From Colombo Public Library to Wave Theory: Deriving the Coastal Wave Formula from First Principles

 

1. A Book, A Stamp, and a Question

A few weeks ago, I borrowed an old coastal engineering text book from the Colombo Public Library:

“Beaches and Coasts” – Cuchlaine A. M. King (1959)

Stamped:

• 📅 20 July 1961
• 📍 Donated by Lanka Salt Ltd

Within just two years, this book had travelled from London to Colombo—without digital systems, without internet, yet with remarkable efficiency.

But what caught my attention was not only the history—it was a formula inside the book, one we still use today in coastal engineering.

That formula describes how waves move.

And it emerges from one of the most beautiful derivations in fluid mechanics.

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2. The Formula Observed in the Book

The book presents:

General wave velocity:

$$C = \sqrt{\frac{gL}{2\pi} \tanh\left(\frac{2\pi h}{L}\right)}$$

Deep water simplification:

$$C = \sqrt{\frac{gL}{2\pi}}$$

And the famous engineering relation:

$$L = 5.12 T^2 \quad (\text{in feet})$$

At first glance, these look empirical.

They are not.

They come directly from differential equations governing fluid motion.

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3. Step 1 — Governing Equation (Laplace Equation)

We begin with ideal assumptions:

• Inviscid fluid (no viscosity)
• Irrotational flow
• Small-amplitude waves (linear theory)

Under these, velocity can be expressed using a potential function $\phi$:

$$\mathbf{u} = \nabla \phi$$

This leads to the governing equation:

$$\nabla^2 \phi = 0$$

This is Laplace’s equation.

👉 This is the foundation of linear wave theory.

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4. Step 2 — Boundary Conditions

To solve Laplace’s equation, we impose physical constraints:

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(a) Free Surface — Kinematic Condition

The surface moves with the fluid:

$$\frac{\partial \eta}{\partial t} = \frac{\partial \phi}{\partial z}$$

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(b) Free Surface — Dynamic Condition

From Bernoulli’s equation (linearized):

$$\frac{\partial \phi}{\partial t} + g\eta = 0$$

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(c) Seabed Condition

No vertical flow through seabed:

$$\frac{\partial \phi}{\partial z} = 0 \quad \text{at } z = -h$$

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5. Step 3 — Assume a Wave Solution

We assume a harmonic wave form:

$$\phi(x,z,t) = A \cosh k(z+h) \, e^{i(kx - \omega t)}$$

Where:

• $k = \frac{2\pi}{L}$ (wave number)
• $\omega = \frac{2\pi}{T}$ (angular frequency)

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6. Step 4 — Apply Boundary Conditions

Substituting into the free surface conditions and eliminating $\eta$, we obtain:

$$\omega^2 = gk \tanh(kh)$$

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🔑 This is the dispersion relation

$\omega^2 = gk \tanh(kh)$

This equation connects:

• Frequency
• Wave number
• Water depth

It is the core of all coastal wave modelling.

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7. Step 5 — Deriving Wave Velocity

Wave celerity:

$$C = \frac{\omega}{k}$$

Substitute dispersion relation:

$$C = \sqrt{\frac{g}{k} \tanh(kh)}$$

Now replace:

$$k = \frac{2\pi}{L}$$

We obtain:

$$C = \sqrt{\frac{gL}{2\pi} \tanh\left(\frac{2\pi h}{L}\right)}$$

✔ This is exactly the formula printed in the 1959 book.

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8. Step 6 — Deep Water Approximation

When:

$$\frac{h}{L} > 0.5$$

Then:

$$\tanh(kh) \approx 1$$

So:

$$C = \sqrt{\frac{gL}{2\pi}}$$

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9. Step 7 — Deriving the Practical Engineering Formula

Using:

$$C = \frac{L}{T}$$

Equate:

$$\frac{L}{T} = \sqrt{\frac{gL}{2\pi}}$$

Solve for $L$:

$$L = \frac{g}{2\pi} T^2$$

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Convert numerically:

• In meters:

$$L \approx 1.56 T^2$$

• In feet:

$$L \approx 5.12 T^2$$

✔ This is the exact number printed in your book.

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10. Physical Interpretation (Important for Learners)

This equation tells us:

• Waves are dispersive
• Longer waves travel faster
• Depth controls wave behaviour through tanh(kh)

Regimes:

Condition	Behaviour
Deep water	Wave speed depends on wavelength
Shallow water	Speed depends on depth only
Intermediate	Both effects combined

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11. Why This Still Matters Today

This same equation is embedded in:

• SWAN
• Delft3D
• MIKE 21
• Offshore design codes (ISO 19901)

👉 What has changed is not the physics
👉 Only the computational scale

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12. Final Reflection

A book printed in 1959, stamped in Colombo in 1961, already contained:

• The full mathematical structure
• The governing physics
• The engineering approximations

The equations we rely on today were already complete decades ago.

What we inherit is not just knowledge.

It is compressed intellectual history.

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Sometimes, an old library book is not outdated.

It is a wave still travelling through time.