# Surface Motion for P-Wave Scattering by an Arbitrary-Shaped Canyon in Saturated Half-Space
## 6 Conclusions  
First, based on [PERSON]'s fluid-saturated porous media theory, this paper applies the scattering wave field potential function, which can automatically satisfy the surface stress freedom in half-space, and then converts the boundary value problem into a series of algebraic equations with infinite terms by applying the moments method according to the stress-free boundary conditions on the boundary of an arbitrary-shaped canyon. The semianalytic solution of P-wave scattering by arbitrary-shape canyon in saturated half-space is obtained by the numerical solution of truncation. On this basis, the influence of the shape of the canyon, porosity, incidence angle, and frequency on surface displacement is investigated. The main conclusions are:  
Figure 7: Influence of incident angle on the displacement response of a trapezoidal canyon (\(n_{s}=0.3\)). The incidence angle has an obvious influence on the displacement amplitude. For the solved cases, almost all of them meet the following rules: for \(x\)-component displacement, when the P-wave is oblique incident at the incidence angle \(\gamma_{1}=30^{\circ}\), the displacement amplitude is the largest, followed by \(60^{\circ}\), \(5^{\circ}\), and \(90^{\circ}\); for \(y\)-component displacement, the amplitude of displacement is the largest when the P-wave is vertically incident at the incident angle \(\gamma_{1}=90^{\circ}\), which is \(60^{\circ}\), \(30^{\circ}\), and \(5^{\circ}\), successively. 2. Compared with the incidence angle, porosity has little influence on the amplitude of displacement, especially for \(y\)-component displacement. The displacement curves of the three kinds of porosity almost coincide on the front side and backside of the canyon, but there is a slight difference in the inner part of the canyon. The effect on the displacement amplitude in the horizontal direction is more significant, and the displacement amplitude is smaller and smaller as the porosity increases on the whole. 3. The influence of the shape of the canyon on displacement amplitude is mainly reflected in the shielding effect on the incident wave. At the same dimensionless frequency, the ground surface displacement of the trapezoidal canyon is more oscillating than that of the semicircular canyon, indicating that the trapezoidal canyon has a more shielding effect on the incident wave. X.D. : validation, formal analysis, data curation, writing--original draft; L.J. : methodology, conceptualization, validation, formal analysis, visualization, writing--review and editing; Z.Z. : supervision, investigation, writing--review and editing. All authors have read and agreed to the published version of the manuscript. This study is supported by the Beijing Natural Science Foundation (Grant No. 8232016) and the China Postdoctoral Science Foundation (Grant No. 2022M/12943), which are gratefully acknowledged. The raw data supporting the conclusions of this article will be made available by the authors. The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.