The discharge through a V- notch varies as (where H is head) A. $${{\text{H}}^{\frac{1}{2}}}$$ B. $${{\text{H}}^{ – \frac{1}{2}}}$$ C. $${{\text{H}}^{\frac{5}{2}}}$$ D. $${{\text{H}}^{\frac{3}{2}}}$$

The correct answer is $\boxed{{\text{H}}^{\frac{1}{2}}}$.

The discharge through a V-notch is given by the following equation:

$$Q = C{\sqrt{2gH}}$$

where $Q$ is the discharge, $C$ is a discharge coefficient, $g$ is the acceleration due to gravity, and $H$ is the head.

The discharge coefficient is a dimensionless number that depends on the geometry of the V-notch. For a sharp-crested V-notch, the discharge coefficient is typically in the range of 0.5 to 0.6.

The head is the vertical distance from the free surface of the water to the crest of the V-notch.

The discharge through a V-notch varies as the square root of the head. This means that if the head is doubled, the discharge will increase by a factor of $\sqrt{2}$.

The other options are incorrect because they do not represent the correct relationship between the discharge and the head.