Question

The buoyancy curve variation of a ship floating in still water and in waves is shown in the following figure. The total area under each curve is the same. The cases 'X' and 'Y' correspond to 

 

• A. X: wave crest is amidships, Y: wave crest is amidships
• B. X: wave trough is amidships, Y: wave trough is amidships
• C. X: wave trough is amidships, Y: wave crest is amidships
• D. X: wave crest is amidships, Y: wave trough is amidships

Hint:

A simple visual mnemonic: - {Crest} amidships lifts the middle \(⇒\) Buoyancy curve 'X' looks like a single large hill in the middle. This causes the ship to {Sag}. - {Trough} amidships drops the middle \(⇒\) Buoyancy curve 'Y' looks like two smaller hills at the ends. This causes the ship to {Hog} (like a hog's back, up at the ends).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The curve shown is a buoyancy distribution curve, which plots the buoyant force per unit length along the ship's length. The total area under this curve represents the total buoyant force, which must equal the ship's weight (displacement). The shape of the curve changes depending on how the ship is supported by the water.
- In still water, the buoyancy distribution is relatively smooth, often fuller in the middle (amidships) and finer at the ends.
- In waves, the distribution of buoyancy changes dramatically.
Step 2: Detailed Analysis of Wave Conditions:
- Wave Crest Amidships (Sagging Condition): When a wave crest is at the midship section, the middle of the ship is lifted by the wave, while the bow and stern are in the adjacent troughs and less supported. This results in an excess of buoyancy amidships and a deficiency of buoyancy at the ends. The buoyancy curve will therefore show a large peak in the middle and be very low at the ends. This corresponds to the curve 'X'. This loading condition creates a "sagging" bending moment on the ship's hull.
- Wave Trough Amidships (Hogging Condition): When a wave trough is at the midship section, the middle of the ship is less supported, while the bow and stern are lifted by adjacent wave crests. This results in a deficiency of buoyancy amidships and an excess of buoyancy at the ends. The buoyancy curve will be low in the middle and higher towards the ends, often showing two peaks near the bow and stern. This corresponds to the curve 'Y'. This loading condition creates a "hogging" bending moment on the ship's hull.
The problem states that the total area under each curve is the same, which is correct because the total buoyancy must always equal the total weight of the ship.
Step 3: Matching Curves to Conditions:
- Curve X shows a large concentration of buoyancy amidships and low buoyancy at the ends. This is consistent with a wave crest amidships.
- Curve Y shows low buoyancy amidships and higher buoyancy concentrated towards the ends. This is consistent with a wave trough amidships.
Therefore, the correct correspondence is X: wave crest is amidships, Y: wave trough is amidships.
Step 4: Why This is Correct:
This matching aligns with the fundamental principles of how wave patterns affect the distribution of buoyant force along a ship's hull. The shapes of curves X and Y are classic representations of the buoyancy distribution in sagging and hogging conditions, respectively. Option (D) correctly identifies this relationship.