Chapter Two
The wave equation and solutions
Abstract
After a brief introduction, the wave equation that governs the propagation of sound in air is derived. The results are for rectangular, cylindrical, and spherical coordinates. This is followed by the solutions to the wave equation, again in rectangular, cylindrical, and spherical coordinates. Forward and backward traveling waves follow, particularly propagation in tubes. Extensively discussed are freely traveling plane, cylindrical, and spherical waves. The chapter ends with solutions to the wave equation in three dimensions—for rectangular, cylindrical, and spherical coordinates.
Keywords
Helmholtz wave equation; Negative direction; One-dimensional wave equation; Positive direction; Steady-state conditions; Three-dimensional wave equation
Part III: The wave equation
2.1. Introduction
We have already outlined the nature of sound propagation in a gas in a qualitative way. In this chapter we shall put the physical principles described earlier into the language of mathematics. The approach is in two steps. First, we shall establish equations expressing Newton's second law of motion, the gas law, and the laws of conservation of mass. Second, we shall combine these equations to produce a wave equation.
The mathematical derivations are given in two ways: with and without use of vector algebra. Those who are familiar with vector notations will appreciate the generality of the three-dimensional vector approach. The two derivations are carried on in parallel; on the left sides of the pages, the one-dimensional wave equation is derived with the use of simple differential notations; on the right sides, the three-dimensional wave equation is derived with the use of vector notations. The simplicity of the vector operations is revealed in the side-by-side presentation of the two derivations.
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