## 4 Discussion  
Conclusively, the parameterization of Nusselt number with frequency and amplitude under sinusoidal and square wave shapes is established in the current study, helping to identify the appropriate operating envelope of vibration characteristics for enhanced thermal performance. Increasing the frequency or amplitude within the appropriate operating envelope region increases the heat sink thermal performance at each vibrational wave shape. However, square wave-shaped vibration is found to be more effective in the enhancement of heat transfer than sinusoidal vibration. Thus, applying square wave-shaped vibration on the forced convective cooling system is more beneficial to its thermal performance and further reduces the cooling system size compared with sinusoidal vibration at similar vibrational frequencies and amplitudes. By using square wave-shaped vibration at \(f=100\) Hz and \(0.005\) m amplitudes, the maximum enhancement of the Nusselt number is 25% compared with the nonvibrating heat sink, which is 11% higher than the Nusselt number recorded by the sinusoidal vibrating heat sink. This finding results in a 42.2% reduction in Reynolds number compared with the Reynolds number of nonvibrating cooling systems, which is 13.4% higher than the reduction in Reynolds number by the sinusoidal vibrating heat sink at similar vibrational frequencies and amplitudes.  
Researchers have been continuously investigating new methods to enhance the cooling of electronic systems. [PERSON], et al. [53] used a fan-integrated microchannel heat sink with rhombus fractal-like units. They achieved an 18% increase in Nusselt number compared with a conventional parallel microchannel heat sink. [PERSON] and [PERSON] [5] applied sinusoidal vibrations on a piezo-actuated fin. They achieved up to a 25.5% improvement in Nusselt number compared with a nonvibrating fin. [PERSON], et al. [54] used plate-fin heat sinks with corrugated half-round pins in a vertical arrangement subject to parallel flow and recorded an approximately 34.5% increase in Nusselt number values compared with the traditional plate-fin heat sink. [PERSON], et al. [55] used an impinging air jet with a crossflow configuration. They recorded an up-to-40% increase in the average Nusselt number compared with the conventional cross-flow configuration. [PERSON], et al. [56] applied a pulsating flow through heated blocks. They recorded increases of up to approximately 23% in Nusselt number values compared with a nonpulsating flow.  
As mentioned, using square-shaped wave vibration achieved an up-to-25% increase in Nusselt number and a 42.2% reduction in Reynolds number compared with nonvibrating fins. This result shows that square shape-waved vibration resulted in a higher cooling effect compared with other methods, making it achieve further reductions in the cooling system size of electronic components.  
It is worth mentioning that the results presented in this research could be assumed applicable to heat sinks with more fins under the same heat flux and air velocity, because past research reported by [PERSON], et al. [57] found that the Nusselt number does not change with the increase in the number of static fins. They also reported that when they varied the number of fins, the flow and thermal contours indicated very small differences between the heat sink fin channels. Since the finding of [57] that the Nusselt number remains unchanged when the number of static fins increases, and in this research, the application of vibration on fins is equal, it is expected that the results presented in this research will be the same when number of fins increases under the same heat flux and air velocity. However, further research is required to confirm this assumption.  
It is also important to mention that special consideration may be required in designing a vibrating system. Therefore, future research could be performed to investigate the reliability of the system demonstrating the mechanical stress factors and fatigue life due to the supplied vibration.