy notions regarding propulsion, propulsive flow versus apparent flows, etc. as developed in Dynamics – Surfing the Force? Hardly. If you buy into my model this is what you get.

What’s next? You actually have the nut, though perhaps not trival, its would not require all that much to start adding the other (vector) terms, such as drag, maybe the propulsive component of kant and toed lateral fins, or the consequences of tail flips and rocker on the propulsive component, even concaves and vees and the other strange things that are done to boards, that is you buy my nonsense about propulsion in surfing.

Much of what now needs to be done however is building functions –e.g. collect data on drag as a function of orientation, or on propulsion as a function of orientation.

But that’s the nut.



I’m sure I’ve made a mistake. Perhaps someone would be kind enough to find it and let me know where.

(edit: I had ‘I sure I’ve…’ my apologies. I’ve corrected it. )

I’d like you to call me Woody.

(I volunteer of course. Why again did you start a new thread? Didn’t like the challenges to your repetitous chanting about how we’re all doing an absolutely unique form of anti-gravity action sport that’s never been seen or experienced by any other life form? )

One, you’re wrong about the propulsion component


why won’t you give data about some studies of measured velocity of the thing itself–these powerful upwave flows versus the waves’ velocity toward beach and then we’ll talk about how the surfer is always going pretty quickly in the opposite direction to them anyway? You say you don’t know those ratios–oh you say you haven’t looked at such data? Oh you’re all pure theory here.

I see.

Two, you’re going on and on, prolific to a rather crank-like extent, and frankly you want a different forum, one for marijuana-smoking mathemetician computer nerds (AKA kooks?) who simply post up stoned theoretical drivel to a communal head-nodding, and don’t bother each other with niggling questions about why nobody in the history of surfing ever noticed that there was this amazingly powerful component of up the wave flow that was actually driving us all in its own opposite direction, and not that it really was just relative to one’s downward trim.

Three, you misused “Prologue,” where you wanted “Epilogue.”

That is all. Back to your vacuous nattering and diagraming, or will you have some (several) measured upflow/crest velocity ratios for us next post?

And I’ll call you Buzz, because I think you’re just the toy for cold (con)fusion, hyperbolic wormhole time travel, and perpetual surfing theory (pro)motion.

Now off with you and be about it all –

to infinity and beyoonnnd!!!

Don’t forget to write!!

BTW: I think your exact errors are substituting a flow of substantial velocity and force for which you have no measured data, in the place of a relatively static body of water over which crests a wave, by which a surfer is pushed and driecting himself across lateral to the pull of gravity, resulting in a directed plane, not a 110 or so degree directional conversion of kinetic energy by some quantum theory

and your use of PROPULSION

instead of REPULSION

See river surfing thread and my last entry on Surfing the Force thread

Ever been well and truly blitzed out the back by the flow as a river surfer is? Or rather you’ve just lost the wave as it rolled under you…


Thanks about the Prologue error, that acutally managed to migrate down the page somehow. I will edit it.

Thank’s again,


Too funny, Janklow!

Reminds of an old post from Laconic1:


But check out LeeDD’s response right below it.


I like a little trim once and awhile.

trim sure feels good. no arguments there.

Woody and Buzz - you guys crack me up.


Slinky Dog.

PS Where’s Mr. Potatohead (Jay); did he go back to Idaho?

taterhead is around. Just talked to him last week about my latest project.


Woody and Buzz - you guys crack me up.


Slinky Dog.

PS Where’s Mr. Potatohead (Jay); did he go back to Idaho?

Are you one of the dog brothers Keith ?




Some Notes

  • The propulsion vector when normalized defines the unit vector of the plane of the bottom of the (effective) board.

  • Modern wave models can used provide velocity data for modeling the relationship between the flow in a wave and propulsion. Basically the model I’ve proposed is that of a flow hitting a flat sheet at a pretty high angle, see Dynamics – Surfing the Force. I’ve no idea about the precise nature of the flow once it hits the bottom, or as it approaches it –i.e. streamlines and such, other than what might be modeled using various conservation approaches and guesses based on work like Stavitsky. I’ve more or less sketched out my guess of what might be going on in Dynamics – Surfing the Force.


Notes on Notes?

My apologies, I keep forgetting to make the following point that ties all this back to the original thread, Dynamics - Surfing the Force.

  • In the thread Dynamics - Surfing the Force, I wrote about that palpable force that seems to be somewhere around your front foot while surfing. That’s basically the propulsion vector, at least that’s its location. The (palpable) force being directed perpendicular to the bottom, up and through your foot. It’s the force produced by the flow of the wave interacting with the bottom of your board.


R=P+W ??

You haven’t told us what R is

And what is P ?

Please explain.


PS I take it that since weight is now part of your equation, that you have now realised that the wave is driving the board with a force which is proportional to weight . . . . just like in other ‘gravity’ sports. . . . . if so you have changed your position, however it is impossible to tell right now as your post is so scattered and confusing. . . no clear propositions or conclusions… . . . and your graphs are incorrect too. . . straight lines aren’t correct.



I said I’d give some feedback on these ideas once I had an idea to chew on them.

I’ve dropped this in this thread, although I talk about ideas from the other thread too.

Honestly, I think you are overestimating the effect of the upward flow.

Even a 40kg grom/board combo has no problems overcoming the effect of this heading straight down the face at, oh 20km/h. That measley 400 kg/m-s momentum has no problems overcoming drag on a shortboard, plus the upflow effect.

A stationary wave (flowrider, river wave, etc) has a lot more of the force you are talking about and I think your model is more appropriate to that.

