Vibroacoustic Simulation

2 Waves in Fluids

2.1 Introduction

The acoustic wave motion is described by the equations of aerodynamics that are linearized because of the small fluctuations that occur in acoustic waves compared to the static state variables. The fluid motion is described generally by three equations:

• Continuity equation – conservation of mass
• Newton’s law – conservation of momentum
• State law – pressure volume relationship.

For lumped systems the velocity is simply the time derivative of the point mass position in space. The same approach can be used in fluid dynamics, but here the continuous fluid is subdivided into several cells and their movement is described by trajectories. This is called the Lagrange description of fluid dynamics. Even if the equation of motions are simpler in that formulation it is quite complicated to follow all coordinates of fluid volumes in a complex flow. Thus, the Euler description of fluid dynamics is used. In this description the conservation equations are performed for a control volume that is fixed in space and the flow passes through this volume.

In this chapter, the three dimensional space is given by the Cartesian coordinates $\mathbf{x}={\left\{\begin{array}{c}x,y,z\end{array}\right\}}^T$ and the velocity of the fluid $\mathbf{v}={\left\{v_{\text{x}},v_{\text{y}},v_{\text{z}}\right\}}^T$.

2.2 Wave Equation for Fluids

2.2.1 Conservation of Mass

For simplicity we consider first the flow in the $x$-direction as in Figure 2.1. The mass flow balance contains the following quantities:

1. The elemental mass $m=\rho dV=\rho A$ with $A=dydz$.
2. Mass flow into the volume ${(\rho v_{\mathrm{x}}A}_{}$