FIG. 9: The stability boundaries for traveling waves according to the amplitude equations (24)– (26), in the special case α ′ = 0, for different values for β $^{′}$: (a) β ′ = 0 . 2, (b) β ′ = 0 . 476, (c) β ′ = 0 . 6, (d) β ′ = 2, (e) β ′ = 4 . 5, (f) β ′ = 5, (g) β ′ = 5 . 0607, (h) β ′ = 6. See the text for more details. From (28), rolls are unstable as long as q > 11 r$_{2}$/ 12 + ˆ v $^{2}$/ 1152. If q < 11 r$_{2}$/ 12 + ˆ v $^{2}$/ 1152, then σ$_{1}$ is purely imaginary and hence σ$_{2}$ must be considered. From (29) we have definite instability if r$_{2}$ > 144 q/ 91. In addition to these rather blunt conditions, the sign of σ$_{2}$ must also be considered in order to determine the stable region. Figure 10 shows the curves q = 11 r$_{2}$/ 12 + ˆ v $^{2}$/ 1152 (solid line), q = 91 r$_{2}$/ 144 (dashed line) and σ$_{2}$ = 0 (dotted lines). Any region of stability must lie between the solid and dashed lines. After checking carefully the signs of the eigenvalues, we find that the stable region (indicated by the asterisks in the figure) lies between the two dotted lines in the upper and lower parts of the graph, and between the dotted and solid lines for a small range of intermediate values of r$_{2}$ (see Fig. 10). Although they appear almost parallel in Fig. 10(a), for large r$_{2}$ , as in Fig. 10(b), the two 19 (a) (b) (c) (d) (e) (f) (g) (h)