ArticlePDF Available
Simulating Ocean Water
Jerry Tessendorf
Copyright c
1999 2004, Jerry Tessendorf
1 Introduction and Goals
These notes are intended to give computer graphics programmers
and artists an introduction to methods of simulating, animating, and
rendering ocean water environments. CG water has become a com-
mon tool in visual effects work at all levels of computer graphics,
from print media to feature films. Several commercial products are
available for nearly any computer platform and work environment.
A few visual effects companies continue to extend and improve
these tools, seeking to generate higher quality surface geometry,
complex interatctions, and more compelling imagery. In order for
an artist to exploit these tools to maximum benefit, it is important
that he or she become familiar with concepts, terminology, a little
oceanography, and the present state of the art.
As demonstrated by the pioneering efforts in the films Water-
world and Titanic, as well as several other films made since about
1995, images of cg water can be generated with a high degree of
realism. However, this level of realism has been mostly limited
to relatively calm, nice ocean conditions. Conditions with large
amounts of spray, breaking waves, foam, splashing, and wakes are
improving and approaching the same realistic look.
Three general approaches are currently popular in computer
graphics for simulating fluid motion, and water surfaces in partic-
ular. The three methods are all related in some way to the basic
Navier-Stokes equations at the heart of many applications. Com-
putational Fluid Dynamics (CFD) is one of the methods which has
recieved a great amount of attention lately. While many versions of
this technology have been in existence for some time, recent papers
by Stam [1], Fedkiw and Foster [2], and many others have demon-
strated that the numerical computation discipline and the computer
graphics discipline have connected well enough to produce useful,
interesting, and sometimes beautiful results. The primary draw-
back of CFD methods is that the computations are performed on
data structured in a 2D or 3D grid, called an Eulerian framework.
This gridded architecture limits the combined extent and flow detail
that can be computed. At present for example, it is practical to sim-
ulate with CFD waves breaking as they approach a shallow beach.
However, that simulation is not able to simulate a bay-sized region
of ocean while also simulating the breaking down to the detail of
spray formation, simply because the number of grid points needed
is prohibitively large.
A second method that is shows great promise is called Smoothed
Particle Hydrodynamics (SPH). This is a completely different ap-
proach to solving the Navier-Stokes equations. SPH imagines that
the volume of a fluid is composed of small overlapping regions, the
center of each region carrying some amount of mass and momen-
tum. The regions are allowed to move about within the fluid and ex-
perience forces due to pressure, strain, gravity, and others. But now
the center of each region acts as a particle, and the Navier-Stokes
equations are converted into equations of motion for discrete par-
ticles. This approach is called a Lagragian framework (as opposed
to the Eulerian framework in CFD). The fluid volume is not bound
to a grid geometry. SPH is a very useful method of simulation for
situations in which there is significant splashing or explosions, and
has even been used to simulated cracking in solids [3]. Using im-
plicit methods to construct a surface for the fluid, standard computer
graphics applications such as pouring water have been achieved [4].
Finally, the third method is the one that is the focus of these
notes, and unlike the CFD and SPH methods, this one is focused
on the more narrow goal of simulating the motion of the surface
of a body of water. Surface simulations are commonly generated
from the CFD and SPH methods, but the surface is generated by
algorithms that are added to those fluid simulations, for example by
tracking a type of implicit surface called a level set. In choosing
to focus on the surface structure and motion, we eliminate much of
the computation and resolution limits in the CFD and SPH meth-
ods. Of course when we eliminate those computations we have lost
certain types of realistic motion, most notably the breakup of the
surface with strong changes in topology. In place of that compu-
tation we substitute a mixture of knowledge of oceanographic phe-
nomenology and computational flexibility to achieve realistic types
of surfaces that cannot practically be achieved by the others. In that
sense, this phemonological approach should be considered comple-
mentary not competitive to the CFD and SPH methods, since
all three work best in different regimes of fluid motion.
Broadly, the reader should come away from this material with
1. an understanding of the important physical concepts for ocean
surface propagation, most notably the concept of dispersion
and types of dispersion relationships.
2. an understanding of some algorithms that generate/animate
water surface height fields suitable for modeling waves as big
as storm surges and as small as tiny capillaries;
3. an understanding of the basic optical processes of reflection
and refraction from a water surface;
4. an introduction to the color filtering behavior of ocean water;
5. an introduction to complex lighting effects known as caus-
tics and godrays, produced when sunlight passes through the
rough surface into the water volume underneath; and
6. some rules of thumb for which choices make nice looking
images and what are the tradeoffs of quality versus compu-
tational resources. Some example shaders are provided, and
example renderings demonstrate the content of the discussion.
Before diving into it, I first want to be more concrete about what
aspect of the ocean environment we cover (or not cover) in these
notes. Figure 1 is a rendering of an oceanscape produced from mod-
els of water, air, and clouds. Light from the clouds is reflected from
the surface. On the extreme left, sun glitter is also present. The
generally bluish color of the water is due to the reflection of blue
skylight, and to light coming out of the water after scattering from
the volume. Although these notes do not tackle the modeling and
rendering of clouds and air, there is a discussion of how skylight
from the clouds and air is reflected from, or refracted through, the
water surface. These notes will tell you how to make a height-field
displacement-mapped surface for the ocean waves with the detail
and quality shown in the figure. The notes also discuss several ef-
fects of the underwater environment and how to model/render them.
The primary four effects are sunbeams (also called godrays), caus-
tics on underwater surfaces, blurring by the scattering of light, and
color filtering.
There are also many other complex and interesting aspects of the
ocean environment that will not be covered. These include break-
ing waves, spray, foam, wakes around objects in the water, splashes
from bodies that impact the surface, and global illumination of the
entire ocean-atmosphere environment. There is substantial research
underway on these topics, and so it is possible that future versions
of this or other lecture notes will include them. I have included a
brief section on advanced modifications to the basic wave height al-
gorithm that produce choppy waves. The modification could feasi-
bly lead to a complete description of the surface portion of breaking
waves, and possibly serve to drive the spray and foam dynamics as
There is, of course, a substantial body of literature on ocean sur-
face simulation and animation, both in computer graphics circles
and in oceanography. One of the first descriptions of water waves in
computer graphics was by Fournier and Reeves[12] , who modeled
a shoreline with waves coming up on it using a water surface model
called Gerstner waves. In that same issue, Darwin Peachey[13]
Figure 1: Rendered image of an oceanscape.
presented a variation on this approach using basis shapes other than
In the oceanographic literature, ocean optics became an inten-
sive topic of research in the 1940s. S.Q. Duntley published[17] in
1963 papers containing optical data of relevance to computer graph-
ics. Work continues today. The field of optical oceanography has
grown into a mature quantitative science with subdisciplines and
many different applications. One excellent review of the state of
the science was written by Curtis Mobley[18].
In these lectures the approach we take to creating surface waves
is close to the one outlined by Masten, Watterberg, and Mareda[11],
although the technique had been in use for many years prior to
their paper in the optical oceanography community. This approach
synthesizes a patch of ocean waves from a Fast Fourier Transform
(FFT) prescription, with user-controllable size and resolution, and
which can be tiled seamlessly over a larger domain. The patch con-
tains many octaves of sinusoidal waves that all add up at each point
to produce the synthesized height. The mixture of sinusoidal am-
plitudes and phases however, comes from statistical, emperically-
based models of the ocean. What makes these sinusoids look like
waves and not just a bunch of sine waves is the large collection of
sinusoids that are used, the relative amplitudes of the sinusoids, and
their animation using the dispersion relation. We examine the im-
pact of the number of sinusoids and resolution on the quality of the
rendered image.
In the next section we begin the discussion of the ocean environ-
ment with a broad introduction to the global illumination problem.
The radiosity equations for this environment look much like those
of any other radiosity problem, although the volumetric character of
some of the environmental components complicate a general imple-
mentation considerably. However, we simplify the issues by ignor-
ing some interactions and replacing others with models generated
by remote sensing data.
Practical methods are presented in section 4 for creating realiza-
tions of ocean surfaces. We present two methods, one based on a
simple model of water structure and movement, and one based on
summing up large numbers of sine waves with amplitudes that are
related to each other based on experimental evidence. This sec-
ond method carries out the sum using the technique of Fast Fourier
Transformation (fft), and has been used effectively in projects for
commercials, television, and motion pictures.
After the discussion of the structure and animation of the water
surface, we focus on the optical properties of water relevant to the
graphics problem. First, we discuss the interaction at the air-water
interface: reflection and refraction. This leaves us with a simple
but effective Renderman-style shader suitable for rendering water
surfaces in BMRT, for example. Next, the optical characteristics of
the underwater environment are explored.
Finally, please remember that these notes are a living
document. Some of the discussion of the various top-
ics is still very limited and incomplete. If you find a
problem or have additional questions, please feel free to
contact me at The
latest version of this course documents are hosted at
2 Radiosity of the Ocean Environment
The ocean environment, for our purposes, consists of only four
components: The water surface, the air, the sun, and the water
volume below the surface. In this section we trace the flow of
light through the environment, both mathematically and schemat-
ically, from the light source to the camera. In general, the radiosity
equations here are as coupled as any other radiosity problem. To a
reasonable degree, however, the coupling can be truncated and the
simplified radiosity problem has a relatively fast solution.
The light seen by a camera is dependent on the flow of light en-
ergy from the source(s) (i.e. the sun and sky) to the surface and
into the camera. In addition to specular reflection of direct sun-
light and skylight from the surface, some fraction of the incident
light is transmitted through the surface. Ultimately, a fraction of the
transmitted light is scattered by the water volume back up through
the interface and into the air. Some of the light that is reflected or
refracted at the surface may strike the surface a second time, pro-
ducing more reflection and refraction events. Under some viewing
conditions, multiple reflections and refractions can have a notice-
able impact on images. For our part however, we will ignore more
than one reflection or refraction from the surface at a time. This
not only makes the algorithms and computation easier and faster,
but also is reasonably accurate in most viewing conditions and pro-
duces visually realistic imagery.
