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At page 10 of "An Introduction to Partial Differential Equation" by Y. Pinchover and J. Rubinstein the authors claim that applying the operator $\vec∇·$ to the equation (2), and substituting the result into the time derivative of the equation (1)

$\rho^1_t + \rho^0\vec∇·\vec u^1 = 0 \tag{1}$

$\vec u_t^1 + \frac{1}{\rho^0}\vec∇ p^1 = 0 \tag{2} $

leads to

$\rho_{tt}^1 - f'(\rho^0)\Delta\rho^1=0 \tag{3}$

where $p=f(\rho)$ and $\vec u$, $\rho$ and $p$ are functions describing a fluid at every point in space-time:

  • $\vec u (x,y,z,t)$ is the velocity;
  • $\rho(x,y,z,t)$ is the density;
  • $p(x,y,z,t)$ is the pressure.

In this particular case:

$\vec u = \epsilon \vec u^1 \tag{4.1}$

$\rho = \rho^0 + \epsilon\rho^1 \tag{4.2}$

$p = p^0 + \epsilon p^1 = f(\rho^0) + \epsilon f'(\rho^0)\rho^1 \tag{4.3}$

where $\vec u^1$, $\rho^1$ and $p^1$ are the pertubations of the unperturbed state

$\vec u = 0 \tag{5.1}$

$\rho = \rho^0 \tag{5.2}$

$p = p^0 = f(\rho^0) \tag{5.3}$

Substituting the equations (4) into the Euler equations, and retaining only the terms that are linear in $\epsilon$, we get the equations (1) and (2).

Can someone show me how the equation (3) follows from the equations (1) and (2)?

Any help is really appreciated!

Aviz
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1 Answers1

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Equation $(3)$ follows from equations $(1)$ and $(2)$ if $\rho^0$ is a constant, that is, $\rho^0_t=0$ and $\vec{\nabla}\rho^0=0$. Under this assumption, the time derivative of $(1)$ and the divergence of $(2)$ yield $$ \rho^1_{tt}+ \rho^0 \vec∇·\vec u^1_t= 0 \tag{A} $$ and $$ \vec{\nabla}\cdot\vec u_t^1 + \frac{1}{\rho^0}\Delta p^1= 0. \tag{B} $$ It follows from $(\text{A})$ and $(\text{B})$ that $$ \rho^1_{tt}-\Delta p^1= 0. \tag{C} $$ According to $(4.3)$, $p^1=f'(\rho^0)\rho^1$, so finally $$ \rho^1_{tt}-f'(\rho^0)\Delta \rho^1= 0. \tag{D} $$

Gonçalo
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