This formulation of acoustic waves is derived under linear approximation assumption of Newtonian fluids.
Such fluids follows the NavierβStokes equations for compressible Newtonian fluids
Ο ( β v β t + ( v β
β ) v ) = β β p + β β
( ΞΌ ( β v + ( β v ) T ) ) + β ( β 2 3 ΞΌ β β
v ) + f β Ο β t + β β
( Ο v ) = 0. \begin{align}
\rho \Bigr(\frac{\partial \boldsymbol{v}}{\partial t}
+ (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} \Bigr)
&= -\nabla p
+ \nabla \cdot (\mu (\nabla \boldsymbol{v}
+ (\nabla \boldsymbol{v})^T))
+ \nabla (-\tfrac{2}{3} \mu \nabla \cdot \boldsymbol{v})
+ \boldsymbol{f} \\[10pt]
\frac{\partial \rho}{\partial t}
+ \nabla \cdot (\rho \boldsymbol{v})
&= 0.
\end{align} Ο ( β t β v β + ( v β
β ) v ) β t β Ο β + β β
( Ο v ) β = β β p + β β
( ΞΌ ( β v + ( β v ) T )) + β ( β 3 2 β ΞΌ β β
v ) + f = 0. β β For small density and pressure variations the speed of sound in the fluid c c c
c = β p β Ο . \begin{align}
c = \sqrt{\frac{\partial p}{\partial \rho}}.
\end{align} c = β Ο β p β β . β β We can neglect the viscosity terms in (1) since we are interesteed only in the region within a few acoustic wavelengths.
Considering an inviscid fluid and no external forces, (1) rewrites as
Ο ( β v β t + ( v β
β ) v ) = β β p β Ο β t + β β
( Ο v ) = 0 , \begin{align}
\rho \Bigr(\frac{\partial \boldsymbol{v}}{\partial t}
+ (\boldsymbol{v} \cdot \nabla) \boldsymbol{v}\Bigr)
= -\nabla p \\[10pt]
\frac{\partial \rho}{\partial t}
+ \nabla \cdot (\rho \boldsymbol{v})
= 0,
\end{align} Ο ( β t β v β + ( v β
β ) v ) = β β p β t β Ο β + β β
( Ο v ) = 0 , β β which can, for tiny perturbations, be linearised around a mean value:
p = p βΎ + Ξ΄ p , Ο = Ο βΎ + Ξ΄ Ο , v = v βΎ + Ξ΄ v = Ξ΄ v , \begin{align}
p = \overline{p} + \delta p,\\
\rho = \overline{\rho} + \delta \rho,\\
\boldsymbol{v} = \overline{\boldsymbol{v}} + \delta \boldsymbol{v} = \delta \boldsymbol{v},
\end{align} p = p β + Ξ΄ p , Ο = Ο β + Ξ΄ Ο , v = v + Ξ΄ v = Ξ΄ v , β β where the overlined quantities are the mean values (constant in space and time) and the Ξ΄ \delta Ξ΄ v βΎ = 0 \overline{\boldsymbol{v}} = 0 v = 0
Ο βΎ β Ξ΄ v β t + Ξ΄ Ο β Ξ΄ v β t + Ο βΎ ( Ξ΄ v β
β ) Ξ΄ v = β β Ξ΄ p β Ξ΄ Ο β t + Ο βΎ β β
Ξ΄ v = 0. \begin{align}
\overline{\rho} \frac{\partial \delta \boldsymbol{v}}{\partial t}
+ \delta \rho \displaystyle\frac{\partial \delta \boldsymbol{v}}{\partial t}
+ \overline{\rho} (\delta \boldsymbol{v} \cdot \nabla) \delta \boldsymbol{v}
= - \nabla \delta p\\[10pt]
\frac{\partial \delta \rho}{\partial t}
+ \overline{\rho} \nabla \cdot \delta \boldsymbol{v}
= 0.
\end{align} Ο β β t β Ξ΄ v β + Ξ΄ Ο β t β Ξ΄ v β + Ο β ( Ξ΄ v β
β ) Ξ΄ v = β β Ξ΄ p β t β Ξ΄ Ο β + Ο β β β
Ξ΄ v = 0. β β By algebraic manipulation (using the product rule and the continuity equation multiplied by Ξ΄ v \delta \boldsymbol{v} Ξ΄ v
Ο βΎ β ( Ξ΄ v ββ£ β
β ) β Ξ΄ v β β Ξ΄ Ο β Ξ΄ v β t . \begin{align}
\overline{\rho}\,(\delta \boldsymbol{v}\!\cdot\nabla)\,\delta \boldsymbol{v}
\approx - \delta \rho \frac{\partial \delta \boldsymbol{v}}{\partial t}.