At any point in the environment above the surface, including at
the camera, the total light intensity (radiance) coming from any di-
rection has three contributions:
with the following definitions of the terms:
ris the Fresnel reflectivity for reflection from a spot on the surface
of the ocean to the camera.
tUis the transmission coefficient for the light LUcoming up from
the ocean volume, refracted at the surface into the camera.
LSis the amount of light coming directly from the sun, through
the atmosphere, to the spot on the ocean surface where it is
reflected by the surface to the camera.
LAis the (diffuse) atmospheric skylight
LUis the light just below the surface that is transmitted through
the surface into the air.
Equation 1 has intentionally been written in a shorthand way that
hides the dependences on position in space and the direction the
light is traveling.
While equation 1 appears to have a relatively simple structure,
the terms LS,LA, and LUcan in principle have complex depen-
dencies on each other, as well on the reflectivity and transmissivity.
There is a large body of research literature investigating these de-
pendencies in detail [19], but we will not at this point pursue these
quantitative methods. But we can elaborate further on the coupling
while continuing with the same shorthand notation. The direct light
from the sun LSis
LS=LT OA exp{−τ},(2)
where LT OA is the intensity of the direct sunlight at the top of the
atmosphere, and τis the “optical thickness” of the atmosphere for
the direction of the sunlight and the point on the earth. Both the
diffuse atmospheric skylight LAand the upwelling light LUcan be
written as the sum of two terms:
A(LS) + L1
U(LS) + L1
These equations reveal the potential complexity of the problem.
While both LAand LUdepend on the direct sunlight, they also
depend on each other. For example, the total amount of light pene-
trating into the ocean comes from the direct sunlight and from the
atmospheric sunlight. Some of the light coming into the ocean is
scattered by particulates and molecules in the ocean, back up into
the atmosphere. Some of that upwelling light in turn is scattered in
the atmosphere and becomes a part of the skylight shining on the
surface, and on and on. This is a classic problem in radiosity. It
is not particularly special for this case, as opposed to other radios-
ity problems, except perhaps for the fact that the upwelling light
is difficult to compute because it comes from volumetric multiple
Our approach, for the purposes of these notes, to solving this
radiosity problem is straightforward: take the skylight to depend
only on the light from the sun, since the upwelling contribution
represents a “tertiary” dependence on the sunlight; and completely
replace the equation for LUwith an empirical formula, based on
scientific observations of the oceans, that depends only on the di-
rect sunlight and a few other parameters that dictate water type and
Under the water surface, the radiosity equation has the schematic
with the meaning
tis the Fresnel transmissivity for transmission through the water
surface at each point and angle on the surface.
LDThe “direct” light from the sun that penetrates into the water.
LIThe “indirect” light from the atmosphere that penetrates into
the water.
LSS The single-scattered light, from both the sun and the atmo-
sphere, that is scattered once in the water volume before ar-
riving at any point.
LMThe multiply-scattered light. This is the single-scattered light
that undergoes more scattering events in the volume.
Just as for the above water case, these terms are all related to each
other is relative complex ways. For example, the single scattered
light depends on the direct and indirect light:
LSS =P(tLI) + P(tLD)(6)
with the quantity Pbeing a linear functional operator of its argu-
ment, containing information about the single scattering event and
the attenuation of the scattered light as it passes from the scatter
Figure 2: Illustration of multiple reflections and transmission
through the air-water interface.
point to the camera. Similarly, the multiply-scattered light is de-
pendent on the single scattered:
LM=G(tLI) + G(tLD).(7)
The functional schematic quantities Pand Gare related, since mul-
tiple scattering is just a series of single scatters. Formally, the two
have an operator dependence that has the form
GPPn1 + P+1
2! PP+1
3! PPP+...o
At this point, the schematic representation may have outlived its
usefullness because of the complex (and not here defined) meaning
of the convolution-like operator , and because the expression for
Gin terms of Phas created an even more schematic view in terms
of an exponentiated P. So for now we will leave the schematic
representation, and journey on with more concrete quantities the
rest of the way through.
The formal schematic discussion put forward here does have a
mathematically and physically precise counterpart. The field of
study in Radiative Transfer has been applied for some time to wa-
ter optics, by a large number researchers. The references cited are
excellent reading for further information.
As mentioned, there is one additional radiosity scenario that can
be important to ocean rendering under certain circumstances, but
which we will not consider. The situation is illustrated in figure 2.
Following the trail of the arrows, which track the direction light
is travelling, we see that sometimes light coming to the surface
(from above or below), can reflect and/or transmit through the sur-
face more than once. The conditions which produce this behavior
in significant amounts are: the wave heights must be fairly high,
and the direction of viewing the waves, or the direction of the light
source must be nearly grazing the surface. The higher the waves
are, the less grazing the light source or camera need to be. This
phenonmenon has been examined experimentally and in computer
simulations. It is reasonably well understood, and we will ignore it
from this point on.
3 Mathematics, Physics, and Experi-
ments on the Motion for the Surface
In this section we take a look at the mathematical problem we are
trying to solve. We simplify the mathematics considerably by ap-
plying a series of approximations. How do we know these approx-
imations are any good? There are decades of oceanographic re-
search in which the ocean surface motion has been characterized
by measurements, simulations, mathematical analysis, and experi-
mentation. The approximations we apply in this section are not per-
fect, and there are many circumstances in the real world in which
they break down. But they work extraordinarily well for most con-
ditions at sea. To give you some idea of how well they work, we
show some experimental work in section 4 that has been done which
clearly shows these approximations at work in the real world.
3.1 Bernoulli’s Equation
The starting point of the mathematical formulation of ocean surface
motion is the incompressible Navier-Stokes equations. Chapter 2
of [14] provides a thorough derivation of Bernoulli’s equation from
Navier-Stokes. We provide here the short version, and refer you to
Kinsman’s or some other textbook for details.
The incompressible Navier-Stokes equations for the velocity
u(x, t)of a fluid a any position xat any time tare
∂t +u· u=−∇p+F(9)
· u(x, t)=0 (10)
In these equations, p(x, t)is the pressure on the fluid, and F(x, t)
is the force applied to the fluid. In our application, the force is
conservative, so F=−∇Ufor some potential energy function
U(x, t).
For ocean surface dynamics due to conservative forces like grav-
ity, it turns out to be worthwhile to restrict the type of motion to a
class called potential flow. This is a situation in which the velocity
has the form of a gradient:
u(x, t) = φ(x, t)(11)
This restriction has the important effect of reducing the number de-
grees of freedom of the flow from the three components of velocity
uto the single one of the potential velocity function φ. In fact, us-
ing this gradient form, the four Navier-Stokes equations transform
into the two equations
∂t +1
2φ(x, t)=0 (13)
Equation 12 is called Bernoulli’s equation. As a fully nonlinear
reduction of the Navier-Stokes equations, Bernoulli’s equation is
capable of simulating a variety of surface dynamics effects, includ-
ing wave breaking in shoaling shallow water (the bottom rises from
deep water up to a beach). For more detail on numerical simulations
of Bernoulli’s equation in 3D, see [23].
3.2 Linearization
For our purposes, we want to reduce the complexity of Bernoulli’s
equation even further by applying two restrictions: linearize the
equations of motion, and limit evaluation of the equations to just
points on the surface itself, ignoring the volume below the sur-
face. This may seem like an extreme restriction, but when com-
bined with some phenomenolical knowledge of the ocean, this re-
strictions work very well.
The first restriction is to linearize Bernoulli’s equation. This is
simply the task of removing the quadratic term 1/2(φ)2. Elimi-
nating this term means that we are most likely restricted to surface
waves that are not extremely violent in their motion, at least in prin-
ciple. So Bernoulli’s equation is reduced further to
∂t =pU(14)
All of the quantities φ,p, and Uare still evaluated at 3D points x
on the surface and in the water volume.
The second restriction is to evaluation quantities only on the wa-
ter surface. To do this we have to first characterize what we mean by
the surface. We will take the surface to be a dynamically changing
height field, h(x, t), that is a function of only the horizontal posi-
tion xand time t. For convenience, we define the mean height of
the wate surface as the zero value of the height. With this definition
of wave height, the gravity-induced potential energy term Uis
U=g h (15)
and gis the gravity constant, usually 9.8m/sec2in metric units.
Restricting to just the water surface has several important conse-
quences. One of the first consequences is for mass conservation. In
the incompressible Navier-Stokes equation, mass is conserved via
the mass flux equation
· u(x, t) = 0 (16)
When we chose to consider only potential flow, this mass conserva-
tion equation became
2φ(x, t) = 0 (17)
If we label the horizontal portion of the position vector as x, so
that x= (x, y), and yis the coordinate pointing down into the
water volume, then the mass conservation equation restricted to the
surface looks in more detail like
∂y2φ(x, t) = 0 (18)
Now, when you look at this equation and see that φnow depends
only on the xon the surface, you might be tempted to throw out
the 2/∂y2part of the equation, because there does not appear to
be a dependence. That would produce useless results. Instead, what
works better in this odd world of partial differential equations is to
allow φto be an arbitrary function (at least with respect to this mass
conservation equation) and to define the y-derivative operator to be
∂y =±p−∇2
so that the operator 2is zero. We will use this approach for any
quantity evaluated on the water surface whenever we need a verti-
cal derivative. Of course, this introduces an unusual operator that
contains a square root function.
Another consequence of restricting ourselves to just the surface
is that the pressure remains essentially constant, and we can choose
to have that constant be 0. With this and the rest of the restrictions,
Bernoulli’s equation has been linearized to
∂φ(x, t)
∂t =g h(x, t)(20)
There is one final equation that must be rewritten for this situ-
ation. Recall that the velocity potential φis used to compute the
3D fluid velocity as a gradient, u=φ. The vertical component
of the velocity must now use equation 19. In addition, the vertical
velocity of the fluid is the same at the speed of the surface height.
Combining these we get
∂h(x, t)
∂t =p−∇2
φ(x, t)(21)
These last two equations, 20 and 21, are the final equations of
motion that are needed to solve for the surface motion. They can
also be converted into a single equation. For example, if we take a
derivative with respect to time of equation 21, and use 20 to sub-
stitute for the time derivative of the velocity potential, we get the
single equation for the evolution of the surface height.
