多种方法求解一维波动方程的教学探讨与实践

Teaching Exploration and Practice of Solving One-Dimensional Wave Equations with Multiple Methods

摘要: 探讨了四种求解一维波动方程的方法，包括达朗贝尔法、傅里叶变换法、拉普拉斯变换法和格林函数法，分析了它们在无界空间上的适用性和物理意义。通过对解法的逐步剖析和案例教学应用的展示，本文旨在帮助学生更好地理解不同方法的原理及其应用场景，提升在不同物理背景下求解偏微分方程的能力。本研究提出的教学策略旨在引导学生掌握多种解法的基本原理与优劣，并在工程与物理实际问题中选择合适的方法。

Abstract: This paper systematically discusses four methods for solving the one-dimensional wave equation, including d’Alembert’s method, Fourier transform method, Laplace transform method, and Green’s function method. It analyzes their applicability and physical significance in unbounded space. Through step-by-step analysis of the solutions and the presentation of case study applications, this paper aims to help students better understand the principles and application scenarios of different methods, and enhance their ability to solve partial differential equations in different physical contexts. The teaching strategies proposed in this study aim to guide students in mastering the basic principles, advantages, and disadvantages of multiple solution methods, and in selecting appropriate methods for solving practical engineering and physical problems.