Hydraulic Structures: Fourth Edition

over one wavelength per unit length of the wave crest. The potential
energy (PE) in a wave is obtained by finding the work done by the wave to
displace water vertically from the still-water level. It is simply

PE

L

0



1

2

^ g^2 dx

withgiven by equation (14.11a); the potential energy (PE) in a wave-
length per unit length of wave crest is

PE

1

4

 ga^2 L. (14.20)

The total energy in a wavelength per unit length parallel to the crest is
therefore

KEPE

1

2

 ga^2 L. (14.21)

The equal division of energy between kinetic and potential energies, i.e.
PEKE, is an essential requirement of free vibrations. The total energy E
per unit plan area for unit length parallel to the crest is

E

1

2

^ ga^2 or 

1

8

^ gH^2. (14.22)

14.2.5 Radiated energy

Radiated energy, R, or energy flux is the rate at which the wave energy
moves in the direction of wave propagation, and it is the rate at which
work is done by the pressure forces: Ris also the wave power. It is given
by

R

0

d

pudy

wherepis the pressure given by the linearized Bernoulli equation (14.6)
without the hydrostatic term gy. Hence

R

0

d

(^) 

∂

∂



t

udy. (14.23)

Integration of the above equation after substituting for ufrom equation
(14.16), and for and hence for ∂/∂tfrom equation (14.12) results in

WAVE MOTION 583