For deep-water gravity waves, the wavelength L can be estimated from the wave period T (in seconds). Which of the following is Bertin’s formula as commonly quoted in coastal engineering texts?

Introduction / Context:
Estimating wavelength from period is essential for preliminary breakwater sizing, harbour resonance checks, and wave transformation calculations. Bertin’s expression (a concise statement of the linear deep-water dispersion relation) links L and T without requiring iteration.

Given Data / Assumptions:

• Deep-water approximation (depth >> L/2).
• g = acceleration due to gravity ≈ 9.81 m/s^2.
• T is the wave period (s), L the wavelength (m).

Concept / Approach:
The linear wave relation gives c = L/T and c^2 = g * L / (2 * pi) in deep water, leading to L = (g * T^2) / (2 * pi). This compact form is widely attributed in exam literature as “Bertin’s formula.” It provides a quick mapping from T to L for early-stage design checks.

Step-by-Step Solution:
Start with deep-water dispersion: c^2 = g * L / (2 * pi).Use c = L / T → (L / T)^2 = g * L / (2 * pi).Rearrange: L = (g * T^2) / (2 * pi).Pick the matching option.

Verification / Alternative check:
Sanity check with T = 8 s: L ≈ (9.81 * 64) / (6.283) ≈ 100 m, consistent with deep-water waves of that period.

Why Other Options Are Wrong:

• Forms linear in T (options a, c) are dimensionally inconsistent for wavelength.
• Multiplying numerator by 2 or omitting 2 * pi yields significant overestimation.
• L = g * T^2 omits the 2 * pi factor and overpredicts.

Common Pitfalls:
Forgetting the deep-water condition; misplacing 2 * pi in numerator/denominator; mixing phase speed and group speed.

Final Answer:
L = (g * T^2) / (2 * pi)