Linear Wave Theory (LWT)
Variables to Know
- H (height) – distance from crest to trough
- T (wave period) – time between passage of wave crests
- L (length) – distance between crests
- h (water depth) – distance from bottom to free surface
- c (wave celerity) – speed of wave
- η (free surface profile) – distance from still water level (SWL) to the crest
- Two common variables are centered on the SWL
- ac – height of crest
- at – depth of trough
Conditions of the LWT
- Intermediate to deep water depth
- Sand or silt bottom
- Wave height to wave length ratio is small
- H/L must be very close to 0
Assumptions (These can only be made if the above conditions are true)
- Incompressible fluid - an increase in pressure does not cause the fluid to decrease in volume
- And so, conservation of mass yields the Laplace equation.
(Laplace Equation is discussed in further detail below.)
- Irrotational flow
- Helps describe velocity potential by giving u and w horizontal and vertical water particle velocities, respectively.
- Horizontal, impermeable, rigid bottom
- No surface stresses
- Pressure
- Wind
- Surface tension
- No currents
Wave Energy in Linear Waves
- E = wave energy density (energy per unit area)
- Group Velocity = speed at which waves propagate
- Wave Celerity is faster than Group Velocity until the waves enter shallow water
(Above is the Ratio of Group Velocity to Wave Celerity)
- Using the two terms above, we can solve for Wave Energy Flux (wave power), which is the rate at which energy is transmitted.
- Wave Energy Flux = Wave Energy Density X Group Velocity
Laplace Equation
- The Laplace Equation is often used in topics such as groundwater flow and magnetic fields.
- Boundary Conditions
- Bottom Boundary Condition (BBC) – no flow into the horizontal, impermeable, rigid bottom
- Kinematic Free Surface Boundary Condition (KFSBC) – velocity of a water particle on the free surface is the same as the velocity of the free surface.
- Dynamic Free Surface Boundary Condition (DFSBC) – assumption of zero stress on the free surface. This is used in the Bernoulli equation.
(Since velocity is related to H/L, and we know that H/L is very close to 0, then we can disregard the v term in this equation.)
- Simple periodic – waves are periodic in time (t) and space (x)
- Example:
- Note: Free surface boundaries are relative to the SWL line (instead of the free surface, η) when referring to the Laplace equation because of the simplicity it provides.
If the above conditions are true for a particular situation, the following equations can be used to make certain wave calculations.
Equations for Linear Wave Theory
References
- Tedesco, McDougal, Ross. Structural Dynamics Theory and Applications. Addison-Wesley, Menlo Park, 1999.
