- #1

bigplanet401

- 104

- 0

## Homework Statement

Show that the density of water at a depth z in the ocean is related to the surface density rho_s by

[tex]

\rho(z) \approx \rho_s [1 + (\rho_s g/B)z]

[/tex]

where B is the bulk modulus of water.

## Homework Equations

**B = -V (dP/dV)**

B = rho (dP/d rho)

3. The Attempt at a Solution

B = rho (dP/d rho)

3. The Attempt at a Solution

**I've been trying to get this problem for 4 hours...aaargh.**

**I started by relating the change in pressure to the change in depth: pressure increases with depth.**

[tex]

\frac{dP}{dz} = \rho(z) g

[/tex]

[tex]

\frac{dP}{dz} = \rho(z) g

[/tex]

**so**

**[tex]**

dP = \rho(z) g \, dz

[/tex]

dP = \rho(z) g \, dz

[/tex]

**Then, substituting this expression for dP into the second formula above, I got**

**[tex]**

B = \rho^2 g \frac{dz}{d\rho}

[/tex]

B = \rho^2 g \frac{dz}{d\rho}

[/tex]

**Then I got**

[tex]

\frac{d\rho}{dz} = \frac{\rho^2 g}{B}

[/tex]

[tex]

\frac{d\rho}{dz} = \frac{\rho^2 g}{B}

[/tex]

**This is a separable differential equation, but I don't think it's the right one. I tried solving it with the initial condition rho(0) = rho_s and got**

[tex]

\rho(z) = \frac{\rho_s}{1 - \frac{\rho_s g}{B}z}

[/tex]

[tex]

\rho(z) = \frac{\rho_s}{1 - \frac{\rho_s g}{B}z}

[/tex]

**which doesn't make sense because density should not become infinite at a certain depth. What did I do wrong?**