X cos(kx t),
Y sin(kx t). (14.18)
In the above equations the mean particle position is given by (x,y), i.e. by
its coordinates before it is perturbed by the wave. It may be easily shown
that, in general, the particles perform elliptical orbits, the horizontal dis-
placements being greater than the vertical displacement (Fig. 14.5). In
deep water, the orbit becomes circular, while in shallow water the particles
tend to move back and forth in a straight line.
Since the particles move in closed orbits, there is no net transport of
mass due to a linear wave. This is not the case with the non-linear waves
shown in Fig. 14.2.
14.2.4 Energy of waves
The kinetic energy (KE) in a wave is obtained by the following integration:
KE
0
d
L
0
2
(u^2 ^2 )dxdy.
The integrand is the kinetic energy of a particle of volume dxdyper unit
length at an instant of time. Substituting for uandfrom equation (14.16)
at an instant, say t0, and performing the double integration, the kinetic
energy is
KE
1
4
ga^2 L (14.19)
asinh[k(yd)]
sinh(kd)
acosh[(k(yd)]
sinh(kd)
582 WAVES AND OFFSHORE ENGINEERING
Fig. 14.5 Elliptical orbits of particles and directions of particle motions
under various phases
