# I Minimum velocity for a surfer to catch a wave

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1. Sep 2, 2016

### MrMann

Hi,
I have a question about surfing, how fast must a surfer be travelling (paddle velocity) in order to catch a wave?

I took the approach to consider the surfer in the wave's frame of reference, then supposed that the surfer would be 'caught' by the wave if the surfer's kinetic energy was less than or equal to the potential energy of the wave.
Ep = m*g*h : Potential energy
Ek = 1/2*m*v^2 : Kinetic energy

v_s' = v_s-v_phase

taking the limit in the wave frame, the Ek of the surfer equalling the Ep of the wave, should give v_s(min), the minium velocity of the surfer to catch the wave, gives:

v_s' = sqrt(2*g*h)

Using this value, translating it back to the rest frame, and then using typical real life ocean wave parameters I found that the wave velocity is negative, the surfer would be travelling towards the wave in the minimum velocity scenario. This infers that the surfer could be stationary and still catch the wave. I know this not to be the actual case

It seemed rather simply initially but I cannot think what I am missing, any thoughts?

v_s : being the surfer velocity in the rest frame
v_s' : being the surfer velocity in the wave frame
v_phase : being the wave's velocity
m: mass of surfer
h : height of wave
g: the acceleration due to gravity

2. Sep 2, 2016

### CWatters

Did you mean greater than?

That assumes the surfers KE is converted to PE as he goes up the front of the wave. I not convinced that the energy to go up the wave comes from the surfer. Why not from the wave?

3. Sep 3, 2016

### MrMann

I mean less than or equal to as if it is greater than the surfer would not be 'caught' and the wave would pass him by.
I mean less than or equal, as if the surfer has more energy than the potential energy of the 'well' created by the waves height he will 'escape' which means he will not be caught by the wave in the rest frame, i.e the wave will pass him by.

well yes i agree it's energy from the wave in the rest frame, but in the wave frame this KE is with the surfer, unless you meant from another process? It is definitely an energy problem here, as energy is lost in the surfers transition in the wave frame

4. Sep 3, 2016

### sophiecentaur

I agree. The surfer must be getting a supply of energy from somewhere else. The water on the surface of a breaking wave (and it has to be breaking for the surfer to 'surf') is not just going up and down, it is moving up the leading slope of the wave (look at film of the spume on the front of the wave), pushing the board upwards. By facing diagonally down the slope, the surfer can get KE from a constant supply of PE as he/she's constantly lifted upwards (resolving the forces favourably). This accumulated KE means that the surfer is travelling faster than the water and can move up fast and then drop down into the trough again at speed, zigzagging along the leading edge of the wave whilst it lasts. Alternatively, the surfer can keep at the same level by maintaining the right angle of the board so that the velocity up is the same as the velocity down, relative to the water but the horizontal velocity is increasing until limited by the hull resistance.
PS look at film of surfers on artificial surf 'slopes'; the water shoots out of horizontal jets from the bottom and flows uphill fast. The surfer is constantly aiming downhill.

5. Sep 3, 2016

### MrMann

Thanks for the response, without going off topic, why must the wave break to surf? I thought these were just more favourable conditions.
Yes I agree he surfer would be travelling faster than the wave in a path non-parraellel to the wavefront if the surfer caught the wave, maintaining the same position, if they choose so, on the wave by having a velocity component equal to the wave velocity in the direction perpendicular to the wavefront.
The water I see gives the buoyancy to the surfer, enabling him to access to the waves potential energy, from its height. Maybe there are additional hydrodynamic/static forces acting here, these could be factored in with ease into the two frames. However, the energy supplied to the surfer from the wave is almost entirely attributable to the potential energy of the wave.
I think my question is more to do with transforming between Galilean frames of reference, as this approach should still give a consistent answer if we assume there is no hydrodynamic effect, just PE to KE. Maybe there is a change in the PE between frames? Could this answer what is the minimum velocity the surfer must have in order to catch the wave

6. Sep 3, 2016

### A.T.

I'm not sure what all the energy talk has to do with this question. To stay on the wave, his velocity component in the direction of the wave propagation needs to be approx. as fast as the wave. Whether he initially accelerates solely by paddling or by converting some potential energy given him by the wave is secondary.

7. Sep 3, 2016

### sophiecentaur

I sort of took the clue from the name 'surf'. If you could gain motive power from normal waves on the surface of the sea, surely people would be using it to power boats (albeit on a one way journey). If you float an object on waves in open water, it just describes a circular trajectory - up and down and back and forth, without making any progress. We've all thrown stones at a floating can and found that the only way to get it to move is for the stone to hit the water right next to the can, displacing water sideways. Once the normal surface mode wave has established itself, there's no flow of water. To get motive power, you need a net gradient of energy. The wind and the tide can give you that but not ordinary waves.
The minimum speed needed to catch a wave will probably be related to the fact that you need to climb up over the peak, which will require you to be going at the wave speed and have enough KE to raise you the extra height (PE) to compensate for the fact that the water behind the peak is dropping as it passes.
Edit: Here's a link which discusses water waves quite well.