\end{align} Ο β ( Ξ΄ v β
β ) Ξ΄ v β β Ξ΄ Ο β t β Ξ΄ v β . β β As a result, the two middle terms in the first relation of (9) cancel and we obtain obtains
Ο βΎ β Ξ΄ v β t + β β Ξ΄ p = 0 β Ξ΄ Ο β t + Ο βΎ β β
Ξ΄ v = 0. \begin{align}
\overline{\rho} \frac{\partial \delta \boldsymbol{v}}{\partial t}
+ \nabla\,\delta p
&= 0 \\[10pt]
\frac{\partial \delta \rho}{\partial t}
+ \overline{\rho} \nabla \cdot \delta \boldsymbol{v}
&= 0.
\end{align} Ο β β t β Ξ΄ v β + β Ξ΄ p β t β Ξ΄ Ο β + Ο β β β
Ξ΄ v β = 0 = 0. β β Taking the divergence of the first relation and the time derivative of the second we get
Ο βΎ β β
β Ξ΄ v β t + Ξ Ξ΄ p = 0 β 2 Ξ΄ Ο β t 2 + Ο βΎ β β
β Ξ΄ v β t = 0 , \begin{align}
\overline{\rho} \nabla \cdot \frac{\partial \delta \boldsymbol{v}}{\partial t}
+ \Delta \delta p
&= 0 \\[10pt]
\frac{\partial^2 \delta \rho}{\partial t^2}
+ \overline{\rho} \nabla \cdot \frac{\partial \delta \boldsymbol{v}}{\partial t}
&= 0,
\end{align} Ο β β β
β t β Ξ΄ v β + Ξ Ξ΄ p β t 2 β 2 Ξ΄ Ο β + Ο β β β
β t β Ξ΄ v β β = 0 = 0 , β β which can be combined into a single equation
β 2 Ξ΄ Ο β t 2 β Ξ Ξ΄ p = 0. \begin{align}
\frac{\partial^2 \delta \rho}{\partial t^2}
- \Delta \delta p
= 0.
\end{align} β t 2 β 2 Ξ΄ Ο β β Ξ Ξ΄ p = 0. β β With the isentropic approximation (3) the acoustic wave equation can be written as
Ξ Ξ΄ p β 1 c 2 β 2 Ξ΄ p β t 2 = 0 , \begin{align}
\Delta \delta p - \frac{1}{c^2} \frac{\partial^2 \delta p}{\partial t^2} = 0,
\end{align} Ξ Ξ΄ p β c 2 1 β β t 2 β 2 Ξ΄ p β = 0 , β β with c c c Ξ΄ p \delta p Ξ΄ p
From now on we will write the Ξ΄ p \delta p Ξ΄ p p p p p β² p^{\prime} p β² Ξ© \Omega Ξ©
β« Ξ© Β ( Ξ p β 1 c 2 β 2 p β t 2 ) Β p β² Β d Ξ© = 0. \begin{align}
\int_{\Omega}\ (\Delta p - \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2})\ p^{\prime}\ d\Omega = 0.
\end{align} β« Ξ© β Β ( Ξ p β c 2 1 β β t 2 β 2 p β ) Β p β² Β d Ξ© = 0. β β We can use the Leibniz rule for a nabla operator to rewrite the Laplace term.
β« Ξ© Β β β
( p β² β p ) Β d Ξ© β β« Ξ© Β β p β
β p β² Β d Ξ© β β« Ξ© Β 1 c 2 β 2 p β t 2 Β p β² Β d Ξ© = 0 , \begin{align}
\int_{\Omega}\ \nabla \cdot (p^{\prime} \nabla p)\ d\Omega
- \int_{\Omega}\ \nabla p \cdot \nabla p^{\prime}\ d\Omega
- \int_{\Omega}\ \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2}
\ p^{\prime}\ d\Omega = 0,
\end{align} β« Ξ© β Β β β
( p β² β p ) Β d Ξ© β β« Ξ© β Β β p β
β p β² Β d Ξ© β β« Ξ© β Β c 2 1 β β t 2 β 2 p β Β p β² Β d Ξ© = 0 , β β and then apply the Divergence theorem on the divergence term to obtain the weak formulation
β« Ξ Β ( β p β
n ) Β p β² Β d Ξ β β« Ξ© Β β p β
β p β² Β d Ξ© β β« Ξ© Β 1 c 2 β 2 p β t 2 Β p β² Β d Ξ© = 0. \begin{align}
\int_{\Gamma}\ (\nabla p \cdot \boldsymbol{n})\ p^{\prime}\ d\Gamma
- \int_{\Omega}\ \nabla p \cdot \nabla p^{\prime}\ d\Omega
- \int_{\Omega}\ \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2}
\ p^{\prime}\ d\Omega = 0.
\end{align} β« Ξ β Β ( β p β
n ) Β p β² Β d Ξ β β« Ξ© β Β β p β
β p β² Β d Ξ© β β« Ξ© β Β c 2 1 β β t 2 β 2 p β Β p β² Β d Ξ© = 0. β β 