In marine exploration, when the seismic data in the exploration area only includes P-wave data collected by hydrophones, traditional acoustic full waveform inversion velocity modeling methods cannot invert S-wave velocity information. This paper focuses on ocean fluid-solid coupled media and derives the acoustic-elastic coupled equations using the boundary coupled method of acoustic-elastic wave equations. We validated the potential of using P-wave data for shear wave modeling through numerical simulation results of a three-layer model. An inaccurate velocity model can lead to the failure of full waveform inversion. To address the dependency issue of velocity model in full waveform inversion, we introduce second-order time integration operation and construct a second-order integration objective function. Then, the second-order integral waveform inversion gradient operators suitable for the acoustic-elastic coupled equations were derived, and a P-and S-wave velocity waveform inversion approach based on pure P-wave data in a marine environment was established. Finally, we conduct inversion tests using modified Marmousi2 models of the horizontal and irregular seabed. The inversion results verified the applicability and accuracy of the second-order integrated wave field acoustic-elastic coupled equation inversion method.