Math 2.0

Fluid mechanics Definition

Water waves

Builds on Potential flowVelocity potentialLaplace's equationIncompressible flow

Definition

Let g>0g > 0 be the acceleration of gravity and h>0h > 0 the depth of the undisturbed water. Use coordinates (x,z)(x, z) with x∈ℝdx \in \R^d horizontal (d=1d = 1 or 22) and zz vertical, the bottom at z=−hz = -h. The water wave problem asks for two functions. The first is η(x,t)\eta(x, t), the elevation of the free surface. The second is ϕ(x,z,t)\phi(x, z, t), the velocity potential, defined on the fluid domain {−h<z<η(x,t)}\{-h < z < \eta(x, t)\}. They must satisfy:

  1. Δϕ=0\Delta \phi = 0 in the fluid domain, where Δ\Delta is the Laplacian in (x,z)(x, z).
  2. ∂zϕ=0\partial_z \phi = 0 on the bottom z=−hz = -h.
  3. ∂tη+∇xϕ⋅∇xη=∂zϕ\partial_t \eta + \nabla_x \phi \cdot \nabla_x \eta = \partial_z \phi on z=η(x,t)z = \eta(x, t) (kinematic condition).
  4. ∂tϕ+12∣∇ϕ∣2+gη=0\partial_t \phi + \tfrac{1}{2}\abs{\nabla \phi}^2 + g \eta = 0 on z=η(x,t)z = \eta(x, t) (dynamic condition).

A solution describes water waves: the motion of an incompressible, inviscid fluid in irrotational motion, with velocity ∇ϕ\nabla \phi, below a free surface at constant pressure.

Remarks

Example: η=0\eta = 0, ϕ=0\phi = 0 is the state of rest, and it satisfies all four clauses, since every term in them vanishes. Non-example: the pair η=acos⁡(kx)\eta = a\cos(kx), ϕ=0\phi = 0 with a≠0a \neq 0, k≠0k \neq 0 satisfies clauses 1 to 3 but fails clause 4, since gη≠0g\eta \neq 0; a raised surface cannot stay at rest. Clause 1 is Laplace's equation, which expresses incompressibility of a potential flow. Clause 4 is the Bernoulli equation at the surface, where the pressure equals the constant atmospheric pressure, absorbed into ϕ\phi. With surface tension σ\sigma and density ρ\rho, clause 4 gains the term −σρ∇x⋅(∇xη1+∣∇xη∣2)-\frac{\sigma}{\rho}\nabla_x \cdot \left(\frac{\nabla_x \eta}{\sqrt{1 + \abs{\nabla_x \eta}^2}}\right) on the left side. For deep water one replaces clause 2 by ∇ϕ→0\nabla \phi \to 0 as z→−∞z \to -\infty.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Incompressible flow, concept, no evidence levelIncompressible flowconceptLaplace's equation, concept, no evidence levelLaplace's equationconceptPotential flow, concept, no evidence levelPotential flowconceptVelocity potential, concept, no evidence levelVelocity potentialconceptWater waves, definition, no evidence levelWater wavesdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Incompressible flow concept
  2. Laplace's equation concept
  3. Potential flow concept
  4. Velocity potential concept
  5. Water waves definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wind_wave
About this page
TagJQ05YFH4CM
URIm2://JQ05YFH4CM
Kinddefinition
WritingW0 stub
MSC76B15, 76B07
DefinesWater waves
UsesPotential flow, Velocity potential, Laplace's equation, Incompressible flow
Changed3 November 2026

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