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.

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