Linear acoustic model prediction

Assume Countable Choice. Let $I\subseteq \mathbb{R}$ be an open time interval measured in seconds with $0\in I$, let $L=1\mathrm{m}$, and let ${\mathrm{\Omega }}_L:={\mathbb{R}}^3\mathrm{/}(2\pi L\mathbb{Z}{)}^3$ be a spatial periodic cell with coordinates in metres and volume measure $dx$ induced from a fundamental cell. Let $p:(0,\mathrm{\infty })\to \mathbb{R}$ be a $C^1$ barotropic pressure law in pascals, let ${\rho }_{\ast }>0$ be a constant reference density in kilograms per cubic metre, and assume $p^{\mathrm{\prime }}({\rho }_{\ast })>0$ in square metres per second squared. The adopted no-force barotropic Euler model is

${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0,\rho ({\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u)+\mathrm{\nabla }p(\rho )=0,$

for periodic fields $\rho \in C^2(I\times {\mathrm{\Omega }}_L;(0,\mathrm{\infty }))$ and $u\in C^2(I\times {\mathrm{\Omega }}_L;{\mathbb{R}}^3)$, where density is in kilograms per cubic metre and velocity is in metres per second. Here $C^{\mathrm{\infty }}$ means that coordinate derivatives of every finite order are continuous, with spatial fields periodic on ${\mathrm{\Omega }}_L$. The uniform state $(\rho ,u)=({\rho }_{\ast },0)$ is an equilibrium. For perturbations $z\in C^{\mathrm{\infty }}(I\times {\mathrm{\Omega }}_L;\mathbb{R})$ and $v\in C^{\mathrm{\infty }}(I\times {\mathrm{\Omega }}_L;{\mathbb{R}}^3)$, define the tangent model as the pointwise first variation at the dimensionless parameter $ϵ=0$ of this equation operator on ${\rho }_{ϵ}={\rho }_{\ast }+ϵz$ and $u_{ϵ}=ϵv$; on each compact time interval these perturbed densities remain positive for sufficiently small $\mathrm{\mid }ϵ\mathrm{\mid }$. For arbitrary initial data $z_0\in C^{\mathrm{\infty }}({\mathrm{\Omega }}_L;\mathbb{R})$ and $v_0\in C^{\mathrm{\infty }}({\mathrm{\Omega }}_L;{\mathbb{R}}^3)$, where these fields have continuous spatial derivatives of every finite order and are periodic, that tangent model is

${\mathrm{\partial }}_{t}z+{\rho }_{\ast }\mathrm{div}v=0,{\mathrm{\partial }}_{t}v+\frac{c_{\ast }^2}{{\rho }_{\ast }}\mathrm{\nabla }z=0,(z,v){\mathrm{\mid }}_{t=0}=(z_0,v_0),c_{\ast }:=\sqrt{p^{\mathrm{\prime }}({\rho }_{\ast })}>0.$

Here $z$ is a density perturbation in kilograms per cubic metre, $v$ is a velocity perturbation in metres per second, and $c_{\ast }$ is the model sound speed in metres per second. For every such initial pair, this linear tangent system has a unique real $C^{\mathrm{\infty }}$ solution $(z,v):\mathbb{R}\times {\mathrm{\Omega }}_L\to \mathbb{R}\times {\mathbb{R}}^3$. Its acoustic energy

$E_{\mathrm{a}\mathrm{c}}(t):=\frac{1}{2}{\int }_{{\mathrm{\Omega }}_L}({\rho }_{\ast }\mathrm{\mid }v{\mathrm{\mid }}^2+\frac{c_{\ast }^2}{{\rho }_{\ast }}{z}^2)dx$

is constant in joules. For a scalar or vector field $f$, take vector coefficients componentwise; for a wavevector $\kappa \in L^{-1}{\mathbb{Z}}^3$, use the Fourier character $e_{\kappa }(x)=e^{i\kappa \cdot x}$ and normalized coefficient $\widehat{f}(\kappa ,t)=((2\pi L{)}^3{)}^{-1}{\int }_{{\mathrm{\Omega }}_L}f(t,x)e^{-i\kappa \cdot x}dx$, with the integral taken over a fundamental cell. For $\kappa \mathrm{\ne }0$, the longitudinal velocity coefficient is ${\widehat{v}}_{\parallel }=(\kappa \cdot \widehat{v})\mathrm{\mid }\kappa {\mathrm{\mid }}^{-2}\kappa$ and the transverse coefficient is ${\widehat{v}}_{\perp }=\widehat{v}-{\widehat{v}}_{\parallel }$. A nonzero longitudinal normal mode is a nonzero Fourier solution with ${\widehat{v}}_{\perp }=0$ and time dependence $e^{-i\omega t}$; its angular frequency $\omega$ in inverse seconds satisfies ${\omega }^2=c_{\ast }^2\mathrm{\mid }\kappa {\mathrm{\mid }}^2$. The transverse coefficient at each $\kappa \mathrm{\ne }0$, and the mean coefficients $\widehat{z}(0,t),\widehat{v}(0,t)$ at $\kappa =0$, are constant in time. These are exact deductions for the specified linear tangent model, not a finite-amplitude approximation theorem, an observation, or a claim that an actual fluid follows the model.

David Tong, Fluid Mechanics, Chapter 3, §4.4.4

François Golse, The Boltzmann Equation and Its Hydrodynamic Limits, Chapter 3, §§2.3 and 7.1.2