2h(x, t)
h(x, t)(22)
This still involves the unusual operator p−∇2
. However, tak-
ing two more time derivatives converts it to a more normal two-
dimensional Laplacian for the equation
4h(x, t)
h(x, t)(23)
This form is frequently the starting point for building mathematical
solutions to the surface wave equation.
3.3 Dispersion
It turns out that the equations we have built for surface height
whether in the form of equations 20 and 21, or equation 22, or equa-
tion23 reduce to one primary lesson about surface wave propaga-
tion. This lesson is embodied in a simple mathematical relationship
called the Dispersion Relation, which is the focus of this section.
Our goal here is to obtain that simple expression from the math-
ematics above, understand some of its meaning, and demonstrate
that, even though it appears unrealistically simplistic, the Disper-
sion Relation is in fact present in natural ocean waves and can be
measured experimentally.
Lets just use the version of the surface evolution equation in
equation 23 for convenience. The other two versions could be used
and arrive at the same answer using a slightly different set of ma-
nipulations. Note that the equation of motion for the surface height
is linear in the surface height. So as with any linear differential
equation, the general solution of the equation is obtained by adding
up any number of specific solutions. So lets find a specific solution.
It turns out that all specific solutions have the form
h(x, t) = h0exp {ik·xiωt}(24)
The 2D vector kand the numbers ωand h0are generic parameters
at this point. If we use this form of a solution in equation 23, it
turns into an algebraic equation like this:
h0ω4g2k2= 0 (25)
( and kis the magnitude of the vector k). For this solution, there
are only two possibilities:
1. h0= 0. Then the surface height is flat, and the solution is not
very interesting.
2. ω=±gk and h0can be anything. This is the interesting
What we have found here is that the entire Navier-Stokes fluid dy-
namics problem, reduced to an evolution equation for the water sur-
face and approximated to something that can be solved relatively
easily, amounts to a single equation imposing the constraint that the
temporal frequency ωof surface height movements is connected to
the spatial extent of a the propagating wave k=|k|. This relation-
ship, ω=±gk is the Dispersion Relation mentioned earlier.
Note in particular that there is no constraint placed on the ampli-
tude h0. But if the Navier-Stokes equation does not have anything
to say about the amplitude, how do we give it a value? One way
is by imposing initial conditions on the height and on its vertical
speed. For ocean surface simulation in the next section, we will use
Figure 3: A slice through a 3D PSD showing that the observed
wave energy follows the deep water dispersion relation very well.
an alternate method, a statistical procedure to generate random real-
izations of the amplitude, guided by measurements of the variance
properties of wave height on the open ocean.
So the question remaining is just how reasonable is the disper-
sion relation for modelling realistic ocean surface waves? This is
where lots of experimental research can come into play. Although
there have been many decades of research on ocean wave proper-
ties using devices placed in the water to directly measure the wave
motion at a point, here we look at some relatively new research that
involves measuring wave properties remotely with a camera in a
The AROSS [24] is a panchromatic camera mounted in a special
hosing on the nose of a small airplane. Attached to the camera is
navigation and GPS instrumentation which allow the camera po-
sition, viewing direction, and orientation to be measured for each
frame. After the plane flys a circular orbit around a spot over the
ocean, this data can be used to remap images of the ocean into a
common reference frame, so that the motion of the aircraft has been
removed (except for lighting variations). This remapping allows
the researchers to use many frames of ocean imagery, typically 1-2
minutes worth, in some data processing to look for the dispersion
The data processing that AROSS imagery is subjected to gen-
erates something called a 3D Power Spectral Density (PSD). This
is obtained by taking the Fourier Transform of a time series of im-
agery in time, as well as Fourier Transforms in the two spatial di-
rections of images. The output of these 3 Fourier Transforms is a
quantity that is closely related to the amplitudes h0(k, ω)for each
spatial and temporal frequency. These are then absolute squared
and smoothed or averaged in some way so that the output is a nu-
merical approximation of a statistical average of |h0(k, ω)|2.
But how does a 3D PSD help us decide whether the dispersion
relation appears in nature? If the imagery found only dispersion
constrained surface waves, then the 3D PSD should have the value
0 for all values of k,ωthat do not satisfy the dispersion relation. So
mostly we would expect the 3D PSD to only have significant values
in a narrow set of k,ωvalues.
Figure 3 show a plot of the 3D PSD generated from AROSS im-
ages [24]. From the 3D volumetric PSD, this plot figure is a plane
sliced through the volume. When sliced like this, the dispersion
relation is a curve on the slice, shown as two dotted curves. The
data is plotted as contours of PSD intensity, color-coded by the key
on the right. PSD levels following the dispersion relation curves
are much higher than in other regions. This shows that the motion
of the surface waves on all scales includes a very strong dispersion
relation style of motion. There are other types of motion certainly,
which the PSD figure shows as intensity levels away from the dis-
Figure 4: Site at which video data was collected in 1986, near Zuma
Beach, California.
persion curves. But the dispersion motion is the strongest feature
of this data.
Relatively simple experiments can be done by anyone with ac-
cess to a video camera and a hilltop overlook of an ocean. For
example, figure 4 is a frame from a video segment showing wa-
ter coming into the beach near Zuma Beach, California. The video
camera was located on hill overlooking the beach, in 1986. In 1993,
the region of video frames indicated in the figure was digitized, to
produce a time series of frames containing just water surface.
Figure 5 shows the actual 3D PSD from the image data. There
are two clear branches along the dispersion relationship we have
discussed, with no apparent modification by shallow water affects.
There is also a third branch that is approximately a straight line
lying between the first two. Examination of the video shows that
this branch comes from a surfactant layer floating on the water in
part of the video frame, and moving with a constant speed. Exclud-
ing the surface layer, this data clearly demonstrates the validity of
the dispersion relationship, and demonstrates the usefulness of the
linearized model of surface waves.
4 Practical Ocean Wave Algorithms
In this section we focus on algorithms and practical steps to build-
ing height fields for ocean waves. Although we will be occupied
mostly by a method based on Fast Fourier Transforms (FFTs), we
begin by introducing a simpler description called Gerstner Waves.
This is a good starting point for several reasons: the mathematics is
relatively light compared to FFTs, several important oceanographic
concepts can be introduced, and they give us a chance to discuss
wave animation. After this discussion of Gerstner waves, we go
after the more complex FFT method, which produces wave height
fields that are more realistic. These waves, called “linear waves” or
“gravity waves” are a fairly realistic representation of typical waves
on the ocean when the weather is not too stormy. Linear waves are
certainly not the whole story, and so we discuss also some meth-
ods by which oceanographers expand the description to “nonlinear
waves”, waves passing over a shallow bottom, and very tiny waves
about one millimeter across called capillary waves.
In the course of this discussion, we will see how quantities like
windspeed, surface tension, and gravitational acceleration come
into the practical implementation of the algorithms.
Figure 5: Slice from a 3D Power Spectral Density grayscale plot,
from processed video data.
4.1 Gerstner Waves
Gerstner waves were first found as an approximate solution to the
fluid dynamic equations almost 200 years ago. There first appli-
cation in computer graphics seems to be the work by Fournier and
Reeves in 1986 (cited previously). The physical model is to de-
scribe the surface in terms of the motion of individual points on the
surface. To a good approximation, points on the surface of the water
go through a circular motion as a wave passes by. If a point on the
undisturbed surface is labelled x0= (x0, z0)and the undisturbed
height is y0= 0, then as a single wave with amplitude Apasses by,
the point on the surface is displaced at time t to
In these expressions, the vector k, called the wavevector, is a
horizontal vector that points in the direction of travel of the wave,
and has magnitude krelated to the length of the wave (λ) by
k= 2π/λ (28)
The frequency wis related to the wavevector, as discussed later.
Figure 6 shows two example wave profiles, each with a different
value of the dimensionless amplitude kA. For values kA < 1, the
wave is periodic and shows a steepening at the tops of the waves as
kA approaches 1. For kA > 1, a loop forms at the tops of the wave,
and the “insides of the wave surface are outside”, not a particularly
desirable or realistic effect.
As presented so far, Gerstner waves are rather limited because
they are a single sine wave horizontally and vertically. However,
this can be generalized to a more complex profile by summing a
set of sine waves. One picks a set of wavevectors ki, amplitudes
Ai, frequencies ωi, and phases φi, for i= 1,...,N, to get the
-5 0 5 10 15 20 25
Wave Amplitude
kA = 0.7
kA = 1.33
Figure 6: Profiles of two single-mode Gerstner waves, with differ-
ent relative amplitudes and wavelengths.
Figure 7 shows an example with three waves in the set. Interest-
ing and complex shapes can be obtained in this way.
4.2 Animating Waves: The Dispersion Relation
The animated behavior of Gerstner waves is determined by the set
of frequencies ωichosen for each component. For water waves,
there is a well-known relationship between these frequencies and
the magnitude of their corresponding wavevectors, ki. In deep wa-
ter, where the bottom may be ignored, that relationship is
ω2(k) = gk . (31)
The parameter gis the gravitational constant, nominally
9.8m/sec2. This dispersion relationship holds for Gerstner waves,
and also for the FFT-based waves introduced next.
There are several conditions in which the dispersion relationship
is modified. When the bottom is relatively shallow compared to
the length of the waves, the bottom has a retarding affect on the
waves. For a bottom at a depth Dbelow the mean water level, the
dispersion relation is
ω2(k) = gk tanh(kD)(32)
Notice that if the bottom is very deep, the behavior of the tanh
function reduces this dispersion relation to the previous one.
A second situation which modifies the dispersion relation is sur-
face tension. Very small waves, with a wavelength of about 1 cm or
less, have an additional term:
ω2(k) = gk(1 + k2L2),(33)
and the parameter Lhas units of length. Its magnitude is the scale
for the surface tension to have effect.
Using these dispersion relationships, it is very difficult to create
a sequence of frames of water surface which for a continuous loop.
-5 0 5 10 15 20 25
Wave Amplitude
Figure 7: Profile of a 3-mode Gerstner wave.
In order to have the sequence repeat after a certain amount of time
Tfor example, it is necessary that all frequencies be multiples of
the basic frequence
However, when the wavevectors kare distributed on a regular lat-
tice, itis impossible to arrange the dispersion-generated frequencies
to also be on a uniform lattce with spacing ω0.