8. Sep 4, 2016

### MrMann

Energy is pretty key, and yes the velocity component in the direction of the wave's propagation needs to be approximately the velocity of the wave; my question is what is this relationship? What is the minimum velocity the surfer must have. The PE is what accelerates the surfer to the wave speed and to a greater velocity

9. Sep 4, 2016

### MrMann

Actually I would suggest the converse, you want to have less KE than the PE of the wave (in the wave frame) hence the wave catches you and brings you along at its velocity until you start falling down the wave where you will then be travelling faster than the wave. Buoyancy will bring you up the wave just as your can in the water example, a surfer just does not want to go up and over the wave as he will not be able to surf down it on the wave's front slope. In fact if you have enough KE to be travelling at the same velocity of the wave and additional amount of KE that was to be used to climb the height of the wave you would never go up the wave as you would be travelling at a greater velocity than the wave itself. This i the amount of energy the surfer would have after he/she has surfed down the wave, entering it at the minimum catch velocity, still can't work out what this is though

10. Sep 4, 2016

### sophiecentaur

As always!!
I suspect that the magic velocity that's needed would be the (horizontal) wave speed plus (vector addition) the (vertical )rate that the water level is dropping behind the peak. The higher the wave, the greater the required speed - which makes sense and also explains how it's easy to be thwarted just when you think you are going fast enough. The wave catches you up and you think that all you need to do is to match that speed but, in fact, you need that extra speed to counteract the drop and the drag on the back of the board; less buoyancy for the back of the board so it will tend to tip upwards and have more drag (energy loss). The additional required KE corresponds to to the PE due to the dropping. In films, the surfers seems to be paddling like mad as the wave starts to overtake them, which means an extra burst of energy is needed at that point. They seem to be paddling even after that start to 'go downhill'.

11. Sep 4, 2016

### sophiecentaur

I haven't quite got what your message is here - I must read it again, I guess - but you seem to be treating the water surface as a slippery, stationary slope. In fact, there is a water flow on the surface producing accelerating and deceleration forces on the board. On the leading edge, the energy transferred to the board is advantageous (lifting) but on the trailing edge, I think that the energy transferred is disadvantageous. (dropping) Again, if you look at passive, floating objects, they are not swept along by a wave that's about to break. It's only when the wave has actually broken that they tend to be pushed along to the beach. That situation is not what the surfer wants because it's uncontrolled.

12. Sep 4, 2016

### MrMann

Yeah, I am looking to neglect these hydrodynamic forces, only recognising that the surfer of course floats. Therefore the energy gain in a surfer form the wave comes from purely the potential energy of the physical displacement of the board. This I thought would be a simpler scenario to establish a minimum velocity, additional factors can be included into the model later.
In terms of wave breaking I thought this was the point of more desirable surfing, as when a wave is about to break, it has it's higher amplitude - most potential energy, and it is moving at its slowest - meaning it is easier for the surfer to reach the required minimum velocity to be watched

13. Sep 4, 2016

### CWatters

I assume we are considering the case where a surfer is paddling towards the shore and an incoming wave catches up with him. If he does nothing the wave will simply pass under him.

If he paddles faster than the wave the wave won't catch him at all so that situation of no interest. The question is what happens if he is going slightly slower than the wave when the leading edge arrives?

Clearly he must be going as fast as the wave by the time the top of the wave reaches him (otherwise it will overtake him and he will fall down the backside/trailing edge).

If we assume the wave can accelerate him then the question becomes.. how much slower can he be going when the leading edge of the wave arrives in order that he will be going fast enough when the top arrives.

Suppose he does nothing until he is half way up the leading face. Then he will have gained some PE that he can convert to KE/increased velocity. If he did nothing until he was at the top then he would have gained a lot of PE but would have no time left to convert it into KE. At this point I'm struggling to express this mathematically.

14. Sep 4, 2016

### sophiecentaur

I think you are missing the crux of this because you are assuming that constant speed and constant power input is needed. If he is paddling along in deep water, the waves are going at a constant speed and the surface water is making no forward net motion. Catching up one of those waves is of no interest to the surfer. Also, if a wave has already broken before it hits him, he can't use it and it will just knock him over.
Surfing involves being at the right place at the right time (sorts the men from the boys). A suitable wave is starting to peak (see that hyperphysics link) and this will be because the floor is shelving upwards. For low amplitude waves, the wave speed would just get less and produce a different wavelength but with a large amplitude wave, the non linearity kicks in and a peak can form. It will fall over because the peak is going faster than the trough (the message about depth hasn't got through yet).
My view is that the surfer needs to climb up that peak as it starts to form. The actual speed he's travelling is not the only important thing; he has to climb up the falling water at the back of the peak. Required speed is the speed relative to the ground plus the speed of the falling wave. So the wording of the OP is not really adequate to describe what's actually necessary. You can go fast enough to beat the wave then you are past the point where / when you can use it but if you are too slow it will leave you behind. Extra energy input is needed at just the right time, to get onto the business part of the breaking wave.
I'm no surfer and it's all a long time ago but I have experienced this happening as I have got up some speed - the wave is only just catching me up - but suddenly found myself left behind by the wave and almost going backwards (very confusing with no solid reference) but the wave seems to be accelerating away from me. I have been falling down the rear slope of the wave.