ABSTRACT
Water wave may be defined as the deformation of free water surface. The wave form propagates far away while the water particle excited by the wave does not propagate along with the wave form. Figure 2.1 indicates the general form of the train of the idealized regular wave. The wave height is defined as the vertical distance from the trough to the crest of the wave while the wave length is the horizontal distance between successive wave crests. The motion of the fluid with the free water surface satisfies three kinds of
boundary condition. The first kind is the dynamic condition which indicates that the pressure on the free water surface is equivalent to the atmospheric pressure. The second kind is the kinematic condition which means that the water particle does not move across the free water surface. The third kind is the seabed condition. Based on the assumptions that water is inviscid,
incompressible and its flow is irrotational, the motion of water particles can be characterized by a quantity known as the velocity potential. The velocity potential that satisfies the Laplace equation in the fluid domain is defined as a function whose derivatives yield the velocity component of the fluid. For small-amplitude waves where the wave height is assumed to be very much smaller than the wave length, the free-surface boundary conditions are expressed as
ζ = − 1 g ∂
∂t , on z = 0 dynamic condition (2.1)
∂
∂z = ∂ζ
∂t , on z = 0 kinematic condition (2.2)
and the sea bed condition is given by
∂
∂z = 0, on z = −h (2.3)
where z is the vertical coordinate measured from the origin which is taken at the still-water, level h the water depth, g the gravitational acceleration, (x, y, z; t) the velocity potential and ζ(x, y; t) the surface elevation. Equations (2.1) and (2.2) may be combined to give
∂t2 + g ∂
∂z = 0, on z = 0 (2.4)
The velocity potential which satisfies the boundary conditions (2.3) and (2.4) is given by
(x, y, z; t)=Re [ igζA ω
cosh k(z + h) cosh kh
(2.5)
where ζA is the amplitude, ω the circular frequency, k the wave number and α the incident angle of the waves with respect to the positive x-axis as shown in Figure 2.2. By substituting Eq. (2.5) into Eq. (2.4), one obtains
g = k tanh kh (2.6)
Equation (2.6) is known as the linear dispersion relation. The surface elevation, or the wave train, obtained from Eqs (2.1) and (2.5)
may be expressed as
ζ(x, y; t)=Re [ ζAe
When discussing waves that propagate only along the negative x-axis, the expression (2.7) may be simply written as
ζ(x; t)=Re [ ζAe
i(kx+ωt)] = ζA cos(kx+ ωt) (2.8) The velocity of the wave is the traveling speed of the wave profile which is called the phase velocity. The expression for the velocity of the wave profile is given by
c= ω k = √ g k tanh kh (2.9)
and it is derived from Eq. (2.6). This indicates the wave velocity depends on just the wave number. On the other hand, the velocity of the harmonic wave can be defined by
c= λ T
(2.10)
where λ is the wave length and T the wave period. The combination of Eqs (2.9) and (2.10) yields
k= 2π λ
(2.11)
which gives the relation between the wave number and the wave length. There are two kinds of velocity for the wave train, namely, the phase
velocity as mentioned earlier and the group velocity. A group of the wave train in Figure 2.1 advances with a group velocity whereas individual waves in the group move along with the phase velocity. Individual waves successively reach the leading wave in the wave train, because the phase velocity is
greater than group velocity. The magnitude of the group velocity is half of the phase velocity for the dispersive wave in deep water. The group velocity for a non-dispersive wave such as swell is equal to the phase velocity. Wave velocity depends on the magnitude of the wave height, and increases
with increasing wave heights. The wave velocity derived from the smallamplitude wave theory is independent of the wave height. The wave velocity, however, is a good approximation for the velocity of a large amplitude wave. Water depth may be regarded as infinite if the water depth is greater than
half of the wave length (i.e., h > λ/2). As h → ∞ in Eqs (2.5) and (2.6), the velocity potential for the incident wave and the dispersion relation are expressed respectively by
(x, y, z; t)=Re [ igζA ω eKz+iK(x cosα+y sinα)+iωt
] (2.12)
g =K (2.13)
In very shallow water (roughly h < λ/20), the wave is usually referred as a long wave and the dispersion relation becomes
g = k2h (2.14)
as h→ 0 in (2.6). This relation gives
c= ω k = √ gh (2.15)
which shows that the wave velocity is independent of the wave length and the long wave is not dispersive. Table 2.1 summarizes the expression for the fundamental properties of
small-amplitude waves. The range of water depth in Table 2.1 is divided into deep water, intermediate depth and shallow water.