The solution to that is to not use the dispersion frequences, but
instead a set that is close to them. For a given wavenumber k, we
use the frequency
¯ω(k) = ω(k)
where [[a]] means take the integer part of the value of a, and ω(k)is
any dispersion relationship of interest. The frequencies ¯ω(k)are a
quantization of the dispersion surface, and the animation of the wa-
ter surface loops after a time Tbecause the quantized frequencies
are all integer multiples of ω0. Figure 8 plots the original disper-
sion curve, along with quantized dispersion curves for two choices
of the repeat time T.
4.3 Statistical Wave Models and the Fourier Trans-
Oceanographic literature tends to downplay Gerstner waves as a re-
alistic model of the ocean. Instead, statistical models are used, in
combination with experimental observations. In the statistical mod-
els, the wave height is considered a random variable of horizontal
position and time, h(x, t).
Statistical models are also based on the ability to decompose
the wave height field as a sum of sine and cosine waves. The
value of this decomposition is that the amplitudes of the waves
have nice mathematical and statistical properties, making it sim-
pler to build models. Computationally, the decomposition uses Fast
Fourier Transforms (ffts), which are a rapid method of evaluating
the sums.
Quantizing the Dispersion Relation
Dispersion Relation
Repeat Time = 100 seconds
Repeat Time = 20 seconds
Figure 8: A comparison of the continuous dispersion curve ω=
gk and quantized dispersion curves, for repeat times of 20 sec-
onds and 100 seconds. Note that for a longer repeat time, the quan-
tized is a closer approximation to the original curve.
The fft-based representation of a wave height field expresses the
wave height h(x, t)at the horizontal position x= (x, z)as the sum
of sinusoids with complex, time-dependent amplitudes:
h(x, t) = X
h(k, t) exp (ik·x)(36)
where tis the time and kis a two-dimensional vector with com-
ponents k= (kx, kz),kx= 2πn/Lx,kz= 2πm/Lz, and
nand mare integers with bounds N/2n < N/2and
M/2m < M/2. The fft process generates the height field
at discrete points x= (nLx/N, mLz/M). The value at other
points can also be obtained by switching to a discrete fourier trans-
form, but under many circumstances this is unnecessary and is not
applied here. The height amplitude Fourier components, ˜
h(k, t),
determine the structure of the surface. The remainder of this sub-
section is concerned with generating random sets of amplitudes in
a way that is consistent with oceanographic phenomenology.
For computer graphics purposes, the slope vector of the wave-
height field is also needed in order to find the surface normal, angles
of incidence, and other aspects of optical modeling as well. One
way to compute the slope is though a finite difference between fft
grid points, separated horizontally by some 2D vector x. While
a finite difference is efficient in terms of memory requirements, it
can be a poor approximation to the slope of waves with small wave-
length. An exact computation of the slope vector can be obtained
by using more ffts:
(x, t) = h(x, t) = X
h(k, t) exp (ik·x).(37)
In terms of this fft representation, the finite difference approach
would replace the term ikwith terms proportional to
exp (ik·x)1(38)
which, for small wavelength waves, does not well approximate the
gradient of the wave height. Whenever possible, slope computation
via the fft in equation 37 is the prefered method.
The fft representation produces waves on a patch with horizontal
dimensions Lx×Lz, outside of which the surface is perfectly peri-
odic. In practical applications, patch sizes vary from 10 meters to 2
kilometers on a side, with the number of discrete sample points as
high as 2048 in each direction (i.e. grids that are 2048 ×2048, or
over 4 million waves). The patch can be tiled seamlessly as desired
over an area. The consequence of such a tiled extension, however, is
that an artificial periodicity in the wave field is present. As long as
the patch size is large compared to the field of view, this periodicity
is unnoticeable. Also, if the camera is near the surface so that the
effective horizon is one or two patch lengths away, the periodicity
will not be noticeable in the look-direction, but it may be apparent
as repeated structures across the field of view.
Oceanographic research has demonstrated that equation 36 is a
reasonable representation of naturally occurring wind-waves in the
open ocean. Statistical analysis of a number of wave-buoy, photo-
graphic, and radar measurements of the ocean surface demonstrates
that the wave height amplitudes ˜
h(k, t)are nearly statistically sta-
tionary, independent, gaussian fluctuations with a spatial spectrum
denoted by
Ph(k) = D
h(k, t)
for data-estimated ensemble averages denoted by the brackets h i.
There are several analytical semi-empirical models for the wave
spectrum Ph(k). A useful model for wind-driven waves larger than
capillary waves in a fully developed sea is the Phillips spectrum
Ph(k) = Aexp 1/(kL)2
where L=V2/g is the largest possible waves arising from a con-
tinuous wind of speed V,gis the gravitational constant, and ˆwis
the direction of the wind. Ais a numeric constant. The cosine factor
k·ˆw|2in the Phillips spectrum eliminates waves that move perpen-
dicular to the wind direction. This model, while relatively simple,
has poor convergence properties at high values of the wavenumber
|k|. A simple fix is to suppress waves smaller that a small length
`L, and modify the Phillips spectrum by the multiplicative fac-
exp k2`2.(41)
Of course, you are free to “roll your own” spectrum to try out
various effects.
4.4 Building a Random Ocean Wave Height Field
Realizations of water wave height fields are created from the prin-
ciples elaborated up to this point: gaussian random numbers with
spatial spectra of a prescribed form. This is most efficiently accom-
plished directly in the fourier domain. The fourier amplitudes of a
wave height field can be produced as
h0(k) = 1
where ξrand ξiare ordinary independent draws from a gaussian
random number generator, with mean 0 and standard deviation 1.
Gaussian distributed random numbers tend to follow the experi-
mental data on ocean waves, but of course other random number
distributions could be used. For example, log-normal distributions
could be used to produce height fields that are vary “intermittent”,
i.e. the waves are very high or nearly flat, with relatively little in
Given a dispersion relation ω(k), the Fourier amplitudes of the
wave field realization at time tare
h(k, t) = ˜
h0(k) exp {(k)t}
0(k) exp {−(k)t}(43)
This form preserves the complex conjugation property ˜
h(k, t) =
h(k, t)by propagating waves “to the left” and “to the right”. In
addition to being simple to implement, this expression is also effi-
cient for computing h(x, t), since it relies on ffts, and because the
wave field at any chosen time can be computed without computing
the field at any other time.
In practice, how big does the Fourier grid need to be? What
range of scales is reasonable to choose? If you want to generate
wave heights faster, what do you do? Lets take a look at these
How big should the Fourier grid be? The values of Nand M
can be between 16 and 2048, in powers of two. For many
situations, values in the range 128 to 512 are sufficient. For
extremely detailed surfaces, 1024 and 2048 can be used. For
example, the wave fields used in the motion pictures Water-
world and Titanic were 2048×2048 in size, with the spacing
between grid points at about 3 cm. Above a value of 2048, one
should be careful because the limits of numerical accuracy for
floating point calculations can become noticeable.
What range of scales is reasonable to choose? The answer to this
question comes down to choosing values for Lx,Lz,M, and
N. The smallest facet in either direction is dx Lx/M or
dz Lz/N. Generally, dx and dz need never go below
2 cm or so. Below this scale, the amount of wave action is
small compared to the rest of the waves. Also, the physics
of wave behavior below 2 cm begins to take on a very differ-
ent character, involving surface tension and “nonlinear” pro-
cesses. From the form of the spectrum, waves with a wave-
length larger than V2/g are suppressed. So make sure that dx
and dz are smaller than V2/g by a substantial amount (10 -
1000) or most of the interesting waves will be lost. The se-
cret to realistic looking waves (e.g. figure 12 (a) compared to
figure 12 (c)) is to have Mand Nas large as reasonable.
How do you generate wave height fields in the fastest time? The
time consuming part of the computation is the fast fourier
transform. Running on a 1+ GHz cpu, 512 ×512 FFTs can
be generated at nearly interactive rates.
4.5 Examples: Height Fields and Renderings
We now turn to some examples of waves created using the fft ap-
proach discussed above. We will show waves in two formats: as
greyscale images in which the grey level is proportional to wave
height; and renderings of oceanscapes using several different ren-
dering packages to illustrate what is possible.
In the first set of examples, the grid size is set to M=N= 512,
with Lx=Lz= 1000 meters. The wind speed is a gale force at
V= 31 meters/second, moving in the x-direction. The small-wave
cutoff of `= 1 meter was also used. Figure 9 is a greyscale rep-
resentation of the wave height: brighter means higher and darker
means lower height. Although produced by the fft algorithms de-
scribed here, figure 9 is not obviously a water height field. It may
help to examine figure 10, which is a greyscale depiction of the
x-component of the slope. This looks more like water waves that
figure 9. What is going on?
Figures 9 and 10 demonstrate a consequence of water surface
optics, discussed in the next section: the visible qualities of the
surface structure tend to be strongly influenced by the slope of the
waves. We will discuss this in quantitative detail, but for now we
willl summarize it by saying that the reflectivity of the water is a
strong function of the slope of the waves, as well as the directions
of the light(s) and camera.
To illustrate a simple effect of customizing the spectrum model,
figure 11 is the greyscale display of a height field identical to figure
Figure 9: A surface wave height realization, displayed in greyscale.
Figure 10: The x-component of the slope for the wave height real-
ization in figure 9.
Figure 11: Wave height realization with increased directional de-
9, with the exception that the directional factor |ˆ
k·ˆw|2in equation
40 has been changed to |ˆ
k·ˆw|6. The surface is clearly more aligned
with the direction of the wind.
The next example of a height field uses a relatively simple shader
in BMRT, the Renderman-compliant raytracer. The shader is shown
in the next section. Figure 12 shows three renderings of water sur-
faces, varying the size of the grid numbers Mand Nand making
the facet sizes dx and dz proportional to 1/M and 1/N . So as we
go from the top image to the bottom, the facet sizes become smaller,
and we see the effect of increasing amount of detail in the render-
ings. Clearly, more wave detail helps to build a realistic-looking
As a final example, figure 13 is an image rendered in the com-
mercial package RenderWorld by Arete Entertainment. This ren-
dering includes the effect of an atmosphere, and water volume scat-
tered light. These are discussed in the next section. But clearly,
wave height fields generated from random numbers using an fft pre-
scription can produce some nice images.
