This Ansatz guarantees that the boundary conditions 6. Substituting this form of the eigenfunctions into the eigenvalue problem 6. Note that the formula for the decay rates is the same independent of whether N is odd or even. Hence, they are either both real numbers or they form a complex conjugate pair, depending on the value of Ra. It is evident form Eq. Hence, for negative Ra propagating modes may appear, i. The presence of propagating modes, branch points and crossing points is evident.
To illustrate this behavior we show in Fig.
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From the 1 3 behavior displayed in Fig. Furthermore, as shown in detail in Sect. Substituting these relationships into Eq. The constants CN are real when the corresponding eigenvalues are two real numbers, while they are purely imaginary when the corresponding N -eigenvalues form a complex conjugate pair propagating modes. For the branch points, we refer to the discussion of propagating modes after Eq.
In any case, upon dividing 6. Since the eigenfunctions 6. Substituting 6. The explicit expressions become simpler if one distinguishes the case when the index N corresponds to two real decay rates from the case when N gives a complex conjugate pair. In the second line of 6.
If the two N -decay rates form a complex conjugate pair, the complex 6. As was already discussed after Eq. For the case of two free boundaries examined in Sect.
We note that the eigenfunctions 6. The odd eigenfunctions have a structure similar to Eq. They satisfy the two 6. Imposing the boundary condition 6. For an analysis of the nonequilibrium structure factor measured in experiments in Sect. For negative Ra, as in Sect. Schmitz and Cohen b analyzed the conditions under which such propagating modes may be present.
We shall not further investigate the propagating modes for the case of rigid boundaries here. In these two asymptotic limits there are no propagating modes; as shown in Fig. For the even ones, we substitute Eqs. The series expansion 6. For the decay rates of the second kind, solving Eq. For the case of the odd decay rates, solving Eq. As was the case for the even decay rates, to obtain the solution of the second kind for the odd ones solving Eq.
For free boundaries we obtained in Sect. Nevertheless, there is a close analogy in the nature of the decay rates for free and rigid boundaries. This fact is true for both rigid and free boundaries, and in both cases the corresponding hydrodynamic mode has even vertical parity. They can be obtained by substituting Eqs.
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After performing the vertical integration in 6. Finally, substitution of 6. However, as will be elucidated in Sect. Since the leading term in the expansion for the decay rates is 6. Next, we determine the normalization constants. Substituting Eqs.
Finally, substitution of the asymptotic expansions 6. For the case of free boundaries, Eq. As is well known, from condition 6.
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Note that the minus sign in the RHS of Eq. The relevance of Eq. As shown by Eq. Values displayed in Table 6. We observe in Fig. From Eqs. The decay rates computed numerically do indeed agree with the expected asymptotic behavior 6. It is interesting that the decay rates 6. Furthermore, this asymptotic behavior for large horizontal wave number is identical to the large-q limit of the decay rates 4. In Fig. Indeed, from Eq. The slowest decay rate, when computed numerically, does indeed agree with Eq. We do not give details, but the procedure is as follows: Upon substitution of Eq.
Such an analysis has been presented by Cross and Cross and Hohenberg We shall further discuss this topic in Sect. We start this program by considering in Sect. Therefore, in Sect. We then proceed in Sect. We conclude this chapter with some comments in Sect. To sum the series 7. We can thus make a rearrangement 7. It is worth noting that the result obtained in order N no longer depends on whether the N -mode is propagating or not, i. Equation 7. The equilibrium part of Eq.
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The second part of Eq. Equations 7. However, Eq. As is well known, Eq.