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Table 1

The long wavelength limit: general characteristic of real solutions of Eq. (57).

Ωτ → −∞ Ωτ < 0 Ωτ = 0 Ωτ > 0 Ωτ → +∞

n1 → n < 0
n2 → 0
two negative roots for Ωτ < Ωτ2
complex conjugate roots for Ωτ2 < Ωτ < Ωτ1
two positive roots for Ωτ1 < Ωτ
n = n0 > 0 n > 0 n → 0+

n1 → n < 0
n2 → 0+
two roots with opposite signs n = n0 < 0 n < 0 n → 0

Notes. The quantities  n0, Ωτ1,2 and n are given by Eqs. (59), (69) and (70), respectively.

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