4.6 Choppy Waves
We turn briefly in this section to the subject of creating choppy
looking waves. The waves produced by the fft methods presented
up to this point have rounded peaks and troughs that give them the
appearance of fair-weather conditions. Even in fairly good weather,
and particularly in a good wind or storm, the waves are sharply
peaked at their tops, and flattened at the bottoms. The extent of this
chopping of the wave profile depends on the environmental condi-
tions, the wavelengths and heights of the waves. Waves that are
sufficiently high (e.g. with a slope greater than about 1/6) eventu-
ally break at the top, generating a new set of physical phenonema
in foam, splash, bubbles, and spray.
The starting point for this method is the fundamental fluid dy-
namic equations of motion for the surface. These equations are ex-
pressed in terms of two dynamical fields: the surface elevation and
the velocity potential on the surface, and derive from the Navier-
Stokes description of the fluid throughout the volume of the water
and air, including both above and below the interface. Creamer
Figure 12: Rendering of waves with (top) a fairly low number of
waves (facet size = 10 cm), with little detail; (middle) a reasonably
good number of waves (facet size = 5 cm); (bottom) a high number
of waves with the most detail (facet size = 2.5 cm).
Figure 13: An image of a wave height field rendered in a commer-
cial package with a model atmosphere and sophisticated shading.
Surface Height
Surface Position
Displacement Behavior
Figure 14: A comparison of a wave height profile with and without
the displacement. The dashed curve is the wave height produced by
the fft representation. The solid curve is the height field displaced
using equation 44.
et al[16] set out to apply a mathematical approach called the ”Lie
Transform technique” to generate a sequence of ”canonical trans-
formations” of the elevation and velocity potential. The benefit of
this complex mathematical procedure is to convert the elevation and
velocity potential into new dynamical fields that have a simpler
dynamics. The transformed case is in fact just the simple ocean
height field we have been discussing, including evolution with the
same dispersion relation we have been using in this paper. Start-
ing from there, the inverse Lie Transform in principle converts our
phenomenological solution into a dynamically more accurate one.
However, the Lie Transform is difficult to manipulate in 3 dimen-
sions, while in two dimensions exact results have been obtained.
Based on those exact results in two dimensions, an extrapolation
for the form of the 3D solution has been proposed: a horizontal dis-
placement of the waves, with the displacement locally varying with
the waves.
In the fft representation, the 2D displacement vector field is com-
puted using the Fourier amplitudes of the height field, as
D(x, t) = X
h(k, t) exp (ik·x)(44)
Using this vector field, the horizontal position of a grid point of
the surface is now x+λD(x, t), with height h(x)as before. The
parameter λis not part of the original conjecture, but is a conve-
nient method of scaling the importance of the displacement vector.
This conjectured solution does not alter the wave heights directly,
but instead warps the horizontal positions of the surface points in
a way that depends on the spatial structure of the height field. The
particular form of this warping however, actually sharpens peaks in
the height field and broadens valleys, which is the kind of nonlin-
ear behavior that should make the fft representation more realistic.
Figure 14 shows a profile of the wave height along one direction
in a simulated surface. This clearly shows that the “displacement
conjecture” can dramatically alter the surface.
The displacment form of the this solution is similar to the algo-
rithm for building Gerstner waves [12] discussed in section 4. In
-5 0 5 10 15 20 25
Figure 15: A time sequence of profiles of a wave surface. From top
to bottom, the time between profiles is 0.5 seconds.
that case however, the displacement behavior, applied to sinusoid
shapes, was the principle method of characterizing the water sur-
face structure, and here it is a modifier to an already useful wave
height representation.
Figure 15 illustrates how these choppy waves behave as they
evolve. The tops of waves form a sharp cusp, which rounds out
and disappears shortly afterward.
One ”problem” with this method of generating choppy waves
can be seen in figure 14. Near the tops of some of the waves, the
surface actally passes through itself and inverts, so that the outward
normal to the surface points inward. This is because the amplitudes
of the wave components can be large enough to create large dis-
placements that overlap. This is easily defeated simply by reducing
the magnitude of the scaling factor λ. For the purposes of computer
graphics, this might actually be a useful effect to signal the pro-
duction of spray, foam and/or breaking waves. We will not discuss
here how to carry out such an extension, except to note that in order
to use this region of overlap, a simple and quick test is needed for
deciding that the effect is taking place. Fortunately, there is such a
simple test in the form of the Jacobian of the transformation from
xto x+λD(x, t). The Jacobian is a measure of the uniqueness of
the transformation. When the displacement is zero, the Jacobian is
1. When there is displacement, the Jacobian has the form
J(x) = JxxJyy Jxy Jy x ,(45)
with individual terms
Jxx(x) = 1 + λDx(x)
Jyy (x) = 1 + λ∂Dy(x)
0 20 40 60 80 100
Water Surface Profiles
Basic Surface
Choppy Surface
Folding Map
Figure 16: Wave height profile with and without the displacement.
Also plotted is the Jacobian map for choppy wave profile.
Jyx(x) = λ Dy(x)
Jxy(x) = λ Dx(x)
∂y =Jyx
and D= (Dx, Dy). The Jacobian signals the presence of the over-
lapping wave bacause its value is less than zero in the overlap re-
gion. For example, figure 16 plots a profile of a basic wave without
displacement, the wave with displacement, and the value of Jfor
the choppy wave (labeled ”Folding Map”). The ”folds” or overlaps
in the choppy surface are clearly visible, and align with the regions
in which J < 0. With this information, it should be relatively easy
to extract the overlapping region and use it for other purposes if
But there is more that can be learned from these folded waves
from a closer examination of this folding criterion. The Jacobian
derives from a 2×2matrix which measures the local uniqueness
of the choppy wave map xx+λD. This matrix can in general
be written in terms of eigenvectors and eigenvalues as:
Jab =Jˆe
b,(a, b =x, y)(46)
where Jand J+are the two eigenvalues of the matrix, ordered
so that JJ+. The corresponding orthonormal eigenvectors are
ˆeand ˆe+respectively. From this expression, the Jacobian is just
The criterion for folding that J < 0means that J<0and
J+>0. So the minimum eigenvalue is the actual signal of the on-
set of folding. Further, the eigenvector ˆepoints in the horizontal
direction in which the folding is taking place. So, the prescrip-
tion now is to watch the minimum eigenvalue for when it becomes
negative, and the alignment of the folded wave is parallel to the
minimum eigenvector.
We can illustrate this phenomenon with an example. Figures 17
and 18 show two images of an ocean surface, one without choppy
waves, and the other with the choppy waves strongly applied. These
two surfaces are identical except for the choppy wave algorithm.
Figure 19 shows the wave profiles of both surfaces along a slice
through the surfaces. Finally, the profile of the choppy wave is
plotted together with the value of the minimum eigenvalue in figure
20, showing the clear connection between folding and the negative
value of J.
Incidentally, computing the eigenvalues and eigenvectors of this
matrix is fast because they have analytic expressions as
2(Jxx +Jyy )±1
2(Jxx Jyy )2+ 4J2
for the eigenvalues and
ˆe±=(1, q±)
p1 + q2
for the eigenvectors.
5 Interactive Waves from Disturbances
The Fourier based approach to water surface evolutiondescribed in
the section 4 has several limitations that make in unworkable for
really interactive applications. Fundamentally, the Fourier method
computes the wave height everywhere in one FFT computation.
You cannot choose to compute wave height in limited areas of a
surface grid without completely altering the calculation. You either
get the whole surface, or you don’t compute it. This makes it very
hard to customize the wave propagation problem from one location
to another. So it you want to put some arbitrarily-shaped object
in the water, move it around some user-constructed path, the FFT
method makes it difficult to compute the wave response to the shape
and movement of the object. So if you want to have an odd looking
craft moving around in the water, and maybe going up and down
and changing shape, the FFT method is not the easiest thing to use.
Also, if you want to have a shallow bottom, with a sloping beach or
underwater sea mound or some other variability in the depth of the
water, the FFT method again is a challenge use.
Fortunately in the last few years an alternative scheme has
emerged which allows fast construction of a water surface in re-
sponse to interactions with objects in the water and/or variable bot-
tom depth. The heart of the method is an approach to computing
the propagation which does not use the FFT method at all. The fun-
damental mathematics of this interactive method is described in the
reference [22], and is refered to as iWave. Here we very quickly run
through the approach and show some examples in action.
At its heart the iWave method returns to the linearized equation
for the wave height, 22. We can turn the time derivatives into finite
time differences for a time step t, and get an explicit expression
for the wave height:
h(x, t + t)=2h(x, t)h(x, t t)
g(∆t)2p−∇2h(x, t)(50)
This form can be used to explicitly advance the surface wave height
from one frame to the next. The hard part of course is figuring out
how to calculate the last term, with the square root of the Laplacian
The solution is to use convolution. The wave height is kept on
a rectangular grid of points (the dimensions of which do not have
to be powers of two), and we make use of the fact that any linear
operation on a grid of data values can be converted into a convolu-
tion of some sort. The details of the numerical implementation are
contained in [22].
But the real, amazing, property of iWave is that wave interaction
with objects on the water surface is evaluated with a very simple
Figure 17: Simulated wave surface without the choppy algorithm
applied. Rendered in BMRT with a generic plastic shader. Figure 18: Same wave surface with strong chop applied. Rendered
in BMRT with a generic plastic shader.
0 5 10 15 20
Water Surface Profiles
Basic Surface
Choppy Surface
Figure 19: Profiles of the two surfaces, showing the effect of the
choppy mapping.
0 5 10 15 20
Water Surface Profiles
Choppy Surface
Minimum E-Value
Figure 20: Plot of the choppy surface profile and the minimum
eigenvalue. The locations of folds of the surface are clearly the
same as where the eigenvalue is negative.
linear operation that amount to an image masking. For this pur-
pose, an object in the water is described as a grid of mask values,
zero meaning the object is present, 1 if the object is not present,
and along the edges of the object values between 0 and 1 are use-
ful as an antialiasing. Applying this mask to the wave height grid,
the waves are effectively removed at grid points that the object is
located at. With just this simple procedure, waves that are incident
on the object reflect off of it in a realistic way. Figure 21 is a frame
from a sequence, showing the wave height computed using iWave
with several irregularly shaped objects.
