## 1 Introduction  
Retaining walls represent one of the most commonly used support structures in geotechnical engineering, and are widely employed for slope stabilization in construction projects, hydraulic engineering, and railway and highway engineering. The 2005 Fukuoka Prefecture West Offshore Earthquake caused severe damage to residential areas on Genkai Island. A survey of 218 retaining walls revealed that 83% of them were damaged, with 61% losing their functionality (collapsed or exhibiting deformation/cracks) [1]. Following the May 12 th Wenchuan Earthquake in 2008, investigations along the Dujiangyan-Yingxiusection of National Highway 213 showed that three types of seismic damage--deformation cracking, sliding, and overturning--accounted for approximately 78% of all retaining wall failures. These failure cases demonstrate that the current seismic design theory for retaining walls still exhibits substantial room for improvement. Proposing appropriate seismic earth pressure theories to study the earth pressure behind retaining walls is a critical component of the seismic design of such structures, as the intensity and distribution of the earth pressure are key factors in retaining wall design [2]. The analysis of forces that act on retaining walls under seismic action is a classical problem in geotechnical engineering. Following the 1923 Great Kanto Earthquake in Tokyo and Yokohama, Japan, [PERSON] [3] and [PERSON] et al. [4] conducted in-depth investigations into the failure modes of retaining walls during earthquakes. Building upon [PERSON]'s static earth pressure theory, they proposed the Mononobe-Okabe method (referred to as the M-O method), also known as the pseudo-static method, for calculating the seismic earth pressure behind retaining walls. As a pseudo-static approach, the M-O method simplifies seismic loading as a static inertial force while ignoring its cyclic characteristics. The M-O method only provides the resultant seismic earth pressure acting on the wall, assuming a linear pressure distribution with the resultant force acting at one-third of the wall height (\(H\)) from the base. Additionally, the M-O method assumes that the backfill soil behaves as a rigid body, implying an infinite shear wave velocity and shear modulus of the soil--an assumption that is inconsistent with reality. The simplified hypothesis of there being a uniform acceleration field within the backfill also neglects the seismic wave amplification effect that occurs in soil masses. Consequently, the M-O method cannot accurately reflect the true dynamic response of the wall-soil system under seismic loading. To address the limitations of the M-O method, [PERSON] and [PERSON] [5] proposed a pseudo-dynamic method, assuming that the seismic acceleration varies sinusoidally with the depth and time, and treating the backfill soil as a non-rigid body with a finite shear modulus and shear wave velocity. By considering the wave periodicity and phase variations, they derived a formula for calculating the seismic active earth pressure on retaining walls under seismic loads. [PERSON] et al. [6] validated the rationality of pseudo-dynamic analysis through comparisons between centrifuge model tests and theoretical results. The principal merit of the pseudo-dynamic approach resides in the explicit consideration of two seismological parameters--shear wave velocity and predominant frequency--which jointly govern the phase variation and cyclical behavior of seismic wavefields. ; thereby, this method reflects the dynamic characteristics of backfill soil and its interaction with seismic waves [7]. In recent years, the pseudo-dynamic method has seen extensive application and development. [PERSON] [8] incorporated vibration amplification effects to calculate the seismic active earth pressure behind non-vertical cantilever retaining walls with zero backfill slope angles. Further, [PERSON] [9] applied the pseudo-dynamic method to cases with non-horizontal backfill but neglected the amplification of seismic acceleration along the wall height.