3.5 Linear Equations with Variable Coeffieients
149
FIGURE 3.8. Misrepresentation of a 4b.x/3 wave as a 4b.x wave when sampled on a
diserete rnesh.
relation
k - {k l +k2 - 27f/llx, ifkl +k2 > nfSx,
(3.91)
-
kl +k2+27f/llx, ifkl +k2 < -rr/Ilx.
There is no aliasing when Ikl + k21 n / Sx , In particular, if both r/J and X are
I
Itl
41lx waves or longer, their product will not be aliased. A graphical diagram
of the
aliasing process may be created by plotting kl + ka and the cutoff wave number
tt / Sx on a number line extending from zero to 27f/ Sx , Since (kl + k2 + IkI)/2 =
n / Ilx, Ikl appears as the reflection of kl + ka about the cutoff wave number,
as illustrated in Fig. 3.9. Note that the product of two extremely short waves is
aliased into a relatively long wave. For example , the product of a 21lx wave and
a 2.51lx wave appears as a IOllx wave.
Unstable Growth Through Aliasing Error
Let us now consider the effects of aliasing error on stability," Suppose that (3.87)
is approximated as the differential-difference equation (3.88) and that a solution
#.-- •• "-..
wave number
I •
I
••
ix (k,+k2)
o
I - - resolved waves - - - . l
FIGURE 3.9. Aliasing of kl + k2 into k sueh that Ikl = 2Jr/ Sx - (kl + k2) appears to be
the symmetrie refleetion of k1 + k2 about the eutoff wave number n / Sx ,
6The following exarnple was anticipated by Miyakoda (1962) and Gary (1979) .