Ordinarily, simulating waves that interact with an object on the
water surface should involve a careful treatment of boundary con-
ditions, matching water motion to the object on the boundary of
the geometry, and enforcing no-slip conditions. This is a complex
and time consuming computation that involves a certain amount of
numerical black art. It is suprising that a simple process like mask-
ing the wave height, as described above, should not be sufficient.
Yet, with the iWave procedure, correct-looking reflection/refraction
waves happen in the simulation. It is not yet clear just how quanti-
tatively accurate these interactions are. Figure 22 shows a rendered
scene with a high resolution calculation of waves interacting with
the hull of the ship.
5.1 Modifications for shallow water
Wave simulations based on the FFT method can simulate shallow
water effects by using the dispersion relationship in equation 32.
This only applies to a flat bottom. It would be nice to simulate wave
propagation onto a beach, or pasta shallow subsurface sea mount, or
over a submarine that is just below the surface. The iWave method
is a great way of doing those things. Recall that the iWave method
converts the mathematical operation
into a convolution, so that effectively g−∇2becomes a convolu-
tion kernel. A shallow bottom with depth Dchanges this term to
(compare with the dispersion relationship)
gp−∇2tanh p−∇2Dh(52)
This operation can also be converted into a convolution, with
g−∇2tanh −∇2Dbecomes a convolution kernel.
How is this applied to a variable-depth bottom? The convolution
kernel is a 13x13 matrix[22]. So we could build a collection of
kernels over a range of depth values D. At each point on the grid, a
custom kernel is constructed for the actual depth at that grid point
by interpolating from the set of prebuilt values. Figure 23 is the
wave height from a simulation using this technique. The bottom
slopes from deep on the right hand side, to a depth of 0 on the left
edge. In addition, there is a subsurface sea mount on the right.
There are three important behaviors in this simulation that occur
in real shallow water propagation:
1. Waves in shallow regions have large amplitudes that in deep
2. As waves approach a beach, they pile up together and have a
higher spatial frequency.
3. The subsurface sea mount causes a diffraction of waves.
In addition to these capabilities, iWave can also compute other
quantities, such as cuspy waves and surface velocity.
Figure 21: A sequence of frames from an iWave simulation, show-
ing waves reflecting off of objects in the water. The objects are the
black regions. In the upper left there is also diffraction taking place
as waves propagate the narrow channel and emerge in the corner
Figure 22: Frame from a simulation and rendering showing waves
interacting with a ship.
Figure 23: Frame from an iWave simulation with a variable depth
shallow bottom.
6 Surface Wave Optics
The optical behavior of the ocean surface is fairly well understood,
at least for the kinds of quiescent wave structure that we consider
in these notes. Fundamentally, the ocean surface is a near perfect
specular reflector, with well-understand relectivity and transmis-
sity functions. In this section these properties are summarized, and
combined into a simple shader for Renderman. There are circum-
stances when the surface does not appear to be a specular reflector.
In particular, direct sunlight reflected from waves at a large distance
from the camera appear to be spread out and made diffuse. This
is due to the collection of waves that are smaller than the camera
can resolve at large distances. The mechanism is somewhat similar
to the underlying microscopic reflection mechnanisms in solid sur-
faces that lead to the Torrance-Sparrow model of BRDFs. Although
the study of glitter patterns in the ocean was pioneered by Cox and
Munk many years ago, the first models of this BRDF behavior that
I am aware of were developed in the early 1980’s. At the end of this
section, we introduce the concepts and conditions, state the results,
and ignore the in-between analysis and derivation.
Throughout these notes, and particularly in this section, we ig-
nore one optical phenomenon completely: polarization. Polariza-
tion effects can be strong at a boundary interface like a water sur-
face. However, since most of computer graphics under considera-
tion ignores polarization, we will continue in that long tradition. Of
course, interested readers can find literature on polarization effects
at the air-water interface.
6.1 Specular Reflection and Transmission
Rays of light incident from above or below at the air-water interface
are split into two components: a transmitted ray continuing through
the interface at a refracted angle, and a reflected ray. The intensity
of each of these two rays is diminished by reflectivity and trans-
missivity coefficients. Here we discussed the directions of the two
outgoing rays. In the next subsection the coefficients are discussed.
6.1.1 Reflection
As is well known, in a perfect specular reflection the reflected ray
and the incident ray have the same angle with respect to the surface
normal. This is true for all specular reflections (ignoring roughen-
ing effects), regardless of the material. We build here a compact
expression for the outgoing reflected ray. First, we need to build up
some notation and geometric quantities.
The three-dimensional points on the ocean surface can be la-
belled by the horizontal position xand the waveheight h(x, t)as
r(x, t) = x+ˆ
yh(x, t),(53)
where ˆ
yis the unit vector pointing straight up. At the point r, the
normal to the surface is computed directly from the surface slope
(x, t) h(x, t)as
nS(x, t) = ˆ
y(x, t)
p1 + 2(x, t)(54)
For a ray intersecting the surface at rfrom direction ˆ
ni, the direc-
tion of the reflected ray can depend only on the incident direction
and the surface normal. Also, as mentioned before, the angle be-
tween the surface normal and the reflected ray must be the same
as the angle between incident ray and the surface normal. You can
verify for yourself that the reflected direction ˆ
nr(x, t) = ˆ
nS(x, t) (ˆ
nS(x, t)·ˆ
Note that this expression is valid for incident ray directions on either
side of the surface.
6.1.2 Transmission
Unfortunately, the direction of the transmitted ray is not expressed
as simply as for the reflected ray. In this case we have two guid-
ing principles: the transmitted direction is dependent only on the
surface normal and incident directions, and Snell’s Law relating the
sines of the angles of the incident and transmitted angles to the in-
dices of refraction of the two materials.
Suppose the incident ray is coming from one of the two media
with index of refraction ni(for air, n= 1, for water, n= 4/3
approximately), and the transmitted ray is in the medium with index
of refraction nr. For the incident ray at angle θito the normal,
sin θi=p1(ˆ
the transmitted ray will be at an angle θtwith
sin θt=|ˆ
Snell’s Law states that these two angles are related by
nisin θi=ntsin θt.(58)
We now have all the pieces needed to derive the direction of trans-
mission. The direction vector can only be a linear combination of
niand ˆ
nS. It must satisfy Snell’s Law, and it must be a unit vector
(by definition). This is adequate to obtain the expression
nt(x, t) = ni
ni+ Γ(x, t)ˆ
nS(x, t)(59)
with the function Γdefined as
Γ(x, t)ni
nS(x, t)
nS(x, t)|21/2
The plus sign is used in Γwhen ˆ
nS<0, and the minus sign is
used when ˆ
6.2 Fresnel Reflectivity and Transmissivity
Accompanying the process of reflection and transmission through
the interface is a pair of coefficients that describe their efficiency.
The reflectivity Rand transmissivity Tare related by the constraint
that no light is lost at the interface. This leads to the relationship
R+T= 1 .(61)
The derivation of the expressions for Rand Tis based on the elec-
tromagnetic theory of dielectrics. We will not carry out the deriva-
tions, but merely write down the solution
nr) = 1
Figure 24 is a plot of the reflectivity for rays of light traveling down
onto a water surface as a function of the angle of incidence to the
surface. The plot extends from a grazing angle of 0 degrees to per-
pendicular incidence at 90 degrees. As should be clear, variation of
the reflectivity across an image is an important source of the “tex-
ture” or feel of water. Notice that reflectivity is a function of the
angle of incidence relative to the wave normal, which in turn is di-
rectly related to the slope of the surface. So we can expect that a
strong contributor to the texture of water surface is the pattern of
slope, while variation of the wave height serves primarily as a wave
hiding mechanism. This is the quantitative explanation of why the
0 10 20 30 40 50 60 70 80 90
Incidence Angle (degrees)
Reflectivity from Above
Incident from Above
Incident from Below
Figure 24: Reflectivity for light coming from the air down to the
water surface, as a function of the angle of incidence of the light.
surface slope more closely resembles rendered water than the wave
height does, as we saw in the previous section when discussing fig-
ure 10.
When the incident ray comes from below the water surface, there
are important differences in the reflectivity and transmissivity. Fig-
ure 25 shows the reflectivity as a function of incidence angle again,
but this time for incident light from below. In this case, the re-
flectivity reaches unity at a fairly large angle, near 41 degrees. At
incidence angles below that, the reflectivity is one and so there is no
transmission of light through the interface. This phenomenon is to-
tal internal reflection, and can be seen just by swimming around in
a pool. The angle at which total internal reflection begins is called
Brewster’s angle, and is given by, from Snell’s Law,
sin θB
= 0.75 (63)
or θB
i= 48.6 deg. In our plots, this angle is 90 θB
i= 41.1 deg.
6.3 Building a Shader for Renderman
From the discussion so far, one of the most important features a ren-
dering must emulate is the textures of the surface due to the strong
slope-dependence of reflectivity and transmissivity. In this section
we construct a simple Renderman-compliant shader using just these
features. Readers who have experience with shaders will know how
to extend this one immediately.
The shader exploits that fact that the Renderman interface al-
ready provides a built-in Fresnel quantity calculator, which pro-
vides R,T,ˆ
nr, and ˆ
ntusing the surface normal, incident direction
vector, and index of refraction. The shader for the air-to-water case
is as follows:
surface watercolorshader(
color upwelling = color(0, 0.2, 0.3);
color sky = color(0.69,0.84,1);
color air = color(0.1,0.1,0.1);
float nSnell = 1.34;
float Kdiffuse = 0.91;
string envmap = "";
0 10 20 30 40 50 60 70 80 90
Incidence Angle (degrees)
Reflectivity from Below
Incident from Above
Figure 25: Reflectivity for light coming from below the water sur-
face, as a function of the angle of incidence of the light.
