The Hydraulic Jump:
The hydraulic jump is a phenomenon that occurs when a high-velocity flow abruptly changes to a lower velocity flow. It is commonly observed in open channel flow, such as rivers and channels. The jump is characterized by a sudden rise in water surface elevation and turbulent flow.

Given Information:
- Depth of water upstream (h1) = 12 m
- Depth of water downstream (h2) = 15 m

Loss of Head:
The loss of head at the hydraulic jump can be calculated using the energy equation. The energy equation states that the sum of the elevation head, pressure head, and velocity head remains constant along a streamline in a steady flow.

In this case, we can assume that the flow is uniform and steady before and after the jump. Therefore, the energy equation can be simplified as:

Elevation head (z1) + Pressure head (P1/ρg) + Velocity head (V1^2/2g) = Elevation head (z2) + Pressure head (P2/ρg) + Velocity head (V2^2/2g)

Since the flow is horizontal and the elevation is the same upstream and downstream, the elevation head can be neglected. The pressure head can also be neglected if the flow is in the atmosphere.

Therefore, the energy equation becomes:

Velocity head (V1^2/2g) = Velocity head (V2^2/2g)

Calculating the Loss of Head:
We can use the velocity head equation to calculate the loss of head (Δh) at the hydraulic jump.

V1^2/2g - V2^2/2g = Δh

The velocity head can be expressed in terms of flow depth using the equation:

V = Q/A
where V is the velocity, Q is the flow rate, and A is the cross-sectional area of flow.

Assuming that the flow area is rectangular, we can write:

V1 = Q/A1 = Q/(b1 * h1)
V2 = Q/A2 = Q/(b2 * h2)

Substituting these expressions into the loss of head equation, we get:

(Q^2/(2g * b1^2 * h1^2)) - (Q^2/(2g * b2^2 * h2^2)) = Δh

Cancelling out the flow rate and rearranging the equation, we have:

Δh = (Q^2/(2g)) * (1/(b1^2 * h1^2) - 1/(b2^2 * h2^2))

Final Calculation:
To calculate the loss of head, we need to know the flow rate and the dimensions of the flow. Unfortunately, the given information does not provide these values. Therefore, it is not possible to calculate the loss of head with the given information alone.

The correct answer provided as '0.038 m' is likely a specific value given in the problem statement or obtained from additional calculations.