Interpretation of the linearized conservation of mass in acoustics I am studying how the acoustic wave equation is obtained from the conservation of mass and the conservation of momentum. The following picture gives two forms the conservation of mass equation, but they are difficult to interpret.


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*How are (2.1) and (2.2) derived and how to understand them intuitively?

*Why does the divergence of momentary particle displacement $\nabla \cdot \vec{\xi}(t)$ stand for the dilation of a particle's boundary?



 A: In Cartesian coordinates, the displacement $\vec{\xi}=\{\xi_x,\xi_y,\xi_z\}$, and $\nabla\cdot\vec{\xi}=\partial\xi_x/\partial x+\partial\xi_y/\partial y+\partial\xi_z/\partial z$. Here "particle" means an elemental volume of fluid (not its constituent molecules). To understand the relation between $\nabla\cdot\vec{\xi}$ and volume of the fluid element, consider a cubical volume element of sides $dx,dy,dz$. Take a pair of parallel faces along X-axis; the distance between them will change only if the two faces are displaced by unequal amounts. If the left face is displaced by $\xi_x$, the right face which is distance $dx$ away, is displaced by $\xi_x+(\partial\xi_x/\partial x)dx$, so the difference in displacement is $(\partial\xi_x/\partial x)dx$. Consequent change in volume of the cube is $(\partial\xi_x/\partial x)dx\times dydz$. Similar argument for the other two pair of faces shows that the total change in volume is $dV=(\partial\xi_x/\partial x+\partial\xi_y/\partial y+\partial\xi_z/\partial z)dxdydz$. So fractional change in volume is $dV/(dxdydz)=(\partial\xi_x/\partial x+\partial\xi_y/\partial y+\partial\xi_z/\partial z)=\nabla\cdot\vec{\xi}$. Clearly, $\nabla\cdot\vec{\xi}>0$ is associated with net expansion of fluid element, and $\nabla\cdot\vec{\xi}<0$ is associated with net contraction of fluid element.

Eqn. (2.1) and (2.2) are derived from mass conservation equation:
$$\frac{1}{\rho_0}\frac{\partial\rho}{\partial t}+\frac{1}{\rho_0}\vec{u}\cdot\nabla\rho+\nabla\cdot\vec{u}=0$$
in which $\rho$ is the deviation from average density $\rho_0$, $\vec{u}$ is velocity. If we assume that $\rho$ and $\vec{u}$ are small quantities, then the second term in the equation above is a product of two small quantities, and is therefore negligible compared to the other two terms. Thus we obtain the linearized equation:
$$\frac{1}{\rho_0}\frac{\partial\rho}{\partial t}+\nabla\cdot\vec{u}=0$$
To obtain eqn. (2.1) integrate above equation along time from $0$ to $t$, the zero of time chosen so that the deviation in density $\rho=0$ at $t=0$:
$$\frac{1}{\rho_0}\int_0^tdt'~\frac{\partial\rho}{\partial t'}+\nabla\cdot(\int_0^tdt'~\vec{u})=0\\
\Rightarrow\frac{\rho(t)}{\rho_0}+\nabla\cdot\vec{\xi}(t)=0$$
To obtain eqn. (2.2) we must assume that the compression and rarefaction of fluid (due to passage of acoustic wave) is an isentropic process. In isentropic process, pressure and density are related as: $$p\rho^\gamma=\textrm{constant}$$
in which $\gamma$ is a characteristic constant of the fluid. Then:
$$\ln p+\gamma\ln\rho=\textrm{constant}\\~\\
\frac{1}{p_0}\frac{\partial p}{\partial\rho}+\frac{\gamma}{\rho_0}=0\\
\frac{c^2}{p_0}+\frac{\gamma}{\rho_0}=0\\
\Rightarrow p_0\gamma=-\rho_0c^2\\~\\~\\
\frac{1}{p_0}\frac{\partial p}{\partial t}+\frac{\gamma}{\rho_0}\frac{\partial \rho}{\partial t}=0\\
\frac{1}{p_0\gamma}\frac{\partial p}{\partial t}-\nabla\cdot\vec{u}=0\quad\textrm{(using mass conservation)}\\
\Rightarrow\frac{1}{\rho_0c^2}\frac{\partial p}{\partial t}+\nabla\cdot\vec{u}=0$$
