Figure 4.

Minimum wave speed and the dispersion relation. (A) Comparison of c_min (κ) from Eq. 26 with the numerically-estimated wave speed for s(x, t) = v_r(x, t) + v_g(x, t). Numerical solutions of Eqs. 12 and 13 are obtained with Δx = 0.1, Δt = 0.001 and $\mathcal{D}=1$. Further, the initial condition is Eq. 9 with ζ = 10. For κ = 0, there is no traveling wave, so we set c = 0. The numerical solutions are considered beginning with κ = 0.06, and then with increasing values of κ from 0.25 to 3, with increments of 0.25. From these numerical solutions we estimate the wave speed for s(x, t) = v_r(x, t) + v_g(x, t) by using linear interpolation to find the position corresponding to s(x, t) = 0.5 on the wave for various times (21). (B) Asymptotic expansions for c_min (κ) as κ → 0 and κ → ∞. (C) Comparison of c from Eq. 35 with the wave speed estimated using numerical solutions and with c_min from Eq. 26. Solutions are given for κ = 1 (blue) and κ = 2 (red). The continuous curves show c from Eq. 35. The dots represent the wave speed from numerical solutions obtained with Δx = 0.1, Δt = 0.001, $\mathcal{D}=1$, and initial conditions of the form Eq. 36 with α = 0.1, 0.2, 0.5, 1, 1.5 and 2. The dotted horizontal lines show c_min from Eq. 26. To see this figure in color, go online.