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Acoustic Theory Summary

Acoustic theory

Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach. Propagation of sound waves in a fluid (such as water) can be modeled by an equation of continuity (conservation of mass) and an equation of motion (conservation of momentum) . With some simplifications, in particular constant density, they can be given as follows: ∂ p ∂ t + κ ∇ ⋅ u = 0 (Mass balance) ρ 0 ∂ u ∂ t + ∇ p = 0 (Momentum balance) {\displaystyle {\begin{aligned}{\frac {\partial p}{\partial t}}+\kappa ~\nabla \cdot \mathbf {u} &=0\qquad {\text{(Mass balance)}}\\\rho _{0}{\frac {\partial \mathbf {u} }{\partial t}}+\nabla p&=0\qquad {\text{(Momentum balance)}}\end{aligned}}} where p ( x , t ) {\displaystyle p(\mathbf {x} ,t)} is the acoustic pressure and u ( x , t ) {\displaystyle \mathbf {u} (\mathbf {x} ,t)} is the flow velocity vector, x {\displaystyle \mathbf {x} } is the vector of spatial coordinates x , y , z {\displaystyle x,y,z} , t {\displaystyle t} is the time, ρ 0 {\displaystyle \rho _{0}} is the static mass density of the medium and κ {\displaystyle \kappa } is the bulk modulus of the medium. The bulk modulus can be expressed in terms of the density and the speed of sound in the medium ( c 0 {\displaystyle c_{0}} ) as κ = ρ 0 c 0 2 . {\displaystyle \kappa =\rho _{0}c_{0}^{2}~.} If the flow velocity field is irrotational, ∇ × u = 0 {\displaystyle \nabla \times \mathbf {u} =\mathbf {0} } , then the acoustic wave equation is a combination of these two sets of balance equations and can be expressed as ∂ 2 u ∂ t 2 − c 0 2 ∇ 2 u = 0 or ∂ 2 p ∂ t 2 − c 0 2 ∇ 2 p = 0 , {\displaystyle {\cfrac {\partial ^{2}\mathbf {u} }{\partial t^{2}}}-c_{0}^{2}~\nabla ^{2}\mathbf {u} =0\qquad {\text{or}}\qquad {\cfrac {\partial ^{2}p}{\partial t^{2}}}-c_{0}^{2}~\nabla ^{2}p=0,} where we have used the vector Laplacian, ∇ 2 u = ∇ ( ∇ ⋅ u ) − ∇ × ( ∇ × u ) {\displaystyle \nabla ^{2}\mathbf {u} =\nabla (\nabla \cdot \mathbf {u} )-\nabla \times (\nabla \times \mathbf {u} )} . The acoustic wave equation (and the mass and momentum balance equations) are often expressed in terms of a scalar potential φ {\displaystyle \varphi } where u = ∇ φ {\displaystyle \mathbf {u} =\nabla \varphi } . In that case the acoustic wave equation is written as ∂ 2 φ ∂ t 2 − c 0 2 ∇ 2 φ = 0 {\displaystyle {\cfrac {\partial ^{2}\varphi }{\partial t^{2}}}-c_{0}^{2}~\nabla ^{2}\varphi =0} and the momentum balance and mass balance are expressed as p + ρ 0 ∂ φ ∂ t = 0 ; ρ + ρ 0 c 0 2 ∂ φ ∂ t = 0 . {\displaystyle p+\rho _{0}~{\cfrac {\partial \varphi }{\partial t}}=0~;~~\rho +{\cfrac {\rho _{0}}{c_{0}^{2}}}~{\cfrac {\partial \varphi }{\partial t}}=0~.}

Acoustic attenuationAcoustic impedanceAcoustic resistanceAcoustic wave equationAcousticsAdiabaticAeroacousticsBasis vectorsBessel equationBessel function
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