The density of water at the surface of the ocean is $\rho$. If the bulk modulus of water is B, what is the density of ocean water at a depth where the pressure is ${\mathrm{nP}}_0$ where ${\mathrm{P}}_0$ is the atmospheric pressure?

Pressure at the surface of the ocean $={\mathrm{P}}_0$

Pressure at a depth $={\mathrm{nP}}_0$

$\mathrm{\Delta P}={\mathrm{nP}}_0-{\mathrm{P}}_0=(\mathrm{n}-1){\mathrm{P}}_0$

$\mathrm{m}=\rho \mathrm{V},\mathrm{\Delta V}=\frac{\mathrm{V\Delta P}}{\mathrm{B}}$

Volume of the same M of water at the given

$\text{ depth is }{\mathrm{V}}^{\text{'}}=\mathrm{V}-\mathrm{\Delta V}=\mathrm{V}-\frac{\mathrm{V}\cdot \mathrm{\Delta P}}{\mathrm{B}}$

${\mathrm{V}}^{\text{'}}=\frac{\mathrm{V}}{\mathrm{B}}(\mathrm{B}-\mathrm{\Delta P})$ ; Density of water at the depth

$\text{ is }{\mathrm{\rho }}^{\text{'}}=\frac{\mathrm{m}}{{\mathrm{V}}^{\text{'}}}$

$=\frac{\mathrm{\rho V}}{{\mathrm{V}}^{\text{'}}}=\frac{\mathrm{\rho V}}{\frac{\mathrm{V}}{\mathrm{B}}(\mathrm{B}-\mathrm{\Delta P})}\Rightarrow {\mathrm{\rho }}^{\text{'}}=\frac{\mathrm{\rho B}}{\mathrm{B}-(\mathrm{n}-1){\mathrm{P}}_0}$