Waves Over Lunch: The Theory of Waves Explained on a Napkin (part 1)

 

“So what are you working on these days?”
“Water waves, mate. The literal ups and downs of life.”
“That sounds deep.”
“About \( h_0 \) metres deep, actually. Here, pass me a napkin.”

Let’s start from first principles: Euler lives here

We live in the world of **dimensional physics**, where the apostrophe —yes, the humble ' — reminds us that everything has units: metres, seconds, pressure in Pascals, etc. This is where water waves are born.

If you want to describe how water moves, you start with the **Euler equations** (no viscosity, because we assume the water is clean, fast, and slick like a dolphin).

1. Momentum (Euler’s equation): \[ \frac{D \tilde{\mathbf{u}}}{D \tilde{t}} = -\frac{1}{\rho} \nabla \tilde{p} + \mathbf{g} \]
2. Incompressibility: \[ \nabla \cdot \tilde{\mathbf{u}} = 0 \]

Here, \( \tilde{\mathbf{u}} = (\tilde{u}, \tilde{w}) \) is the velocity field, \( \tilde{p} \) is the pressure, and \( \mathbf{g} = (0, -g) \) is gravity pulling downward.

Why ignore viscosity?

Because surface gravity waves are all about inertia and pressure. Viscous effects are tiny unless you're dealing with boundary layers or mud. So we keep things ideal: no stickiness, no turbulence. Just beautiful rolling water.

Why irrotational?

We assume the flow has no vorticity: \( \nabla \times \tilde{\mathbf{u}} = 0 \). That’s not always true in the real ocean, but for clean wave motion, it’s a good approximation — especially in the bulk of the water column. This allows us to write: \[ \tilde{\mathbf{u}} = \nabla \tilde{\phi} \] where \( \tilde{\phi} \) is the **velocity potential**. That one move — assuming irrotational flow — collapses the whole momentum equation into one neat line: \[ \nabla^2 \tilde{\phi} = 0 \] Boom. Now we’re solving **Laplace’s equation** instead of vector PDE chaos.

Boundary conditions: what happens at the edges

We still need to tell the fluid what to do at the surface and bottom.

1. At the surface \( \tilde{z} = \tilde{h}_0 + \tilde{\eta} \):
• Kinematic condition — the fluid can’t punch through the surface: \[ \frac{\partial \tilde{\eta}}{\partial \tilde{t}} = \tilde{w} - \tilde{u} \frac{\partial \tilde{\eta}}{\partial \tilde{x}} \]
• Dynamic condition — energy is conserved: \[ \frac{\partial \tilde{\phi}}{\partial \tilde{t}} + \frac{1}{2} |\nabla \tilde{\phi}|^2 + g \tilde{\eta} = 0 \]
2. At the seabed \( \tilde{z} = \tilde{b}(\tilde{x}) \):
• Bottom kinematic — water follows the slope: \[ \tilde{w} = \tilde{u} \frac{\partial \tilde{b}}{\partial \tilde{x}} \]

Now we step into another world: the dimensionless one

We want to stop thinking in metres and seconds, and start thinking in ratios: how tall is the wave compared to the depth? How shallow is the water compared to the wavelength?

We introduce scalings:

• \( x = \tilde{x} / \tilde{\lambda} \)
• \( z = \tilde{z} / \tilde{h}_0 \)
• \( t = \tilde{t} \sqrt{g/\tilde{h}_0} / \tilde{\lambda} \)
• \( \eta = \tilde{\eta} / \tilde{a} \)
• \( \phi = \tilde{\phi} / (\tilde{a} \tilde{\lambda} \sqrt{g/\tilde{h}_0}) \)

Out pop two dimensionless parameters:

• \( \varepsilon = \tilde{a} / \tilde{h}_0 \): amplitude parameter
• \( \delta = \tilde{h}_0 / \tilde{\lambda} \): shallowness parameter

The new wave world: unit depth and clean equations

Once rescaled, the undisturbed surface is at \( z = 1 \), and the total depth is no longer \( \tilde{h}_0 \), it’s just 1. This makes the math and the physics much more transparent.

1. Laplace equation: \[ \phi_{zz} + \delta^2 \phi_{xx} = 0 \]
2. Surface kinematic condition at \( z = 1 + \varepsilon \eta \): \[ \phi_z = \delta^2 \left( \frac{\partial \eta}{\partial t} + \varepsilon \phi_x \eta_x \right) \]
3. Surface Bernoulli condition: \[ \phi_t + \eta + \frac{\varepsilon}{2} \left( \phi_x^2 + \frac{1}{\delta^2} \phi_z^2 \right) = 0 \]
4. Bottom boundary at \( z = b(x) \): \[ \phi_z = \delta^2 \phi_x b_x \]

Two classic limits

• Linear limit: \( \varepsilon \to 0 \). Small waves, linear theory, works for any depth.
• Shallow water limit: \( \delta \to 0 \). Fully nonlinear, no dispersion. Perfect for tsunamis and river bores.

🧾 Napkin Summary

- Start with Euler and strip it down
- Assume inviscid and irrotational flow to get Laplace’s equation
- Adimensionalise using a wave-based scaling
- The domain now has unit depth
- Two key parameters: ε (amplitude), δ (shallowness)
- Two golden limits: ε → 0 (linear), δ → 0 (nonlinear shallow water)

— “Mate, that was actually fascinating.”
— “Told you. Want me to show you how KdV shows up next time?”
— “Only if you bring the napkins.”