Symmetry in the interval wave problem (Q1081438)

We will consider the problem of the perturbation spectrum in a layered (stratified) heavy fluid. Denoting the set of eigenvalues by \(\{\) \(E\}\), we will investigate the invariance properties of \(\{\) \(E\}\) under the transformation of the density profile within the layer \(\rho (\xi)\to [\rho (-\xi)]^{-1}\). In what follows this transformation is called the R inversion. It is shown that the \(\{\) \(E\}\) of a layer with a free and a rigid boundary is invariant under the R inversion in the case of a stepped profile with number of steps \(N\leq 4\); the hypothesis that this result holds good for arbitrary N is advanced; it is shown that if the assertion concerning invariance under the R inversion is valid in the case of a layer with a free and rigid boundary, then it is also valid in an unbounded fluid and vice versa; the spectrum of the internal waves and Rayleigh-Taylor modes is calculated for a stepped profile.