The standard finite-difference time-domain (FDTD) wave solver employs a regular grid that approximates vocal tract boundaries in a stair-stepped manner, making it less effective for modeling complex and dynamic geometry. We present a novel 2D wave solver that integrates a unique immersed boundary method with FDTD, enabling precise approximation of wave propagation in acoustic tubes. The solver uses Lagrangian points to define vocal tract boundaries on a regular grid, eliminating their stair-stepped discretization. The boundary and the flow equations interact via additional forcing terms characterized by boundary immittances. The results show that the formant frequencies of a single-segment tube closely align with its analytical solutions, exhibiting a percentage deviation of less than 3%. Similarly, the frequency responses of a two-segment tube and actual vocal tract geometries show good agreement with existing state-of-the-art 2D and 3D wave solvers, except for a few discrepancies.