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Chapter 3: Problem 12

Assuming the bulk modulus is constant for seawater, derive an expression for the density variation with depth, \(h\) below the surface. Show that the result may be written \\[ \rho \approx \rho_{0}+b h \\] where \(\rho_{0}\) is the density at the surface. Evaluate the constant \(b .\) Then, using the approximation, obtain an equation for the variation of pressure with depth below the surface. Determine the depth in feet at which the error in pressure predicted by the approximate solution is 0.01 percent.

Short Answer

Expert verified
The density variation with depth is given by \(\rho = \rho_0 + b h\), where \(b = - \rho_0 g/B\). The pressure variation with depth is given by \(P = \rho_0 g h + b g h^2\). The specific depth at which the error in pressure amounts to 0.01 percent needs to be calculated by comparing the approximate and exact pressure results for different depths.

Step by step solution

01

Derive the Density Variation Function

Using the given, we can set up the hydrostatic pressure equation, which is \(P = \rho g h\), where \(P\) is the pressure, \(\rho\) is the density, \(g\) is acceleration due to gravity, and \(h\) is depth below the surface. The bulk modulus \(B\) is defined as the change in pressure divided by the fractional change in volume, i.e., {\(B = -P/(\Delta V/V)\)}. Since density and volume are inversely proportional, we can rewrite this as {\(B = -P (\Delta \rho/ \rho)\)}. Using these definitions, we can solve for \(\Delta \rho\) to get {\(\Delta \rho = - \Delta P \rho / B = - \rho g h / B\)}.
02

Evaluate Constant \(b\)

Rearranging the previous equation gives us the variation of density with depth, which comes out to be {\(\Delta \rho = \rho - \rho_0 = - \rho_0 g h / B\)}, which can be approximated as {\(\rho = \rho_0 + b h\)}, as stated in the exercise. Comparing these two, we find that \(b = - \rho_0 g / B\). We can substitute standard values for \(\rho_0\) (seawater density at the surface, \(1025 \, kg/m^3\)), \(g\) (acceleration due to gravity, \(9.8 \, m/s^2\)), and \(B\) (bulk modulus of seawater, \(2.2 \times 10^9 \, Pa\)) to calculate \(b\).
03

Obtain Equation for Pressure Variation

Using the derived density function, we can now obtain the equation for how pressure changes with depth. Substituting for \(\rho\) in the hydrostatic pressure equation provides us \(P = (\rho_0 + b h) g h = \rho_0 g h + b g h^2\).
04

Calculate Error Depth

The approximate solution for pressure from Step 3 and the exact solution from our Step 1 equation are compared to find the depth at which the error is 0.01 percent. Solving the equation \(0.01/100 = (P_{approx} - P_{exact}) / P_{exact}\) will provide the depth \(h\) for which our approximation starts to deviate by more than 0.01 percent from the exact pressure value.

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