Question

The term  \(\frac{∂^2 \overline{u}}{∂x^2}+\frac{∂^2 \overline{u}}{∂y^2}\) in terms of r and θ is given by _______

(a) \(\frac{∂^2 \overline{u}}{∂x^2}+\frac{∂^2 \overline{u}}{∂y^2}=\frac{∂^2 \overline{u}}{∂r^2}+\frac{1}{r}  \frac{∂\overline{u}}{∂r}-\frac{1}{r^2}\frac{∂^2 \overline{u}}{∂θ^2}\)

(b) \(\frac{∂^2 \overline{u}}{∂x^2}+\frac{∂^2 \overline{u}}{∂y^2}=\frac{∂^2 \overline{u}}{∂r^2}+\frac{1}{r}  \frac{∂\overline{u}}{∂r}+\frac{1}{r^2}\frac{∂^2 \overline{u}}{∂θ^2}\)

(c) \(\frac{∂^2 \overline{u}}{∂x^2}+\frac{∂^2 \overline{u}}{∂y^2}=\frac{∂^2 \overline{u}}{∂r^2}-\frac{1}{r}  \frac{∂\overline{u}}{∂r}-\frac{1}{r^2}\frac{∂^2 \overline{u}}{∂θ^2}\)

(d) \(\frac{∂^2 \overline{u}}{∂x^2}+\frac{∂^2 \overline{u}}{∂y^2}=\frac{∂^2 \overline{u}}{∂r^2}-\frac{1}{r}  \frac{∂\overline{u}}{∂r}+\frac{1}{c^2}\frac{∂^2 \overline{u}}{∂θ^2}\)

I have been asked this question by my college professor while I was bunking the class.

The above asked question is from Consolidation Equation in Polar Coordinates topic in portion One Dimensional Consolidation of Geotechnical Engineering I

Answer 1

The correct choice is (b) \(\frac{∂^2 \overline{u}}{∂x^2}+\frac{∂^2 \overline{u}}{∂y^2}=\frac{∂^2 \overline{u}}{∂r^2}+\frac{1}{r}  \frac{∂\overline{u}}{∂r}+\frac{1}{r^2}\frac{∂^2 \overline{u}}{∂θ^2}\)

To explain I would say: The second order of the partial differentiation of excess hydrostatic pressure u with respect to x is,

\(\frac{∂^2 \overline{u}}{∂x^2}=(\frac{∂}{∂r} cosθ-\frac{1}{r} sinθ \frac{∂}{∂θ})(\frac{∂\overline{u}}{∂r} cosθ-\frac{1}{r}\frac{∂\overline{u}}{∂θ} θsinθ)  ————-(1)\)

also, the second order of the partial differentiation of excess hydrostatic pressure \overline{u} with respect to y is,

\(\frac{∂^2 \overline{u}}{∂y^2}=(\frac{∂}{∂r} cosθ-\frac{1}{r} sinθ \frac{∂}{∂θ})(\frac{∂\overline{u}}{∂r} cosθ-\frac{1}{r}\frac{∂\overline{u}}{∂θ} θsinθ)\)  ———-(2)

∴ adding (1) and (2),

\(\frac{∂^2 \overline{u}}{∂x^2}+\frac{∂^2 \overline{u}}{∂y^2}=\frac{∂^2 \overline{u}}{∂r^2}+\frac{1}{r}  \frac{∂\overline{u}}{∂r}+\frac{1}{r^2}\frac{∂^2 \overline{u}}{∂θ^2}.\)