fluid_mechanics

10.2 Surface Waves 539
__ c 2
gy
1.0
c = √gy
shallow layer
c =
g __ λ
√ 2 π
deep layer
y > λ
0
0 __y λ
F I G U R E 10.5
of wavelength.
Wave speed as a function
c, m/s
150
y >> l
100
50
0
0 2 4 6 8 10
l, km
This result, shown in the figure in the margin, follows from Eq. 10.4, since tanh12pyl2 S 1 as
yl S . Note that waves with very long wavelengths [e.g., waves created by a tsunami (“tidal
wave”) with wavelengths on the order of several kilometers] travel very rapidly. On the other hand,
if the fluid layer is shallow 1y l, as often happens in open channels2, the wave speed is given
by c 1gy2 12 , as derived for the solitary wave in Fig. 10.2. This result also follows from Eq. 10.4,
since tanh12pyl2 S 2pyl as yl S 0. These two limiting cases are shown in Fig. 10.5. For
moderate depth layers 1y l2, the results are given by the complete Eq. 10.4. Note that for a given
fluid depth, the long wave travels the fastest. Hence, for our purposes we will consider the wave
speed to be this limiting situation, c 1gy2 12 .
F l u i d s i n t h e N e w s
Tsunami, the nonstorm wave A tsunami, often miscalled a “tidal
wave,” is a wave produced by a disturbance (for example, an earthquake,
volcanic eruption, or meteorite impact) that vertically displaces
the water column. Tsunamis are characterized as shallowwater
waves, with long periods, very long wavelengths, and
extremely large wave speeds. For example, the waves of the great
December 2005, Indian Ocean tsunami traveled with speeds to
500–1000 m/s. Typically, these waves were of small amplitude in
deep water far from land. Satellite radar measured the wave height
less than 1 m in these areas. However, as the waves approached
shore and moved into shallower water, they slowed down considerably
and reached heights up to 30 m. Because the rate at which a
wave loses its energy is inversely related to its wavelength,
tsunamis, with their wavelengths on the order of 100 km, not only
travel at high speeds, they also travel great distances with minimal
energy loss. The furthest reported death from the Indian Ocean
tsunami occurred approximately 8000 km from the epicenter of
the earthquake that produced it. (See Problem 10.14.)
V – c
V > c
Stationary
V = c
c – V
V < c
10.2.2 Froude Number Effects
Consider an elementary wave traveling on the surface of a fluid, as is shown in the figure in the
margin and Fig. 10.2a. If the fluid layer is stationary, the wave moves to the right with speed c
relative to the fluid and the stationary observer. If the fluid is flowing to the left with velocity
V 6 c, the wave 1which travels with speed c relative to the fluid2 will travel to the right with a
speed of c V relative to a fixed observer. If the fluid flows to the left with V c, the wave will
remain stationary, but if V 7 c the wave will be washed to the left with speed V c.
The above ideas can be expressed in dimensionless form by use of the Froude number,
Fr V1gy2 12 , where we take the characteristic length to be the fluid depth, y. Thus, the
Froude number, Fr V1gy2 12 Vc, is the ratio of the fluid velocity to the wave speed.
The following characteristics are observed when a wave is produced on the surface of a
moving stream, as happens when a rock is thrown into a river. If the stream is not flowing, the
wave spreads equally in all directions. If the stream is nearly stationary or moving in a tranquil
manner 1i.e., V 6 c2, the wave can move upstream. Upstream locations are said to be in hydraulic
communication with the downstream locations. That is, an observer upstream of a disturbance can
tell that there has been a disturbance on the surface because that disturbance can propagate upstream