Mechanics of Fluids

when the wave was formed. The length of the surface is
� λ
which for small amplitude is
� λ
0
� λ
≈
0
{1 + (dη/dx) 2 } 1/2 dx
�
1 + a 2 m 2 cos 2 �1/2 {m(x − ct)} dx
0
�
1 + 1
2 a2m 2 cos 2 �
{m(x − ct)} dx = λ + 1
4 a2m 2 λ
so the free surface energy over one wavelength is 1
4 γ a2 m 2 λ.
The total energy = gravitational energy + kinetic energy + free surface
energy.
Hence the total energy divided by the width of the wave, over one
wavelength
= 1
2 a2 �
λ ϱg + λm 2�
A (10.58)
Although we assumed that the bed is horizontal (h uniform) variations of
h have no appreciable effect on the result provided that h exceeds (say) λ/2.
10.13.6 Rate of energy transmission
Although the wave shape moves with velocity c this is not necessarily the
velocity with which energy is transmitted through the liquid. The energy is
carried by the particles of liquid and, as we have seen, these do not all move
at velocity c.
The quantity p∗ + 1
2ϱ(u2 + v2 ) is a measure of the amount of energy
carried by a small element of liquid when it moves divided by the volume of
the element. Here p∗ denotes the piezometric pressure and u, v the horizontal
and vertical components of velocity. In any fixed vertical plane perpendicular
to the x direction the volume flow rate through a small element of height δz
and unit breadth is uδz and so the rate at which the total amount of energy
is transferred across that plane is given by
� η �
p ∗ + 1
2 ϱ(u2 + v 2 �
) udz (10.59)
−h
Since the flow is assumed irrotational Bernoulli’s equation may be applied
between any two points in steady flow. One of these points may be taken on
the free surface at η = 0; then, for the steady conditions and waves of small
amplitude.
p ∗ + 1
2 ϱ
�
(u − c) 2 + v 2�
= 1
2 ϱc2
A (10.60)
Oscillatory waves 473