Page 120 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Wave Energy Chapter | 5 111

TABLE 5.1 Wave Properties Based on Linear Wave Theory
Wave Property Equation
Velocity potential Φ = H g cosh k(d+z) sin(kx − σt)
2 σ cosh kd
Wave celerity C = L = σ
T k
Horizontal component of velocity u = ∂Φ = H gk cosh k(d+z) cos(kx − σt)
∂x 2 σ cosh kd
Vertical component of velocity v = ∂Φ = H gk sinh k(d+z) sin(kx − σt)
∂z 2 σ cosh kd
H
Wave surface displacement η = cos(kx − σt)
2
cosh k(d+z) H
Pressure (hydrostatic + dynamic) p =−ρgz + ρg cos(kx − σt) =
cosh kd 2
p o + p D
Horizontal acceleration a x = ∂u = gkH cosh k(d+z) sin(kx − σt)
∂t 2 cosh kd
Vertical acceleration a z = ∂w =− gkH sinh k(d+z) cos(kx − σt)
∂t 2 cosh kd
2
Dispersion C = g tanh kd
k

Using linear wave theory, all of the hydrodynamic variables, including
velocity, pressure, water surface elevation, and wave celerity, can be derived
from the potential function and implementation of the boundary conditions.
Table 5.1 summarizes the wave properties based on linear wave theory.

5.1.2 Relationship Between Wave Celerity, Wave Number, and
Water Depth: The Dispersion Equation
Wave celerity is dependent on water depth and wave length, and important
wave processes such as wave refraction can be explained by the dependence of
wave celerity on depth (Section 5.2.2). The relation of wave celerity and water
depth can be derived by implementing the (kinematic) free surface boundary
condition, which implies that the vertical speed of water particles at the water
dη ∂η ∂η
surface is equal to the speed of the free surface (i.e. v = = + u at the
dt ∂t ∂t
water surface). Assuming a small amplitude wave, it results in
g
2 2
C = tanh kd ⇒ σ = gk tanh kd (5.13)
k
which is called the dispersion equation, because waves of different celerities
will be dispersed according to their wave length. Looking at Fig. 5.2, the
hyperbolic tangent function (tanh kd) in the dispersion equation approaches 1 for
large depth values (deep water waves) and kd for small values of depth (shallow
water waves). Therefore,