Exploring the Essential Spectrum of Normal Vibrations in Internal Waves: Insights from Specific Geometries and Explicit Eigenvalues

Aims: For various models of three-dimensional fluid which describe the flows in the Atmosphere and the Ocean in the gravity field with the stratification of vertical density, we investigate a relation between the essential spectrum of normal vibrations of internal waves and non-uniqueness of the limit amplitude of vibrations induced by external mass forces. To make the study more detailed and descriptive, we find the explicit spectrum for some particular domains.

Methodology: Fourier Transform, Spectral Analysis of Self-Adjoint Operators in Hilbert Spaces.

Results: We establish a direct relation between the frequency of the induced vibrations, the essential spectrum, and the non-uniqueness of the limit amplitude. We also find the explicit eigenfunctions and the eigenvalues of the spectrum for rectangular, spherical and cylindrical domains.