mechanical power is the sum of the induced power and the profile power
induced power is the cost of lift
profile power is the cost of drag during flight
profile power proportional to the product of wingbeat frequency and amplitude cubed
there is a lot more math involved in these calculations. see Lehmann & Dickson 1997 for more specifics
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references
Lehmann, F. O., & Dickinson, M. H. (1997). The changes in power requirements and muscle efficiency during elevated force production in the fruit fly Drosophila melanogaster. The Journal of Experimental Biology, 200(Pt 7), 1133–1143. https://doi.org/10.1242/jeb.200.7.1133