Displacement thickness and momentum thickness are two important parameters used to describe the boundary layer over a flat plate. The ratio of displacement thickness to momentum thickness depends on the velocity distribution over the plate.

To understand the concept, let's break down the key points:

1. Displacement Thickness:
- Displacement thickness (\(\delta^*\)) represents the additional distance by which the boundary layer would have to be displaced to accommodate the same mass flow rate if the flow were inviscid (i.e., no boundary layer).
- It is a measure of the reduction in effective cross-sectional area due to the presence of the boundary layer.
- Displacement thickness can be calculated using the integral momentum equation and is given by the equation: \(\delta^* = \int_0^\infty (1-\frac{u}{U})dy\), where \(u\) is the local velocity and \(U\) is the free-stream velocity.

2. Momentum Thickness:
- Momentum thickness (\(\theta\)) represents the increase in the boundary layer's overall momentum due to the presence of viscosity.
- It is a measure of the momentum deficit in the boundary layer compared to an inviscid flow.
- Momentum thickness can be calculated using the integral momentum equation and is given by the equation: \(\theta = \int_0^\infty \frac{u}{U}(1-\frac{u}{U})dy\).

3. Linear Velocity Distribution:
- In the case of a linear velocity distribution over a flat plate, the velocity profile is given by \(u = U\left(1-\frac{y}{\delta}\right)\), where \(y\) is the distance from the plate and \(\delta\) is the boundary layer thickness.
- The boundary layer thickness in this case is given by \(\delta = \frac{\delta^*}{2}\).
- Substituting these values into the equations for displacement thickness and momentum thickness, we get: \(\delta^* = \frac{4}{3}\delta\) and \(\theta = \frac{2}{3}\delta\).

4. Ratio of Displacement Thickness to Momentum Thickness:
- The ratio of displacement thickness to momentum thickness for a linear velocity distribution over a flat plate is given by \(\frac{\delta^*}{\theta} = \frac{\frac{4}{3}\delta}{\frac{2}{3}\delta} = \frac{4}{2} = 2\).

Therefore, the correct answer is option 'D' - 3.0.