In this paper, we study the spectrum of the Rayleigh equation, which models the Rayleigh–Taylor instability in a fluid of variable density ρ(x), where ρ(x) goes to ρ− at −∞ and ρ+ at +∞:
$-h^{2}{\frac{\mathrm{d}}{\mathrm{d}x}}\bigl(\rho(x){\frac{\mathrm{d}u}{\mathrm{d}x}}\bigr)+(\rho(x)+\delta \rho'(x))u=0,\quad u\in L^{2}(\mathbb{R} ).$
The behavior of the smallest value δ(h) of δ>0 for which there exists a nontrivial solution u is investigated in terms of h for the two regimes h→0 (called the semi‐classical regime) and h→+∞. The so‐called growth rate of the Rayleigh–Taylor instability is
$\sqrt{h/\delta(h)}$
.
When ρ−ρ−∈L2(
$\mathbb{R} $
−) and ρ−ρ+∈L2(
$\mathbb{R} $
+), we prove that δ(h)/h→(ρ++ρ−)/(ρ−−ρ+), as h→+∞, generalizing the result of Lord Rayleigh [23].
We then investigate the expansion of δ(h)/h in terms of 1/h and properties of ρ. In particular, we identify the number of terms n of the expansion in function of the behavior of ρ(x)−ρ± at ±∞, extending the results of Cherfils, Lafitte and Raviart [6].
