Game Development Reference
In-Depth Information
Chapter 15 Summary
The Wave Equation
The two-dimensional wave equation for a surface experiencing a viscous damp-
ing force is
2
2
2
z
z
z
z
2
=
c
+
μ
.
t
2
x
2
y
2
t
The constant c is the speed at which waves propagate through the medium, and
the constant
represents the viscosity of the medium.
μ
Approximating Derivatives for a Fluid Surface
The first derivative of a function
()
zx can be approximated by the formula
(
)
(
)
d zx
dx
zx d zx d
+− −
()
,
2
d
where d represents some constant step size. The second derivative of
zx can be
()
approximated by the formula
(
)
( )
(
)
d
2
zx d zx zx d
+−
2
+ −
zx
()
.
dx
2
d
2
Evaluating Fluid Surface Displacement
The future displacement
(
)
of a point on the surface of a fluid after a
time t has passed is calculated using the equation
zijk
,,
+
1
48
ct d
22
2
μt
2
(
)
(
)
(
)
zijk
,,
+=
1
zijk
,,
+
zijk
,,
1
μt μt
ct d zi jk zi jk zij k zij k
μt
+
2
+
2
22
2
2
[
]
(
)
(
)
(
)
(
)
+
+
1,
,
+
1,
,
+
,
+
1,
+
,
1,
,
+
2
where d is the distance between neighboring vertices in the triangle mesh.
Stability of the Numerical Method for a Fluid
Given a constant time step t , the wave speed c must satisfy
d
0
<<
c
μt
+
2
.
2
t