OCE421 Marine Structure Designs Lecture

1
OCE421 Marine Structure DesignsLecture 2
(wave theory)

• Fall, 2003

2
Linear Wave Theory

• The most elementary wave theory, referred to as
  small-amplitude or linear wave theory, was
  developed by Airy (1845).
• Mathematically, the Airy theory can be considered
  a first approximation of a complete theoretical
  description of wave behavior.
• For some situations, waves are better described
  by these higher order theories, which are usually
  referred to as finite-amplitude theories.

3
Two-Dimensional Surface Waves

• The x-axis is the still water position, with the
  wave crest at the origin. The bottom is at z
  -d
• Two useful dimensionless parameters wave
  steepness (H/L) and relative depth (d/L)

4
Two-Dimensional Periodic Wave
5
Governing Differential Equation
6
Other Governing Assumptions

• The water is homogeneous and incompressible and
  surface tension forces are negligible.
• The bottom is horizontal, impermeable, and
  stationary.
• The pressure along the air-water interface is
  constant.

7
Bernoulli Equation (for unsteady flow)
8
Three Boundary Conditions
9
Validity of Linear Wave Theory

• Wave amplitude is small relative to the wave
  length and the water depth.
• Particle velocities must be small compared to the
  wave celerity.
• For high waves at sea or for waves propagating in
  shallow areas where these assumptions do not
  strictly hold, the small amplitude wave theory is
  of more limited accuracy.

10
Velocity Potential
Employing the Laplace equation, the BBC, and the
linearized FSDBC (without using the linearized
FSKBC)
progressive wave
(the water surface needs to be assumed in a
certain form)
11
Wave Elevation / Dispersion Relationship
12
Fundamental Parameters
13
Alternate Forms for Linear Dispersion Relationship
an iterative technique is often used
14
Hunts Approximated Solution
less than 1 error
15
Matlab Function ldis.m
meter, second
Hunts formula as initial value
iterative procedure
16
Wave propagates from deep water toward the shore

• Wave period will remain constant
• Other characteristics such as the height, length,
  celerity, surface profile, internal pressure
  field, and particle kinematics change

17
Wave Classification by Relative Depth

• Deep, intermediate (transitional), and shallow
  water wave.
• The classification is based on the local relative
  depth d/L, A tide wave having a period around 12
  h is so long that it is a shallow water wave in
  the deepest part of the ocean.
• The relative depth limits of 0.5 and 0.05 for
  deep and shallow water are somewhat arbitrarily
  chosen.

18
Asymptotic Form of Hyperbolic Functions
d/L gt0.5
d/L lt0.05
19
Deep Water Wave Length and Celerity
(a function of wave period only)
20
Shallow Water Wave Celerity
Wave length
Wave celerity
(a function of depth only)
21
An Example Problem

• A tsunami is detected at 1200 h on the edge of
  the continental shelf by a warning system. At
  what time can the tsunami be expected to reach
  the shoreline?

22
Solution Explanation
how much time is required to travel a distance dx?
depth at x
wave speed at x