Numerical Computation of 1D Wave Equation

The one-dimensional wave equation is a classic partial differential equation describing vibration and wave phenomena, with broad applications in physics and engineering. Numerical solution using finite difference method provides a common and effective approach.

The core concept of finite difference method involves discretizing continuous differential equations by replacing derivatives with finite difference approximations, transforming the equation into a system of algebraic equations. For the 1D wave equation, central difference schemes are typically employed, which ensure computational accuracy while maintaining numerical stability. The selection of time step and spatial step is critical and must satisfy the CFL condition to prevent numerical divergence.

The implementation process generally includes grid initialization, boundary condition setting, and iterative solution updates. At each time step, the computation relies on results from the two previous time steps, forming a recurrence relation. Through proper parameter configuration, physical phenomena like wave propagation and reflection can be effectively simulated. Key implementation components include: defining spatial and temporal grids using linspace/meshgrid functions, implementing central difference operators using vectorized operations, and applying boundary conditions through array indexing modifications.

This numerical solution method is not only suitable for theoretical research but can also be extended to practical engineering problems such as acoustics modeling and seismic wave simulation. The algorithm can be implemented using matrix operations for efficient computation, with potential optimizations like parallel processing for large-scale simulations.