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.