This work provided analytical investigations of internal gravity waves affected by the presence of two localized chemicals. Based on the coupling between gravity wave equations under the Boussinesq approximation and the continuity equations for two chemicals, the study presented a mathematical model modeling gravity-wave interactions with the two atmospheric chemicals. The linearized version of this model was analytically investigated for two cases: one with constant mean flow and the other one with a nonconstant mean flow, resulting in a critical level effect. It was observed that in both cases the presence of the two chemicals led to a significant impact on gravity waves. Analytical investigations of the vertically varying mean flow case reveal that the wave-reduction behavior was observed in the vicinity of the critical layer.

This study aims to derive criteria for examining the asymptotic and oscillatory behavior of solutions to functional differential equations with delayed damping. By employing the Riccati technique together with an improved approach, we establish new criteria that complement the existing literature while distinguishing themselves by accounting for the delay effect in the damping term and by providing the well-known sharp criterion for Euler-type equations. Numerical examples illustrate the theoretical findings and clarify how the delay in the damping term affects the solution dynamics.

The objective of this work was to provide sufficient conditions for testing the oscillatory performance of solutions of the neutral second-order differential equation with distributed deviation arguments. We derived improved relations that influence the oscillation parameters of the equation under study. We used comparison with lower-order equations and Riccati techniques to derive the oscillation parameters. Comparing our findings with earlier pertinent findings allows us to assess the advancements made in oscillation theory. Furthermore, some numerical solutions for a special case of the equation under study are presented, and the numerical and theoretical results are compared.