Acoustic Array Systems

2

Theoretical Preliminaries of Acoustics

2.1 Fundamentals of Acoustics

Sound is a mechanical wave that propagates through a compressible medium. Sound is transmitted in the form of longitudinal waves described by the wave equation. Acoustics is rooted in fluid dynamics and can also be regarded as a simplified version of fluid dynamics that deals primarily with small perturbation quantities in a non-viscous medium. The acoustic wave equation can be derived with the considerations of mass and momentum conservation and the adiabatic thermodynamic process. The Eulerian and Lagrangian coordinates are two commonly used coordinate systems in fluid dynamics. Eulerian coordinates assume a fixed control volume in the space, while Lagrangian coordinates assume a control volume flowing with the fluid, as illustrated in Figure 2.1. Lagrangian coordinates are particularly useful in analysis involving conservation laws. The so-called material derivative in Lagrangian coordinates reads

(2.1)

where M denotes the total mass of the control volume, t is the time variable, u is the velocity of fluid, symbolizes the inner (vector) product, D symbolizes the total differential operator and symbolizes the ...