1. Modeling Ocean Waves
Gokul Varadhan

2. Outline What do we want from our model ? Different approaches
Fournier-Reeves model
Peachey model

3. What are we looking for ? realistic shape of the waves
realistic wave motion
range of waves
adapt to varying conditions - winds, shallow sea bottoms etc
effects of approaching the shore

4. Simple Wave a sinusoid x = distance from origin C = propagation speed
L = wavelength
T = L/C is the period
Wave number

5. Previous work
Nelson Max modeled wave by a collection of superimposed sinusoidal waves [Max 1981]
Ts’o and Barsky model the sea surface as a height field from Stokes model and fit it with splines [Ts’o & Barsky 1987]
Perlin used spherical cycloidal waves [Perlin 1985]
Bump mapping approaches
Problem with intersection with obstacles
Self-shadowing

6. Height Field Model
for each pair (x,y), we have z = f (x,y)

7. Modeling fluid motion Eulerian method Lagrangian method
Consider a point (x,y,z) and properties of the fluid at this point as a function of time
Lagrangian method
Follow trajectory of a point at reference position (x0,y0,z0) with time
We are going to use Lagrangian method. Its more convenient

8. Fournier-Reeves Model
Coordinate system
X-Y is the plane of sea at rest
Z points up
Model
Ocean consists of particles which are in motion
Motion is along XZ plane
All points with equal Y exhibit identical motion

9. Particles move in circular orbits around the rest position
(x0,z0) - Rest position
- Wave number
- Angular velocity
Note this is not a height field.

10. A wave ! Locus of the particles with same z0 at a time t
is a Trochoid which represent a wave in the model

11. Properties Periodicity in space and time Not a height field
No net transport of matter with time
Choice of circular orbits is not arbitrary
Based on Gerstner or Rankine model in oceanography
Gerstner or Rankine model is a model in oceanography.

12. Parameters in our Model
Rest position (x0,z0)
Amplitude (r)
Wave Number (k)
Angular velocity (w)
Phase (kx0-wt)
Wavelength (2*Π/k)
Period (2*Π/w)
Propagation speed (w/k)
Parameters can be adjusted to give us different shapes of waves

14. Effects of Wind
To model the effects of the wind on top of the crest, make phase angle

15. Effects of Wind contd … Wind Velocity Wind direction
Wind => amplitude ++
amplitude = function (wind velocity)
Wind direction
Expected behavior - particles move in the direction of the wind
Rotate the orbit plane
orbit plane = plane (wind velocity, Z)

16. More Effects Water depth depth ++ => wavelength ++
= function (particle depth)
where h is the particle depth

17. Depth dependence

18. Depth dependence contd …
Phase depends on cumulative depth information at each point
which is approximately

19. Wave Refraction Waves get refracted Obey Snell’s law of refraction
are propagation
speeds

20. Breaking of waves on shore
Replace circular orbits by ellipses
Ellipses align themselves with the shore
Complicated equations

21. Wave/particle motion
Distinguish motion of wave from motion of the water particles.
orbital speed
H – height
= 2 * r

22. Spray and Foam Cause Spray Foam Surface curvature is large
diff = Orbital speed – Wave propagation speed
if diff >= Threshold => Spray
otherwise => Foam
Spray
rendered using particle systems
follows gravity
Foam
Foam sent sliding along the wave surface

23. Wave trains Regularity of waves is a problem
Wave train - a group of waves with similar properties
rectangular box on the surface of sea
Shape the train using height envelopes
X and Y envelopes
Multiply amplitude of the wave by the height envelopes
Vary wavelength based on the age of the wave train
Removed the constraint that all particles with same Y exhibit the same motion

24. Rendering
Interpolate the data as a set of bi-linear or bi-cubic Catmull-Rom patches
Do a ray trace
Color and shading: Environmental map - a natural choice

25. Breaking of waves at shore

26. Spray and Foam

27. Spray, Foam and Wind
Notice that this is not a height field

28. Next : Peachey model for ocean waves

29. Simple Wave x = distance from origin C = propagation speed
a sinusoid
x = distance from origin
C = propagation speed
L = wavelength
T = L/C is the period
Wave number
Lets look at the simple sinusoidal wave again. Our model is based on it.

30. Airy Model Sinusoidal waves of small magnitude
Propagation speed, C and wavelength L depend on depth d as follows:
Airy model tells us about dependence of C and wavelength on depth
where T = period

31. Basic Model
Represent ocean surface as a height field, a single-valued function of three variables
Disadvantage: Can’t produce waves whose crests curl forward
Advantage: Can superpose numerous waves easily

32. Basic Model contd … f is the sum of several linear waveforms, Wi
is a wave profile
is the phase function
We have to compute the phase first and the apply the wave profile function.

33. Phase Computation Phase depends on: time t position x,y
depends on depth and
is complex
Explain how we compute the phase function.
Design a beach first
Use Airy model to compute
Ci as a f(depth)
Use Numerical integration

34. Wave Profiles Linearly blend sinusoid
and a steep quadratic wave such as:
according to steepness
Also depends on the ratio between depth of water and deep water wavelength
To simulate breaking of waves near the shore
Explain the whole process:
First design a beach. That gives us the depth at different points
Compute the phase function at each point using numerical integration
Apply the wave profile function

35. Spray Criterion Particle generation where S = H/L is the steepness
initial position = crest of the wave
initial velocity is Qavg in the direction of wave motion plus a stochastic perturbation
trajectory follows gravity
Number of particles increase with the difference between Qavg and C
where S = H/L is the steepness
Wave breaks when Qavg > C i.e. if S exceeds a threshold

36. Spray contd
Similar particle system to simulate spray resulting from striking obstacles

37. Rendering Raytracing is too expensive, suffers from aliasing A-buffer
Why not Z-buffer ?
Aliasing, memory consumption
Scanline rendering
Reduce memory consumption
Reduce Virtual Memory page faults
Render wave patches as a collection of polygons
Split large wave patches into smaller ones
Explain A-buffer

38. Rendering spray
Difficult to render particle systems in a scanline order
Use sparse Z-buffer
Combine with the A-buffer

39. Breaking waves

40. Breaking waves with obstacle

41. Sunset

42. Late afternoon

43. References
[Fournier & Reeves 1986] Fournier, A., Reeves, W.T., A Simple Model of Ocean Waves, ACM Computer Graphics (SIGGRAPH '86 Proceedings) , pp , 1986.
[Peachey 1986] Peachey, D., "Modeling of Waves and Surf," Proceedings of SIGGRAPH `86, in Computer Graphics, Vol. 20, No. 4, pp , 1986.
[Max 1981] Max, Nelson L. "Vectorized Procedural Models for Natural Terrain: Waves and Islands in the Sunset." SIGGRAPH '81, p

44. References contd …
[Ts’o & Barsky 1987] Tso, P., and Barsky, B., "Modeling and Rendering Waves: Wave Tracing Using Beta-Splines and Reflective and Refractive Texture Mapping", ACM Transactions on Graphics (6), pp , 1987.
[Perlin 1985] K. Perlin, An Image Synthesizer, Computer Graphics, v19, n3, p , 1985