All that said I reckon the effect you are talking about is still there on an ocean wave and has some impact, just less that you think.

A 1m wave travelling at 12sec will displace water from the trough to the peak at somewhere in something like 1sec. Like I said, that flow has an effect, especially if you bury a rail in it… But most of the time we don’t want to do that.

I think it is part of trim tho, because it is a factor we want to control… But I think it’s something we use as a control mechanism, balancing it with the board planing and other factors.

Naturally I may be quite wrong… And I am interested in what everyone else thinks.

Still an interesting avenue of thought.



R = P + W

Is a vector equation (underlining variables is often used to indicate a vector, bolding is also used), P is the propulsive force, and you’re correct W is weight, which is -mgk , k being the unit vector in the postive vertical direction, so the negative sign means it points down. R is the resultant force you get when you add the two. The propulsive force is that force which arises from the impact of the flow from the wave on the bottom of the board.

Stand in waist deep water and let a wave hit you. For the first few moments, if you eyes were closed you’d have a hard time telling if it was a wave or some fool throwing a bucket of water at you. As soon as the wave envelops you’d of course know it wasn’t a bucket of water, but even then you still might not know it was a wave. I know its not the way people ‘see’ waves, especially if they’re riding them.

The propulsion derived then produces motion, what I have called apparent motion, perhaps not the best term as would suggest that the motion isn’t real, and it is. A major point of my arguement was that to design for the ‘apparent motion’, which can be interpreted as a flow, as if it was the source of propulsion, will likely cost you - for you just generated that motion.


I think your “location” of the “palpable force” (personally I don’t think those terms are quite correct, but who cares) will be different depending on the board (e.g.'s knee, thruster, fish, mid-length, long, SUP, paddle) and the surfer’s style (e.g., front-footed, rear-footed, balanced).

I understand you idea, but calling that vector the “propulsion vector” I am also not sure about. Propulsion suggests drive or thrust, which are vectorless and physics usually associates with objects that have their own inbuilt power source (so they are often somewhat simpler). I’d suggest that you mean velocity, which is a vector of speed and direction.

Anyway, sorry to nitpick. Again, it’s possible that I am wrong and I’d be happy for anyone to point out where and why.

Just my 2c, again.


Yes, I agree about the palpable force, it depends – but I would argue it’s still there.

No, force is vector. Here propulsion is just a name, the name I gave to the force that propels the surfboard, seemed appropriate at the time - it was the vector representation of the force that propels a surfboard.

I’m not great a choosing names for things. For example, I wish I never called the generated motion ‘apparent motion’ in one of essays in Dynamics – Surfing the Force. Like I mentioned in my reply to Roy, it’s a real motion, to call it apparent risks suggesting that it might be otherwise.


Dear Kooky CAD,

There are no propulsion elements in surfing except the surfer’s muscles and the wave’s forward speed. Gravity which provides the forward motion in the first place is an attractive force, not a propulsive one.

I now switch to BOLD so you don’t miss this next IMPORTANT DISTINCTION:::::Upwave flow is in any case a repulsive force.

Any case you make, you cannot escape this fact–

your own modeling shows it as a repulsive force relative to the bias of overall surfing direction.

You deflect against it.

ON a north bound wave,

any upwave flow is definitively south bound

opposite north wave direction and gravity’s pull north,


surfer’s stable lateral deflection off the water under him relative the gravity fall line gives trim,

and he can push off the water ahead of him given any turn’s centrifugal momentum,

you see.

You’re proving one thing about guys with diagrams Buzz. Thus I find it very useful your persistence. (Did you mean to give the impression you kind of leaned rightward politically with the police firehosing the demonstrator? I can’t stop foiling you (!) – I find wrongway-microthinking amusing.)

Until next time Buzz. You go boy! To infinity and beyooonnnnd! Actually, you should be in the middle East, engineering their peace plan! You could engineer it or a sellable facsimile with calculus and a captive, easily-tiring audience, I’m sure.


Sorry, I forgot about the PS portion.

I think you’d be hard pressed to find were I suggested that weight was not involved, or that it wasn’t important. And yes you can interpret the trim equation as a force that is proportional to weight. But no, it doesn’t mean it’s like other gravity sports. There maybe a sport on the planet that doesn’t involve gravity in some way, but at the moment I can’t think of one (even if its just to keep the cards on the table.) But just because gravity shows up in the equations doesn’t mean a sport is a 'gravity sport’ in the usual way 'gravity sport’ is used -e.g. skiing, snowboarding, sleighing, etc. In gravity sports, gravity is the sole means of propulsion; and it can be modeled using the same mechanical treatment as a block sliding down an inclined plane. That is not to say, a luge is merely a block sliding down an inclined plane, or that a snowboarder is either - but in both cases their sole source of energy is from the downhill run.

In surfing, it is possible to adjust roll, pitch and yaw such that the vertical component of the propulsion vector is not balanced by weight. Two cases: it exceeds weight so the surfer climbs, or it is less than weight and the surfer ‘falls’, but in either case he will still have the other components of propulsion operating, so he can go down or up the wave at an angle, say. In fact surfers, if they are on the wave, are generally moving towards the beach, to that towards-the-beach component is almost always present.

I haven’t changed my position. I apologize if my posts appear to be scattered and confused. I’m not sure I can do better. I will try in the future.


Since when did we start underlining vectors? Is this a ME thing? I know EE’s don’t use this notation.

We tend to think of waves as the wave function’s frequency is below the cutoff mode of the beach boundary condition.

Closeouts: infinite phase velocity relative to the shoreline. :slight_smile: