Water Waves

Question 1

Describe the set up for water waves

Answer 1

Flat bed at z=-h and undisturbed surface at z=0.

Small perturbations at surface z = η(x, y, t)

Question 2

What conditions do we impose on the water surface (kinematic boundary conditions)

Answer 2

p = 0 on z = η(x, y, t)

u · n = 0 where n is normal to z = η(x, y, t)

Question 3

What assumptions do we make for water waves?

Answer 3

That water is an ideal, irrotational, incompressible fluid, with body force f = -g eZ

Question 4

What does incompressible and irrotational impart on our velocity

Answer 4

u = ∇φ

For some potential which satisfies
∆φ = 0 in −h < z < η

Question 5

How do we linearize?

Answer 5

Neglect second order terms in four equations.

Taylor expand the derivatives of φ with boundary conditions at η to only keep linear terms.

Question 6

State the basic plane wave solutions for travelling waves

Answer 6

φ(x, z, t) = X(x − ct)Z(z)

Question 7

How do we express the boundary conditions?

Answer 7

Using Bernoulli (z=η) for p=0

Kinematic (z=η) Dη/Dt = Dz/Dt (expand)

Kinematic (z=-h) u· n = 0 gives dφ/dt = 0

Question 8

What characterises the dispersion relation?

Answer 8

ω=ck