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Acoustic theory
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<p>Acoustic theory is a scientific field that relates to the description of <a href="/facts/Sound/DlKXlPuF">sound waves</a>. It derives from <a href="/facts/Fluid_dynamics/y7uzSCfg">fluid dynamics</a>. See <a href="/facts/Acoustics/2GpQUymy">acoustics</a> for the <a href="/facts/Engineering/RFVg8H3I">engineering</a> approach. </p><p>For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have </p> ∂ ρ ′ ∂ t + ρ 0 ∇ ⋅ v + ∇ ⋅ ( ρ ′ v ) = 0 (Conservation of Mass) ( ρ 0 + ρ ′ ) ∂ v ∂ t + ( ρ 0 + ρ ′ ) ( v ⋅ ∇ ) v + ∇ p ′ = 0 (Equation of Motion) {\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} +\nabla \cdot (\rho '\mathbf {v} )&=0\qquad {\text{(Conservation of Mass)}}\\(\rho _{0}+\rho '){\frac {\partial \mathbf {v} }{\partial t}}+(\rho _{0}+\rho ')(\mathbf {v} \cdot \nabla )\mathbf {v} +\nabla p'&=0\qquad {\text{(Equation of Motion)}}\end{aligned}}} <p>In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as </p> ∂ ρ ′ ∂ t + ρ 0 ∇ ⋅ v = 0 ∂ v ∂ t + 1 ρ 0 ∇ p ′ = 0 {\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} &=0\\{\frac {\partial \mathbf {v} }{\partial t}}+{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}} <p>Where v ( x , t ) {\displaystyle \mathbf {v} (\mathbf {x} ,t)} is the perturbed velocity of the fluid, p 0 {\displaystyle p_{0}} is the pressure of the fluid at rest, p ′ ( x , t ) {\displaystyle p'(\mathbf {x} ,t)} is the perturbed pressure of the system as a function of space and time, ρ 0 {\displaystyle \rho _{0}} is the density of the fluid at rest, and ρ ′ ( x , t ) {\displaystyle \rho '(\mathbf {x} ,t)} is the variance in the density of the fluid over space and time. </p><p>In the case that the velocity is <a href="/facts/Irrotational/MoS3V9aV">irrotational</a> ( ∇ × v = 0 {\displaystyle \nabla \times \mathbf {v} =0} ), we then have the acoustic wave equation that describes the system: </p> 1 c 2 ∂ 2 ϕ ∂ t 2 − ∇ 2 ϕ = 0 {\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}-\nabla ^{2}\phi =0} <p>Where we have </p> v = − ∇ ϕ c 2 = ( ∂ p ∂ ρ ) s p ′ = ρ 0 ∂ ϕ ∂ t ρ ′ = ρ 0 c 2 ∂ ϕ ∂ t {\displaystyle {\begin{aligned}\mathbf {v} &=-\nabla \phi \\c^{2}&=({\frac {\partial p}{\partial \rho }})_{s}\\p'&=\rho _{0}{\frac {\partial \phi }{\partial t}}\\\rho '&={\frac {\rho _{0}}{c^{2}}}{\frac {\partial \phi }{\partial t}}\end{aligned}}}