This study investigates the stability of wave modes in zonal shear currents on a beta-plane. The main research methods are analytical analysis and numerical modeling, utilizing MATLAB to solve the governing equations and visualize the results. The main result is the identification of stable ranges of wavenumbers for both sinusoidal and varicose modes that manifest within the flow. Numerical analysis reveals a β-destabilizing effect, which leads to the emergence of “β-destabilized” modes. It is shown that the isofrequencies in phase space consist of two branches: one corresponding to “flow waves modified by the β-effect” and the other to “Rossby waves modified by the current.” The distinction between “flow waves modified by the β-effect” (wave number k > 0) and “Rossby waves modified by the current” (wave number k < 0) reflects the dominant factor governing their properties: the β-effect for flow waves and the current for Rossby waves. These waves play a key role in long-distance energy transfer, influencing the redistribution of heat and salinity in the ocean. It is hypothesized that the spatial functions corresponding to the sinusoidal mode represent a pair of monopoles of the same sign, while the varicose mode corresponds to dipoles. A pair of monopoles with opposite signs forms a vortex pair. Studying of the β-destabilizing effect and transverse energy radiation provides deeper insights into the mechanisms of oceanic current dynamics and their impact on global climate processes. This is particularly relevant in global warming, where changes in ocean circulation can have far-reaching consequences for the planet's climate and ecosystems. While specific parameterizations for existing climate model related to β-destabilizing effect are beyond the scope of this study, our results contribute to that fundamental understanding.