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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