I want to calculate the under water pressure, taking into account the compression of water. I derived a formula, but apparently the function has a vertical asymptote, meaning that the pressure approaches infinity at a certain depth under water. What is my mistake?

Let's say the atmospheric pressure is $p_o$, and $h$ is the depth of water, and $\beta$ is the bulk modulus. The increase in pressure in an infinitesimal increase in depth($dh$) can be measured by $\rho g.dh$. So:

$dp = \rho g.dh$
$\beta = -\dfrac{dp}{dV}V => -\dfrac{dV}{V}=\dfrac{dp}{\beta}$
$\therefore V(p)=V_0 \times e^{\dfrac{-p}{\beta}}$
$\rho = \dfrac{mass}{volume} => \rho = \rho_0 \times e^{\dfrac{p}{\beta}}$
$\rho_0$ is the density of water at $h=0$
$dp=\rho_0 \times e^{\dfrac{p}{\beta}}\times g .dh$
Now I integrated the both sides and solved for $p$ as a function of $h$. The result was:
$p = -\beta \times ln(e^{\dfrac{-p_0}{\beta}}-\dfrac{\rho_0 gh}{\beta})$

But this function has a vertical asymptote, which means pressure approaches infinity, obviously a wrong result.

Thanks!

I want to calculate the under water pressure, taking into account the compression of water. I derived a formula, but apparently the function has a vertical asymptote, meaning that the pressure approaches infinity at a certain depth under water. What is my mistake?

Let's say the atmospheric pressure is $p_o$, and $h$ is the depth of water, and $\beta$ is the bulk modulus. The increase in pressure in an infinitesimal increase in depth($dh$) can be measured by $\rho g.dh$. So:

$dp = \rho g.dh$
$\beta = -\dfrac{dp}{dV}V => -\dfrac{dV}{V}=\dfrac{dp}{\beta}$
$\therefore V(p)=V_0 \times e^{\dfrac{-p}{\beta}}$
$\rho = \dfrac{mass}{volume} => \rho = \rho_0 \times e^{\dfrac{p}{\beta}}$
$\rho_0$ is the density of water at $h=0$
$dp=\rho_0 \times e^{\dfrac{p}{\beta}}\times g .dh$
Now I integrated the both sides and solved for $p$ as a function of $h$. The result was:
$p = -\beta \times ln(e^{\dfrac{-p_0}{\beta}}-\dfrac{\rho_0 gh}{\beta})$

But this function has a vertical asymptote, which means pressure approaches infinity, likely a wrong result.

Thanks!

The only explanation I could come up with is that the bulk modulus is not a constant number.