{float reflectivity;
vector nI = normalize(I);
vector nN = normalize(Ng);
float costhetai = abs(nI . nN);
float thetai = acos(costhetai);
float sinthetat = sin(thetai)/nSnell;
float thetat = asin(sinthetat);
if(thetai == 0.0)
{reflectivity = (nSnell - 1)/(nSnell + 1);
reflectivity = reflectivity * reflectivity;
{float fs = sin(thetat - thetai)
/ sin(thetat + thetai);
float ts = tan(thetat - thetai)
/ tan(thetat + thetai);
reflectivity = 0.5 * ( fs*fs + ts*ts );
vector dPE = P-E;
float dist = length(dPE) * Kdiffuse;
dist = exp(-dist);
if(envmap != "")
{sky = color environment(envmap, nN);
Ci = dist * ( reflectivity * sky
+ (1-reflectivity) * upwelling )
+ (1-dist)* air;
There are two contributions to the color: light coming downward
onto the surface with the default color of the sky, and light coming
upward from the depths with a default color. This second term will
be discussed in the next section. It is important for incidence angles
that are high in the sky, because the reflectivity is low and transmis-
sivity is high.
Figure 26: Simulated water surface with a generic plastic surface
shader. Rendered with BMRT. Figure 27: Simulated water surface with a realistic surface shader.
Rendered with BMRT.
This shader was used to render the image in figure 27 using the
BMRT raytrace renderer. For reference, the exact same image has
been rendered in 26 with a generic plastic shader. Note that the
realistic water shader tends to highlight the tops of the the waves,
where the angle of incidence is nearly 90 degrees grazing and the
reflectivity is high, while the sides of the waves are dark, where
angle of incidence is nearly 0 that the reflectivity is low.
7 Water Volume Effects
The previous section was devoted to a discussion of the optical be-
havior of the surface of the ocean. In this section we focus on the
optical behavior of the water volume below the surface. We begin
with a discussion of the major optical effects the water volume has
on light, followed by an introduction to color models researchers
have built to try to connect the ocean color on any given day to un-
derlying biological and physical processes. These models are built
upon many years of in-situ measures off of ships and peers. Fi-
nally, we discuss two important effects, caustics and sunbeams, that
sometimes are hard to grasp, and which produce beautiful images
when properly simulated.
7.1 Scattering, Transmission, and Reflection by
the Water Volume
In the open ocean, light is both scattering and absorbed by the vol-
ume of the water. The sources for these events are of three types:
water molecules, living and dead organic matter, and non-organic
matter. In most oceans around the world, away from the shore lines,
absorption is a fairly even mixture of water molecules and organic
matter. Scattering is dominated by organic matter however.
To simulate the processes of volumetric absorption and scatter-
ing, there are five quantities that are of interest: absorption coef-
ficient, scattering coefficient, extinction coefficient, diffuse extinc-
tion coefficient, and bulk reflectivity. All of these coefficients have
units of inverse length, and represent the exponential rate of atten-
uation of light with distance through the medium. The absorption
coefficient ais the rate of absorption of light with distance, the
scattering coefficient bis the rate of scattering with length, the ex-
tinction coefficient cis the sum of the two previous ones c=a+b,
and the diffuse extinction coefficient Kdescribes the rate of loss
of intensity of light with distance after taking into account both ab-
sorption and scattering processes. The connection between Kand
the other parameters is not completely understood, in part because
there are a variety of ways to define Kin terms of operational mea-
surements. Different ways change the details of the dependence.
However, there is a condition called the asymptotic limit at very
deep depths in the water, at which all operational definitions of K
converge to a single value. This asymptotic value of Khas been
modeled in a variety of ways. There is a mathematically precise
result that the ratio K/c depends only on b/c, the single scatter
albedo, and some details of the angular distribution of individual
scattering distributions. Figure 28 is an example of a model of
K/c for reasonable water conditions. Models have been gener-
ated for the color dependence of K, most notably by Jerlov. In
1990, Austin and Petzold performed a revised analysis of spectral
models, including new data, to produce refined models of Kas a
function of color. For typical visible light conditions in the ocean,
Kranges in value from 0.03/meter to 0.1/meter. It is generally true
that a < K < c.
One way to interpret these quantities for a simulation of water
volume effects is as follows:
1. A ray of sunlight enters the water with intensity I(after los-
ing some intensity to Fresnel transmission). Along a path un-
derwater of a length s, the intensity at the end of the path is
0 0.2 0.4 0.6 0.8 1
1-x + sqrt(mu*x*(1-x))
Figure 28: Dependence of the Diffuse Extinction Coefficient on the
Single Scatter Albedo, normalized to the extinction.
Iexp(cs), i.e. the ray of direct sunlight is attenuated as fast
a possible.
2. Along the path through the water, a fraction of the ray is scat-
tered into a distribution of directions. The strength of the scat-
tering per unit length of the ray is b, so the intensity is propor-
tional to bI exp(cs).
3. The light that is scattered out of the ray goes through poten-
tially many more scattering events. It would be nearly im-
possible to track all of them. However, the sum whole out-
come of this process is to attenuate the ray along the path from
the original path to the camera as bI exp(cs) exp(Ksc),
where scis the distance from the scatter point in the ocean to
the camera.
A fifth quantity of interest for simulation is the bulk reflectivity
of the water volume. This is a quantity that is intended to allow
us to ignore the details of what is going on, treat the volume as a
Lambertian reflector, and compute a value for bulk reflectivity. That
number is sensitive to many factors, including wave surface condi-
tions, sun angle, water optical properties, and details of the angular
spread. Nevertheless, values of reflectivity around 0.04 seem to
agree well with data.
7.2 The Underwater POV: Refracted Skylight,
Caustics, and Sunbeams
Now that we have underwater optical properties at hand, we can
look at two important phenomena in the ocean: caustics and sun-
7.2.1 Caustics
Caustics, in this context, are a light pattern that is formed on sur-
faces underwater. Because the water surface is not flat, groups of
Figure 29: Rendering of a caustic pattern at a shallow depth (5
meters) below the surface.
light rays incident on the surface are either focussed or defocussed.
As a result, a point on a fictitious plane some depth below the ocean
surface receives direct sunlight from several different positions on
the surface. The intensity of light varies due to the depth, orig-
inal contrast, and other factors. For now, lets write the intensity
of the pattern as I=Ref I0, with I0as the light intensity just
above the water surface. The quantity Ref is the scaling factor that
varies with position on the fictitious plane due to focussing and de-
focussing of waves, and is called a caustic pattern. Figure 29 shows
an example of the caustic pattern Ref. Notice that the caustic pat-
tern exhibits filaments and ring-like structure. At a very deep depth,
the caustic pattern is even more striking, as shown in figure 30.
One of the important properties of underwater light that produce
caustic patterns is conservation of flux. This is actually a simple
idea: suppose a small area on the ocean surface has sunlight passing
through it into the water, with intensity Iat the surface. As we
map that area to greater depths, the amount of area within it grows
or shrinks, but most likely grows depending on whether the area
is focussed or defocussed. The intensity at any depth within the
water is proportional to inverse of the area of the projected region.
Another way of saying this is that if a bundle of light rays diverges,
their intensities are reduced to keep the product of intensity time
area fixed.
Simulated caustic patterns can actually be compared (roughly)
with real-world data. In a series of papers published throughout the
1970’s, 1980’s, and into the 1990’s, Dera and others collected high-
speed time series of light intensity[21]. As part of this data collec-
tion and analysis project, the data was used to generate a probability
distribution function (PDF) for the light intensity. Figure 31 shows
two PDFs taken from one of Dera’s papers. The two PDF’s were
collected for different surface roughness conditions: rougher wa-
ter tended to suppress more of the high magnitude fluctuations in
Figure 32 shows the pdf at two depths from a simulation of the
ocean surface. These two sets do not match Dera’s measurements
because of many factors, but most importantly because we have
not simulated the environmental conditions and instrumentation in
Dera’s experiments. Nevertheless, the similarity of figure 32 with
Figure 30: Rendering of a caustic pattern at great depth (100 me-
ters) below the surface.
Figure 31: PDF’s as measured by Dera in reference [21].
Dera’s data is an encouraging point of information for the realism
of the simulation.
7.2.2 Godrays
Underwater sunbeams, also called godrays, have a very similar ori-
gin to caustics. Direct sunlight passes into the water volume, fo-
cussed and defocussed at different points across the surface. As the
rays of light pass down through the volume, some of the light is
scattered in other directions, and a fraction arrives at the camera.
The accumulated pattern of scattered light apparent to the camera
are the godrays. So, while caustics are the pattern of direct sun-
light that penetrates down to the floor of a water volume, sunbeams
are scattered light coming from those shafts of direct sunlight in
the water volume. Figure 33 demonstrates sunbeams as seen by a
camera looking up at it.
[1] Jos Stam, “Stable Fluids,” Proceedings of the 26th annual
conference on Computer graphics and interactive techniques,
pp 121-128, (1999).
Intensity / Mean Intensity
Figure 32: Computed Probability Density Function for light inten-
sity fluctuations in caustics. (upper curve) shallow depth of 2 me-
ters; (lower curve) deep depth of 10 meters.
Figure 33: Rendering of sunbeams, or godrays, as seen looking
straight up at the light source.
[2] Nick Foster and Ronald Fedkiw, “Practical animation of liq-
uids,” Proceedings of the 28th annual conference on Com-
puter graphics and interactive techniques, pp 23-30, (2001).
[3] B.A. Cheeseman and C.P.R. Hoppel,“SIMULATING THE
[4] Simon Premoze, “Particle-Based Simulation of Fluids,” Euro-
graphics 2003, Vol 22, No. 3.
[5] Jeff Odien, “On the Waterfront”, Cinefex, No. 64, p 96,
[6] Ted Elrick, “Elemental Images”, Cinefex, No. 70, p 114,
[7] Kevin H. Martin, “Close Contact”, Cinefex, No. 71, p 114,
[8] Don Shay, “Ship of Dreams”, Cinefex, No. 72, p 82, (1997)
[9] Kevin H. Martin, “Virus: Building a Better Borg”, Cinefex,
No. 76, p 55, (1999)
[10] Simon Premoˇ
ze and Michael Ashikhmin, “Rendering Natu-
ral Waters, Eighth Pacific Conference on Computer Graphics
and Applications, October 2000.
