This chapter of the book "Finite Element Analysis of Acoustic Scattering" by Frank Ihlenburg provides an overview of the governing equations and analytical solutions for time-harmonic wave propagation, particularly focusing on the Helmholtz equation. The book aims to address the numerical simulation of wave propagation and fluid-structure interaction, with a specific emphasis on the Helmholtz equation, which is a fundamental equation in mathematical physics. The chapter covers the following key topics: 1. **Governing Equations of Time-Harmonic Wave Propagation**: This section outlines the equations governing acoustic, elastic, and electromagnetic waves, including the Sommerfeld condition and Maxwell's equations. 2. **Analytical and Variational Solutions of Helmholtz Problems**: It discusses methods for solving exterior Helmholtz problems, including separation of variables, functional analysis, and variational formulations. 3. **Discretization Methods for Exterior Helmholtz Problems**: This part introduces techniques for discretizing exterior domains, such as the introduction of artificial boundaries, Dirichlet-to-Neumann operators, absorbing boundary conditions, and the finite element method (FEM). 4. **Finite Element Error Analysis and Control for Helmholtz Problems**: This section delves into the convergence of Galerkin FEM, stability estimates, quasi-optimal convergence, preasymptotic error estimates, and the influence of large wave numbers. It also covers the analysis of the hp version of FEM and generalized FEM. 5. **Computational Simulation of Elastic Scattering**: This chapter includes computational results for elastic scattering from spheres and cylinders, providing detailed implementation and comparison with experimental data. The book is intended for mathematicians, physicists, and computational engineers working on scattering problems, and it aims to be self-contained and accessible, with numerous numerical examples to aid understanding.This chapter of the book "Finite Element Analysis of Acoustic Scattering" by Frank Ihlenburg provides an overview of the governing equations and analytical solutions for time-harmonic wave propagation, particularly focusing on the Helmholtz equation. The book aims to address the numerical simulation of wave propagation and fluid-structure interaction, with a specific emphasis on the Helmholtz equation, which is a fundamental equation in mathematical physics. The chapter covers the following key topics: 1. **Governing Equations of Time-Harmonic Wave Propagation**: This section outlines the equations governing acoustic, elastic, and electromagnetic waves, including the Sommerfeld condition and Maxwell's equations. 2. **Analytical and Variational Solutions of Helmholtz Problems**: It discusses methods for solving exterior Helmholtz problems, including separation of variables, functional analysis, and variational formulations. 3. **Discretization Methods for Exterior Helmholtz Problems**: This part introduces techniques for discretizing exterior domains, such as the introduction of artificial boundaries, Dirichlet-to-Neumann operators, absorbing boundary conditions, and the finite element method (FEM). 4. **Finite Element Error Analysis and Control for Helmholtz Problems**: This section delves into the convergence of Galerkin FEM, stability estimates, quasi-optimal convergence, preasymptotic error estimates, and the influence of large wave numbers. It also covers the analysis of the hp version of FEM and generalized FEM. 5. **Computational Simulation of Elastic Scattering**: This chapter includes computational results for elastic scattering from spheres and cylinders, providing detailed implementation and comparison with experimental data. The book is intended for mathematicians, physicists, and computational engineers working on scattering problems, and it aims to be self-contained and accessible, with numerous numerical examples to aid understanding.