Small Amplitude Wave Theory (SAWT) Flashcards
What is the wave number k and how is it defined?
k = 2π / λ, where λ = wavelength
What is the equation for surface elevation in SAWT?
η(x, t) = a · sin(ωt − kx)
The phase velocity or wave celerity defines the velocity of the wave ________.
crest
True or False: In SAWT, the wave amplitude is considered large compared to water depth.
False – it is assumed small compared to both depth and wavelength.
What are the 6 assumptions in Small Amplitude Wave Theory (SAWT)?
- Mass continuity (incompressible fluid)
- Irrotational motion
- Unsteady Bernoulli applies
- Wave motion is periodic in space and time
- a ≪ λ
- a≪h
What is the irrotationality condition in 2D (x-z plane)?
∂w/∂x − ∂u/∂z = 0
What is the form of Laplace’s equation after applying both continuity and irrotationality?
∂²u/∂x² + ∂²u/∂z² = 0
General solution for horizontal velocity in wave motion?
u = F(z) · sin(ωt − kx)
Final expression for vertical velocity w?
w = aω · [sinh(k(z + d)) / sinh(kd)] · cos(ωt − kx)
Final expression for horizontal velocity u?
u = aω · [cosh(k(z + d)) / sinh(kd)] · sin(ωt − kx)
True or False: Maximum horizontal and vertical velocities occur at the surface.
True
Match velocity location dependence with depth:
- Surface →
- Bed →
- Surface → Max velocity
- Bed → Min velocity (due to decay with depth)
What is the non-dimensional wave number kd for deep water?
kd > π (deep water condition)
What is the non-dimensional wave number kd for shallow water?
kd < π/10
What type of decay occurs in the boundary layer?
Exponential
What forces are important in the boundary layer?
Above the boundary layer only the inertial forces (or gravitational forces) are
important.
What happens to pressure under a wave crest at the seabed in SAWT?
Pressure increases and reaches a maximum under a wave crest, decaying with depth.
What is the final expression for pressure p in SAWT?
p = ρg · (a · [cosh(k(z + d)) / cosh(kd)] · cos(ωt − kx) + z)
Total pressure at depth is the sum of dynamic pressure and __________ pressure.
hydrostatic
What is the final dispersion relation for linear waves?
ω² = gk · tanh(kd)
True or False: ω² = gk is valid for all water depths.
False – only valid for deep water where tanh(kd)≈1
What is the phase speed
c of a wave?
c = ω/k
Rewrite the dispersion relation in terms of c.
c² = (g / k) · tanh(kd)
