Physics of Surfing

Energetics of a SurferDavid T. Sandwell

(http://topex.ucsd.edu)

• Four styles of surfing

• Waves– Big swell coming? (storms)

– What makes “sets”? (dispersion)

– Why is Blacks Beach good for surfing?(refraction)

• Riding waves– “catching” the wave (speed)

– “dropping-in” (energy conservation)

– “tube riding” (tapping wave energy)

– “need more speed” (surfboard drag)

1 2 kg 1 kg

Which ball is going faster when it reaches the bottom?

(a) (b)

Which ball is going faster when it reaches the bottom?

(a) (b) (c)

conservation of energy(assume no friction)

kinetic energy + potential energy = constant

�

mgH =1

2mv

2m

m

v

H

mv

optimal skateboard ramp(The brachistochrone problem)

What is the best ramp shape for the minimum time down?

60 ft

35 mph

24 ft

22 mph15 ft

18 mph4.5 ft

10 mph

ocean depth and breaker height - empirical

Hb - height of breaker

db - depth where wave

breaks

�

db

=1.28Hb

Mavericks video

shallow water waves

�

cs = gd

breaking waves

db

surfers

empirical relation

Hb - height of breaker

db - depth where wave

breaks�

db

=1.28Hb

“catching the wave”

db

surfers

paddle speed

Hb - height of breaker

db - depth where wave

breaks�

cb = 1.28gHb

paddle speed <= wave speed

60 ft

35 mph

24 ft

22 mph15 ft

18 mph4.5 ft

10 mph

“dropping in”

Hb

top of wave

bottom of wave

KEbottom = KEtop + PE

�

1

2mvd

2=1

2mcb

2+ mgHb

vd2

= cb2

+ 2gHb = 3.28gHb

g - acc. Gravity

cb - wave speed

vd - surfer speed after drop

“dropping in”

59.827.217.023tow-in

35.216.010.08tube

27.912.77.95fun

15.26.94.3 1.5longboard

vd

(mph)

vd

(m/s)

cb

(m/s)

Hb

(m)

style

correction for surfer height

H' Hb

H' = Hb - 1/2 Hs

effective height = breaker height - 1/2 surfer height

- overhead waves are good

- waist high waves are bad

- short surfers and heavy boards have an advantage in small waves

Pipeline video

dropping in

“dropping in”

Hb

top of wave

bottom of wave

KEbottom = KEtop + PE

�

1

2mvd

2=1

2mcb

2+ mgHb

vd2

= cb2

+ 2gHb = 3.28gHb

g - acc. Gravity

cb - wave speed

vd - surfer speed after drop

“dropping in”

59.827.217.023tow-in

35.216.010.08tube

27.912.77.95fun

15.26.94.3 1.5longboard

vd

(mph)

vd

(m/s)

cb

(m/s)

Hb

(m)

style

“cutting across”

back of wave

front of wave

vd !

cb

�

cb

2

vd

2=1.28

3.28= cos

2!

! =~ 50˚

top

view

This angle is independent of

wave height or wave speed!

“riding the wave”

Suppose the surfer remains on the steepest part of the wave having

a slope s. What is the rate of potential energy increase supplied to

the surfer?

cb s

�

P ˙ E = sgcbside view

�

v(t)2

= vd2

+ sgcbo

t

! dt

v(t)2

= 3.28gHb + tsg 1.28gHb

How does speed increase

with time?

speed video

“the drag”

�

v(t)2 = 3.28gHb + 1.13g

3 / 2 s Hb

1/ 2 t ! drag

total

velocity= paddle

drop in+ -slope of

“the curl”breaker

height

duration

of ride

energy

dissipation

Future Research

– Are the best surf spots in areas of narrow continental margin?

– Are “sets” real? What are the statistical properties of sets? Do “sets”become amplified when they reach shallow water?

– How does the shape of the bottom translate into the “perfect wave”?

– What is the terminal velocity for a given breaker height? (Can we establishthe magnitude of the drag term?)

– Need to instrument surfers with intertial sensors.