Introduction:
The question pertains to the hydraulic efficiency of triangular and rectangular channel sections when water is flowing at the same depth. The hydraulic radius, which is a measure of the efficiency of a channel section, is compared between the two types of sections.

Explanation:

1. Hydraulic Radius:
The hydraulic radius (Rh) is a parameter used to evaluate the efficiency of a channel section in conveying water. It is defined as the ratio of the cross-sectional area (A) of flow to the wetted perimeter (P) of the channel. Mathematically, it can be expressed as:

Rh = A / P

2. Triangular Channel Section:
In a triangular channel section, the cross-sectional area (A) can be calculated using the formula:

A = (b * h) / 2

where b is the base width of the triangle and h is the height of the triangle.

The wetted perimeter (P) can be calculated as the sum of the base width and the two sloping sides of the triangle:

P = b + 2 * sqrt(b^2 + h^2)

3. Rectangular Channel Section:
In a rectangular channel section, the cross-sectional area (A) can be calculated as:

A = b * h

where b is the base width of the rectangle and h is the height of the rectangle.

The wetted perimeter (P) can be calculated as the sum of the base width and the two sides of the rectangle:

P = 2 * (b + h)

4. Comparison:
Now, let's compare the hydraulic radius (Rh) of the triangular and rectangular channel sections when the water is flowing at the same depth.

For the triangular section: Rh(triangular) = A / P = [(b * h) / 2] / [b + 2 * sqrt(b^2 + h^2)]

For the rectangular section: Rh(rectangular) = A / P = (b * h) / [2 * (b + h)]

5. Calculation:
Now, consider that the water is flowing at the same depth in both channel sections. This implies that the height of the triangular section (h) would be equal to the height of the rectangular section (h).

Let's substitute h = h in the above equations and simplify:

For the triangular section: Rh(triangular) = (b * h) / [b + 2 * sqrt(b^2 + h^2)]

For the rectangular section: Rh(rectangular) = (b * h) / [2 * (b + h)]

6. Simplification:
Since h = h, let's cancel out the common terms:

Rh(triangular) = b / [1 + 2 * sqrt(b^2 + h^2) / h]

Rh(rectangular) = b / [2 + 2 * b / h]

7. Final Comparison:
Now, let's compare the ratio of the hydraulic radius of the triangular section to that of the rectangular section:

Rh(triangular) / Rh(rectangular) = [b / (1 + 2 * sqrt(b^2 + h^2) / h)] / [b / (2 + 2 * b / h)]

Rh(triangular) / Rh(rectangular

For most hydraulic efficient triangular channel