If the waves are also long crested, the fluid motion is two dimen-
sional. Since waves can travel over large distances without significant
decay of energy, it can be assumed that wave motion is irrotational. Thus,
in linear wave analysis, the study of wave motion makes use of inviscid
fluid theory.
The equation of continuity for an incompressible flow is
∂
∂
u
x
∂
∂
y
^0 (14.2)
whereuis the horizontal velocity, is the vertical velocity and x,yare the
coordinate axes, as shown in Fig. 14.1. Note that the positive y-axis is
upwards from the still water level. A velocity potential , is defined as
u∂/∂x, ∂/∂y. (14.3)
After substitution for u and from equation (14.3), equation (14.2)
becomes
∂
∂
2
x
2
∂
∂
2
y
2 0. (14.4)
Equation (14.4) is the Laplace equation for the velocity potential. The
advantage of introducing the potential is that is the only property of the
field to be determined (instead of the two velocities); the penalty is a
second-order partial differential equation.
Bernoulli’s equation for unsteady flow, expressing the conservation
of energy, can be written as
∂
∂
t
u^2
2
^2
p
gy^0 (14.5)
wherepis the pressure, is the density of the fluid, gis the acceleration
due to gravity and tis the time. For waves of very small height relative to
the wavelength and depth, the velocity-squared terms in equation (14.5)
are only of second-order importance and hence can be neglected. Con-
sequently, equation (14.5) becomes
∂
∂
t
p
gy0. (14.6)
At the horizontal bed, with dthe water depth in the undisturbed state,
y
∂
∂
y
y d
0, (14.7)
578 WAVES AND OFFSHORE ENGINEERING
