Neglecting the gravitational term in an equation of acoustic wave

Question

It appears that one can derive a wave equation if the gravitational force is considered in its derivation by using $$\frac{\partial p_0}{\partial x} = -\rho_0 g.$$ Here, $\rho_0 = \rho(p_0)$ and $p_0 = p(\rho_0)$ are the density and pressure respectively in the equilibrium state. The wave equation has the following form: $$c^{-2}\frac{\partial \tilde{p}}{\partial t^2} = \nabla^2\tilde{p} + g\frac{\partial}{\partial x}\left(\frac{\tilde{p}}{c^2(x)}\right),$$ where $$c(x) = \left(\frac{\partial\rho}{\partial p}(p_0(x))\right)^{-1/2}.$$ How come that the gravitational term can be neglected if $\lambda \ll c^2/g$?

I know that $$\tilde{p} = - \rho_0 \frac{\partial \phi}{\partial t}$$ where $\phi$ is the acoustic velocity potential. So, $$\nabla^2\tilde{p} = -\rho_0\left(\frac{\partial\phi}{\partial x\partial t} + \frac{\partial\phi}{\partial y\partial t} + \frac{\partial\phi}{\partial z\partial t}\right)$$ and $$g\frac{\partial}{\partial x}\left(\frac{\tilde{p}}{c^2(x)}\right) = -g\frac{\partial}{\partial x}\left(\frac{\rho_0}{c^2(x)}\frac{\partial\phi}{\partial t}\right).$$ Now, I have no idea how to show that the gravitational term is indeed negligible if $\lambda \ll c^2/g$. Anyone has an idea?

Answer 1

Straight from Euler equation you may write: $$ \rho_{0} \frac{\partial v}{\partial t}=-\mathbf{\nabla} p=-\rho_{0} g - \nabla \tilde{p}$$ So you end up comparing $\rho_{0} g$ with $\nabla \tilde{p}$. With the state equation, you may write: $$ \tilde{p}= \rho_{0}c^{2}$$ So $\nabla \tilde{p}$ is of the order of $\tilde{p} / \lambda =\rho_{0} c^{2}/ \lambda$, and the condition $\rho_{0} g \ll \nabla \tilde{p} $ yields: $$ \lambda \ll \frac{c^2}{g}$$

Answer 2

We make a scaling $x\rightarrow \lambda x, c\rightarrow c_0c, t\rightarrow c_0/\lambda t,p\rightarrow \tilde{p}/\rho_0c_0^2$, then the equation for $p$ takes the form $$c^{-2}p_{tt}-\nabla^2p=\epsilon \partial_x(p/c^2)$$ where $\epsilon =g\lambda/c_0^2$. Now we need just put $\epsilon \rightarrow 0$.