Abstract: Existing analytical models for the longitudinal transient seepage field around subsea lined tunnels under wave loading are limited to two-dimensional cross-sectional analyses with constant hydraulic head boundaries and therefore fail to capture complex longitudinal hydraulic gradients. To address this limitation, an explicit analytical solution for the longitudinal total hydraulic head and flow velocity in the soil-lining system is derived using the method of separation of variables and the orthogonality of trigonometric functions, assuming a third-kind (Robin) seepage boundary at the inner lining surface of a deep tunnel and periodic wave loading at the ground surface. The feasibility and applicability of the proposed solution are verified by comparison with numerical results. The results indicate the following. (1) The wave-induced hydraulic head and flow velocity exhibit antisymmetric distributions over half a wave period in the temporal domain and about the tunnel mid-section in the spatial domain. Both variables complete two fluctuation cycles within a single wave period, exhibiting a frequency-doubling characteristic, with fluctuations initiating near the wave crest. (2) The relative permeability, lining thickness-to-radius ratio, wave period, and inner boundary seepage parameter significantly influence the spatiotemporal characteristics of the hydraulic response. Specifically, the relative permeability and inner boundary seepage parameter primarily govern the initial phase lag of the total hydraulic head. The lining thickness-to-radius ratio has a substantially greater influence on the hydraulic response amplitude at the outer lining boundary than at the inner boundary, whereas the wave period primarily controls the spatial distribution of the hydraulic head at the outer boundary and the variation in flow velocity at the inner boundary.

Key words: tunnel, wave loading, longitudinal hydraulic response, third-kind (Robin) boundary condition, method of separation of variables, transient seepage field, analytical solution