Aeroacoustics/Linear Acoustics

Introduction
In chapter Wave Equation and Green's function, we mentioned that the motion of small amplitude acoustic waves in fluids is governed by the wave equation. In this chapter we intend to formally prove this statement.

Linear Acoustic Waves in an Ideal Fluid at Rest
Let's start from the governing equations of fluids motion, namely conservation of mass (continuity) and momentum (Navier-Stokes).

The continuity equation for a compressible fluid is

$$\frac{\partial \rho}{\partial t}+\frac{\partial (\rho v_i)}{\partial x_i} =0 $$

Navier-Stokes equation is also given by

$$\rho\left[\frac{\partial v_i} {\partial t} + v_j \frac{\partial v_i}{\partial x_j}\right]= \frac{\partial }{\partial x_j} ( -p\delta_{ij} + \tau_{ij}) $$

where $$\delta_{ij}$$ is the Kronecker delta and $$\tau_{ij}$$ is the deviatoric stress tensor. For the time being let's assume that we are dealing with wave propagation of an ideal (inviscid) fluid at rest. Later, we will extend our analysis to consider the moving, viscous fluid. Further assume that the motion of acoustic waves cause small amplitude fluctuations such that the instantaneous density, pressure, and velocity components at any point can be written as

$$p=p_0+p'$$

$$\rho = \rho_0 +\rho'$$

$$v_{i}=0+v'_{i}$$

where $$p_0$$ and $$\rho_0$$ are the mean pressure and density, respectively, which are independent of time and position. It should be noted that mean velocity has been set to zero, because we have assumed that the fluid is at rest. Substituting these quantities in the conservation of mass and momentum, and neglecting second and higher order fluctuation terms, we obtain

$$\frac{\partial \rho'}{\partial t}+\rho_0\frac{\partial v'_i}{\partial x_i}=0$$

and

$$\rho_0\frac{\partial v_i'}{\partial t}-\frac{\partial p'}{\partial x_i}=0$$

Now let's take the space derivative of the linearized momentum equation and subtract it from the time derivative of the linearized continuity equation to obtain

$$\frac{\partial^2 \rho'}{\partial t^2}-\frac{\partial^2 p'}{\partial x_i\partial x_i }= 0$$.

We can obtain the pressure fluctuation from a Taylor expansion

$$p'=\left(\frac{\partial p}{\partial \rho}\right)_{s}\rho'+\left(\frac{\partial p}{\partial s}\right)_{\rho}s'$$

Since we assumed that the fluid is inviscid and fluctuations are small in amplitude, it is safe to assume that the motion of acoustic waves does not generate entropy and is an isentropic process. Thus,

$$p'=\left(\frac{\partial p}{\partial \rho}\right)_{s}\rho'=c_0^2\rho'$$

Hence,

$$\frac{\partial^2 \rho'}{\partial t^2}-c_{0}^2\frac{\partial^2 \rho'}{\partial x_i\partial x_i }=0$$.

which is our celebrated wave equation. Exactly similar equations can be obtained for pressure and velocity fluctuations as well. It is interesting to note that the acoustic perturbations propagate at speed

$$c_{0}=\sqrt{\left(\frac{\partial p}{\partial \rho}\right)_{s}}$$

which is the famous speed of sound.

Velocity Potential and Wave Equation
In the previous section we obtained a wave equation in terms of density fluctuation. An alternate formulation can be obtained in terms of velocity potential.

Taking the curl of linearized momentum (Euler) equation, and noting that the curl of a gradient is zero, we obtain

$$\frac{\partial}{\partial t}\left(\mathbf{\nabla\times v'}\right) = 0$$

which means vorticity is constant in time. If we consider the initial vorticity to be zero, the velocity vector can be written as the gradient of a potential function at any moment of time

$$\mathbf{v'}(x,t) = \mathbf{\nabla}\Phi(x,t)$$

The linearized continuity and momentum equation can be used then to obtain

$$\frac{\partial^2 \Phi(x,t)}{\partial t^2}-c_{0}^2\nabla^2 \Phi(x,t) =0$$

$$p'=-\rho_0\frac{\partial \Phi}{\partial t}$$

$$\rho'=p'/c_{0}^2$$