Hydraulic Structures: Fourth Edition

14.2.3 Particle kinematics

The velocities uandvare obtained from their definitions in terms of the
potential (equation (14.3)). They are

u sin(kx t), (14.16)

 cos(kx t).

For the Airy wave the fluid particles do not make large excursions from
the mean position, the coordinates of which are the same as the point
occupied by the particle before the start of the wave motion. The horizon-
tal and vertical displacements of the particle from the mean position
respectively,X,Y, are

X

t

0

udt, Y

t

0

dt. (14.17)

Substituting for velocities uandvfrom equation (14.16), the temporal
variations of the x-wise and y-wise displacements XandYrespectively are

asinh[k(yd)]



sinh(kd)

acosh[k(yd)]



sinh(kd)

WAVE MOTION 581

Fig. 14.4 Wave period as a function of wavelength and depth