arXiv:2610.37247v1 Announce Type: cross Abstract: We construct explicit quantum circuits for the velocity-stress formulation of elastic wave equations in homogeneous media using Schr\"odingerisation. After rewriting the equations in symmetric matrix form(SMF), we use Fourier spectral discretization for source-driven periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundary conditions. The Schr\"odingerisation method transforms any linear PDEs into a system of Schr\"odinger-type PDEs -with unitary evolution- using the warped phase transformation that maps the equations in one higher dimension. For periodic problems, we develop circuits by combining controlled Fourier-mode rotations. For physical boundaries, interleaved encoding swaps the spatial and field-component registers, enabling a more efficient circuit construction using shift operators. We analyze $p$th-order Trotter-Suzuki formulas and derive splitting-error and CNOT-gate estimates, under the assumptions of smooth auxiliary initialization and solution-state recovery. Comparison with classical partitioned splitting methods identifies a polynomial improvement in spatial-resolution dependence for the quantum state-evolution subroutine. Noiseless circuit simulations of a forced periodic problem and an impedance problem illustrate the constructions and quantify their errors against exact solutions and classical reference computations.