A semi-analytical solution to incident plane P waves scattering by saturated river valley with arbitrary shapes containing water

River valley terrain can lead to local amplification effects of site seismic responses and significant spatial variation of strong ground motions. This paper presented a semi-analytical solution for the scattering of incident plane P-waves by a saturated river valley with arbitrary shapes containing water, using the moment method and the wave function expansion method. The influence of factors such as valley geometry shapes, soil porosity, wave incident angle and frequency on the seismic response of valley topography is further investigated. Parameter analysis shows that these factors have a significant impact on the site seismic response. Soil porosity has a greater impact on the horizontal response of the site than on the vertical response. Moreover, the soil porosity is directly related to the amplitude of the pore pressure, while the incidence angle and frequency of the incident wave determine the distribution position of the pore pressure amplitude. This semi-analytical solution can be seen as an attempt to solve the scattering problem of incident P wave by saturated river valleys of arbitrary shapes containing water. The purpose of this paper is to provide a reference and a possible way for subsequent scholars to study the analytical solution to this problem. The conclusions drawn from the parameter analysis in this paper can also serve as a theoretical basis for seismic fortification in saturated valley areas.