[11] Gary A. Mastin, Peter A. Watterger, and John F. Mareda,
“Fourier Synthesis of Ocean Scenes”, IEEE CG&A, March
1987, p 16-23.
[12] Alain Fournier and William T. Reeves, “A Simple Model of
Ocean Waves”, Computer Graphics, Vol. 20, No. 4, 1986, p
[13] Darwyn Peachey, “Modeling Waves and Surf”, Computer
Graphics, Vol. 20, No. 4, 1986, p 65-74.
[14] Blair Kinsman, Wind Waves, Their Generation and Propaga-
tion on the Ocean Surface, Dover Publications, 1984.
[15] S. Gran, A Course in Ocean Engineering,De-
velopments in Marine Technology No. 8, El-
sevier Science Publishers B.V. 1992. See also
[16] Dennis B. Creamer, Frank Henyey, Roy Schult, and Jon
Wright, “Improved Linear Representation of Ocean Surface
Waves. J. Fluid Mech, 205, pp. 135-161, (1989).
[17] Seibert Q. Duntley, “Light in the Sea,” J. Opt. Soc. Am., 53,2,
pg 214-233, 1963.
[18] Curtis D. Mobley, Light and Water: Radiative Transfer in
Natural Waters, Academic Press, 1994.
[19] Selected Papers on Multiple Scattering in Plane-Parallel At-
mospheres and Oceans: Methods, ed. by George W. Kattawar,
SPIE Milestones Series, MS 42, SPIE Opt. Eng. Press., 1991.
[20] R.W. Austin and T. Petzold, “Spectral Dependence of the Dif-
fuse Attenuation Coefficient of Light in Ocean Waters: A Re-
examination Using New Data, Ocean Optics X, Richard W.
Spinrad, ed., SPIE 1302, 1990.
[21] Jerzy Dera, Slawomir Sagan, Dariusz Stramski, “Focusing of
Sunlight by Sea Surface Waves: New Measurement Results
from the Black Sea,” Ocean Optics XI, SPIE Proceedings,
[22] Jerry Tessendorf, “Interactive Water Surfaces, Game Pro-
gramming Gems 4 , ed. Andrew Kirmse, Charles River Media,
[23] St´
ephan T. Grilli home page, grilli/.
[24] AROSS - Airborne Remote Optical Spotlight System,
8 Appendix: Sample code for interactive
water surfaces
The code listed in this appendix is a working implementation of the
iWave algorithm. As listed below, iwave paint looks like a crude
paint routine. The user can paint in two modes. When iwave paint
starts up, it begins running an iWave simulation on a small grid
(200x200 with the settings listed). This grid is small enough that
the iWave simulation runs interactively on cpus around 1 GHz and
faster. The running of the simulation is not apparent since there are
no disturbing waves at start up. The interface looks like figure 34.
In the default mode at start up, the painting generates obstructions
that show up in black and block water waves. Figure 35 shows
an example obstruction that has been painted. Hitting the ’s’ key
changes the painting mode to painting a source disturbance on the
water surface. As you paint the disturbance, iwave paint propagates
the disturbance, reflecting off of any obstructions you may have
painted. Figure 36 shows a frame after source has been painted
inside the obstructed and allows a brief period of time to propagate
inside the obstruction and exit, creating a diffraction pattern at the
mouth of the obstruction.
A few useful keyboard options:
oPuts the paint mode into obstruction painting. This may be se-
lected at any time.
sPuts the paint mode into source painting. This may be selected
at any time.
// iwave_paint
// demonstrates the generation and interaction of
// waves around objects by allowing the user to
// paint obstructions and source, and watch iwave
// propagation.
// author: Jerry Tessendorf
// August, 2003
// This software is in the public domain.
// usage:
// iwave_paint is an interactive paint program
// in which the user paints on a water surface and
// the waves evolve and react with obstructions
// in the water.
// There are two paint modes. Typing ’o’ puts the
// program in obstruction painting mode. When you
// hold down the left mouse button and paint, you
// will see a black obstruction painted. This
// obstruction may be any shape.
// Typing ’s’ puts the program in source painting
// mode. Now painting with the left mouse button
// down generates a source disturbance on the water
// surface. The waves it produces evolve if as you
// continue to paint. The waves bounce of off any
// obstructions that have been painted or are
// subsequently painted.
// Typing ’b’ clears all obstructions and waves.
// Typing ’=’ and ’-’ brightens and darkens the display
// of the waves.
// Pressing the spacebar starts and stops the wave
// evolution. While the evolution is stopped, you
// can continue painting obstructions.
// This code was written and runs under Linux. The
// compile command is
// g++ iwave_paint.C -O2 -o iwave_paint -lglut -lGL
#include <cmath>
Figure 34: The iwave paint window at startup.
Figure 35: The iwave paint window with obstruction painted.
Figure 36: The iwave paint window after painting source inside the
obstruction and letting it propagate out.
#ifdef __APPLE__
#include <GLUT/glut.h>
#include <GL/gl.h> // OpenGL itself.
#include <GL/glu.h> // GLU support library.
#include <GL/glut.h> // GLUT support library.
#include <iostream>
using namespace std;
int iwidth, iheight, size;
float *display_map;
float *obstruction;
float *source;
float *height;
float *previous_height;
float *vertical_derivative;
float scaling_factor;
float kernel[13][13];
int paint_mode;
bool regenerate_data;
bool toggle_animation_on_off;
float dt, alpha, gravity;
float obstruction_brush[3][3];
float source_brush[3][3];
int xmouse_prev, ymouse_prev;
// Initialization routines
// Initialize all of the fields to zero
void Initialize( float *data, int size, float value )
{for(int i=0;i<size;i++ ) { data[i] = value; }
// Compute the elements of the convolution kernel
void InitializeKernel()
{double dk = 0.01;
double sigma = 1.0;
double norm = 0;
for(double k=0;k<10;k+=dk)
{norm += k*k*exp(-sigma*k*k);
for( int i=-6;i<=6;i++ )
for( int j=-6;j<=6;j++ )
{double r = sqrt( (float)(i*i + j*j) );
double kern = 0;
for( double k=0;k<10;k+=dk)
{kern += k*k*exp(-sigma*k*k)*j0(r*k);
}kernel[i+6][j+6] = kern / norm;
void InitializeBrushes()
{obstruction_brush[1][1] = 0.0;
obstruction_brush[1][0] = 0.5;
obstruction_brush[0][1] = 0.5;
obstruction_brush[2][1] = 0.5;
obstruction_brush[1][2] = 0.5;
obstruction_brush[0][2] = 0.75;
obstruction_brush[2][0] = 0.75;
obstruction_brush[0][0] = 0.75;
obstruction_brush[2][2] = 0.75;
source_brush[1][1] = 1.0;
source_brush[1][0] = 0.5;
source_brush[0][1] = 0.5;
source_brush[2][1] = 0.5;
source_brush[1][2] = 0.5;
source_brush[0][2] = 0.25;
source_brush[2][0] = 0.25;
source_brush[0][0] = 0.25;
source_brush[2][2] = 0.25;
void ClearObstruction()
{for(int i=0;i<size;i++ ){ obstruction[i] = 1.0; }
void ClearWaves()
{for(int i=0;i<size;i++ )
height[i] = 0.0;
previous_height[i] = 0.0;
vertical_derivative[i] = 0.0;
void ConvertToDisplay()
{for(int i=0;i<size;i++ )
{display_map[i] = 0.5*( height[i]/scaling_factor + 1.0 )*obstruction[i];
// These two routines,
// ComputeVerticalDerivative()
// Propagate()
// are the heart of the iWave algorithm.
// In Propagate(), we have not bothered to handle the
// boundary conditions. This makes the outermost
// 6 pixels all the way around act like a hard wall.
void ComputeVerticalDerivative()
{// first step: the interior
for(int ix=6;ix<iwidth-6;ix++)
{for(int iy=6;iy<iheight-6;iy++)
{int index = ix + iwidth*iy;
float vd = 0;
for(int iix=-6;iix<=6;iix++)
{for(int iiy=-6;iiy<=6;iiy++)
int iindex = ix+iix + iwidth*(iy+iiy);
vd += kernel[iix+6][iiy+6] * height[iindex];
vertical_derivative[index] = vd;
void Propagate()
{// apply obstruction
for( int i=0;i<size;i++ ) { height[i] *= obstruction[i]; }
// compute vertical derivative
// advance surface
float adt = alpha*dt;
float adt2 = 1.0/(1.0+adt);
for( int i=0;i<size;i++ )
{float temp = height[i];
height[i] = height[i]*(2.0-adt)-previous_height[i]-gravity*vertical_derivative[i];
height[i] *= adt2;
height[i] += source[i];
height[i] *= obstruction[i];
previous_height[i] = temp;
// reset source each step
source[i] = 0;
// Painting and display code
void resetScaleFactor( float amount )
{scaling_factor *= amount;
void DabSomePaint( int x, int y )
{int xstart = x - 1;
int ystart = y - 1;
if( xstart < 0 ){ xstart = 0; }
if( ystart < 0 ){ ystart = 0; }
int xend = x + 1;
int yend = y + 1;
if( xend >= iwidth ){ xend = iwidth-1; }
if( yend >= iheight ){ yend = iheight-1; }
if( paint_mode == PAINT_OBSTRUCTION )
{for(int ix=xstart;ix <= xend; ix++)
{for( int iy=ystart;iy<=yend; iy++)
{int index = ix + iwidth*(iheight-iy-1);
obstruction[index] *= obstruction_brush[ix-xstart][iy-ystart];
else if( paint_mode == PAINT_SOURCE )
{for(int ix=xstart;ix <= xend; ix++)
{for( int iy=ystart;iy<=yend; iy++)
{int index = ix + iwidth*(iheight-iy-1);
source[index] += source_brush[ix-xstart][iy-ystart];
// GL and GLUT callbacks
void cbDisplay( void )
glDrawPixels( iwidth, iheight, GL_LUMINANCE, GL_FLOAT, display_map );
// animate and display new result
void cbIdle()
{if( toggle_animation_on_off ) { Propagate(); }
void cbOnKeyboard( unsigned