Game Development Reference
In-Depth Information
Solving for c , we find
d
0
<<
c
μt
+
2
.
(15.30)
2
t
This tells us that for any given distance d between adjacent vertices and any time
interval t between consecutive iterations of Equation (15.25), the wave velocity c
must be less than the maximum value imposed by Equation (15.30).
Alternatively, we may calculate a maximum time interval t given the distance
d and the wave velocity c . Multiplying both sides of Equation (15.29) by
(
)
μt
+
2
and simplifying yields
2
4
c
2
0
<
t μt
<
+
2
.
(15.31)
2
d
The left inequality simply requires that
, a condition that we would naturally
impose in any case. The right inequality yields the quadratic expression
>
t
0
2
4
c
2
t μt
−−<
2
0
.
(15.32)
2
d
Using the quadratic equation, the roots of the polynomial are given by
2
2
2
μ μ cd
±+
32
t
=
.
(15.33)
2
2
8
cd
Since the coefficient of the quadratic term in Equation (15.32) is positive, the
corresponding parabola is concave upward, and the polynomial is therefore nega-
tive when t lies in between the two roots. The value under the radical in Equation
(15.33) is larger than μ , so the lesser of the two roots is negative and can be dis-
carded. We can now express the restriction on the time interval t as
2
2
2
μ μ cd
++
32
0
<<
t
.
(15.34)
2
2
8
cd
Using a value for the wave velocity c falling outside the range given by
Equation (15.30) or a value for the time interval t falling outside the range given
by Equation (15.34) results in an exponential explosion of the vertex dis-
placements.