The correct answer is $\boxed{\text{D}}$.
The ratio of the width and depth of a rectangular beam cut from a circular log for the strongest beam is $\frac{1}{\sqrt{2}} \approx 0.707$. This is because the beam will be strongest when the forces acting on it are evenly distributed. If the width of the beam is too small, the forces will be concentrated at the ends of the beam, and the beam will be more likely to break. If the depth of the beam is too small, the forces will be concentrated in the middle of the beam, and the beam will also be more likely to break. When the width and depth of the beam are in the ratio $\frac{1}{\sqrt{2}}$, the forces will be evenly distributed, and the beam will be the strongest.
Option A is incorrect because it is too small. If the width of the beam is too small, the forces will be concentrated at the ends of the beam, and the beam will be more likely to break.
Option B is incorrect because it is too large. If the depth of the beam is too small, the forces will be concentrated in the middle of the beam, and the beam will also be more likely to break.
Option C is incorrect because it is not a ratio of the width and depth of the beam.