Physical Basis of Acoustics

J.P. Lefebvre

Introduction

Acoustics is concerned with the generation and space-time evolution of small mechanical perturbations in a fluid (sound waves) or in a solid (elastic waves). Then equations of acoustics are simply obtained by linearization of the equations of the mechanics of continua.

The main phenomenon encountered in acoustics is wave propagation. This phenomenon is the only one that occurs in an infinite homogeneous medium. A second important phenomenon is scattering, due to the various obstacles and inhomogeneities encountered by the wave. A third, more tenuous, phenomenon is absorption and dispersion of waves, due to dissipation processes.

The first and second phenomena need only a simple methodology: derivation of a wave equation and of a boundary condition from the linearized equations of the mechanics of continua for a homogeneous, steady, perfect simple fluid or elastic solid. That simple methodology allows us to solve a lot of problems, as other chapters of the book show. The third phenomenon, more subtle, needs many more conceptual tools, since it must call upon thermodynamics.

In a first step, we recall basic equations of the mechanics of continua for the rather general case of a thermo-viscous fluid or a thermo-elastic solid. One obtains all material necessary for derivation of the acoustic equations in complex situations.

In a second step we restrict ourselves to perfect simple fluids or elastic solids and linearize equations around an initial homogeneous steady state, leading to basic equations of elementary acoustics and elastic waves. Development of these simple cases are then proposed.

1.1 Review of Mechanics of Continua

Since acoustics is defined as small dynamic perturbations of a fluid or a solid, it is useful to make a quick recapitulation of the mechanics of continua in order to establish equations that are to be linearized. Those equations are conservation equations, the state equation and behaviour equations.

1.1.1 Conservation equations

Conservation equations describe conservation of mass (continuity equation), momentum (motion equation) and energy (from the first law of thermodynamics).

Let us consider a portion of material volume Ω filled with a (piecewise) continuous medium (for greater generality we suppose the existence of a discontinuity surface ∑ – a shock wave or an interface – moving at velocity →); the equations of conservation of mass, momentum, and energy are as follows.

Mass conservation equation (or continuity equation). The hypothesis of continuous medium allows us to introduce the notion of a (piecewise continuous) density function ρ, so that the total mass M of the material volume Ω is M = ∫Ω ρ dΩ; the mass conservation (or continuity) equation is written

dt∫ΩρdΩ=0

where d/dt is the material time derivative of the volume integral.

Momentum conservation equation (or motion equation). Let → be the local velocity; the momentum of the material volume Ω is defined as ∫Ωυ→ dΩ; and, if σ is the stress tensor and → the supply of body forces per unit volume (or volumic force source), the momentum balance equation for a volume Ω of boundary S with outward normal → is written

dt∫Ωρυ→dΩ=∫Sσ.n→dS+∫ΩF→dΩ

Energy conservation equation (first law of thermodynamics). If e is the specific internal energy, the total energy of the material volume Ω is ∫Ω(e→+12υ→2) dΩ; and, if → is the heat flux vector and r the heat supply per unit volume and unit time (or volumic heat source), the energy balance equation for a volume Ω of boundary S with outer normal → is written

dt∫Ωρ(e+12υ→2)dΩ=∫S(σ·υ→−q→)·n→dS+∫Ω(F→·υ→+r)dΩ

Using the lemma on derivatives of integrals over a material volume Ω crossed by a discontinuity ∑ of velocity →:

ϕ;ddt∫ΩϕdΩ=∫Ω(∂ϕ∂t+∇·(ϕυ→))dΩ+∫Σ[ϕ(υ→−V→)·n→]ΣdΩ

where [Φ]∑ designates the jump Φ(2) – Φ(1) of the quantity Φ at the crossing of the discontinuity surface ∑; or

dt∫ΩϕdΩ=∫Ω(dϕdt+ϕ∇·υ→)dΩ+∫Σ[ϕ(υ→−V→)·n→]Σdσ

with

˙≡dϕdt=∂ϕ∂t+υ→·∇ϕ

the material time derivative of the function ϕ. Using the formula

U→:∫SU→·n→dS=∫Ω∇·U→dΩ+∫Σ[U→·n→]Σdσ

one obtains

∫Ω(dρdt+ρ∇·υ→)dΩ+∫Σ[ρ(υ→−V→)·n→]Σdσ=0∫Ω(ddt(ρυ→)+ρυ→∇·υ→)dΩ+∫Σ[(ρυ→⊗(υ→−V→))·n→]Σdσ=∫Ω∇·σdΩ+∫Σ[σ·n→]Σdσ+∫ΩF→dΩ∫Ω(ddtρ(e+12υ→2)+ρ(e+12υ→2)∇·υ→)dΩ+∫Σ[ρ(e+12υ→2)(υ→−V→)·n→]Σdσ=∫Ω∇·(σ·υ→−q→)dΩ+∫Σ[(σ·υ→−q→)·n→]Σdσ+∫Ω(F→·υ→+r)dΩ

or

∫Ω(dρdt+ρ∇·υ→)dΩ+∫Σ[ρ(υ→−V→)·n→]Σdσ=0∫Ω(ddt(ρυ→)+ρυ→∇·υ→−∇·σ−F→)dΩ+∫Σ[(ρυ→⊗(υ→−V→)−σ)·n→]Σdσ=0∫Ω(ddt(ρ(e+12υ→2))+ρ(e+12υ→2)∇·υ→−∇·(σ·υ→−q→)−(F→·υ→+r))dΩ+∫Σ[(ρ(e+12υ→2)(υ→−V→)−(σ·υ→−q→))·n→]Σdσ=0

The continuity hypothesis states that all equations are true for any material volume Ω. So one finds the local forms of the conservation equations:

dρdt+ρ∇·υ→=0[ρ(υ→−V→)·n→]Σ=0ddt(ρυ→)+ρυ→∇·υ→−∇·σ−F→=0[(ρυ→⊗(υ→−V→)−σ)·n→]Σ=0ddt(ρ(e+12υ→2))+ρ(e+12υ→2)∇·υ→−∇·(σ·υ→−q→)−(F→·υ→+r)=0[(ρ(e+12υ→2)(υ→−V→)−(σ·υ→−q→))·n→]Σ=0

So at the discontinuities, one obtains

[ρ(N→−V→)·n→]Σ=0[(ρυ→⊗(υ→−V→)−σ)·n→]Σ=0[(ρ(e+12N→2)(N→−V→)−(σ·N→−q→))·n→]Σ=0

Away from the discontinuity surfaces combining the first three local forms of the conservation equations above, one...