Geoscience Reference
In-Depth Information
l
For simple harmonic motion of
angular velocity, v , the
displacement of the still water
level over time, t , is given by:
Wave
adv ance
Wave speed, c
Crest
H
x
Trough
Still water
level
y = H sin vt
y
Depth, h ,
> l
The equations of motion for
an inviscid fluid can be solved
to give the following useful
expression for wave speed, c :
/2
Every water
particle rotates
about a
time-mean
circular motion
c = gl / 2 p
Since the coefficients are
constant, for SI units we have:
Arrows show instantaneous
motion vectors at each
arrowhead
c = 1.25
l
Fig. 4.42 Deep water wave parameters, circular orbitals, and instantaneous water motion vectors. Deep water waves are sometimes called short
waves because their wavelengths are short compared to water depth.
Note: In nature, individual waves pass through wave groups traveling at speed C 2 with energy transmitting at this rate.
simple mathematical guide to our study of wave physics
(Cookie 14). It is a common mistake to imagine deep
water waves as heaps and troughs of water moving along a
surface: it is just wave energy that is transferred, with no
net forward water motion.
The simplest approach is to set the shape of the wave-
form along an xz graph and consider that the periodic
motion of z will be a function of distance x , wave height,
H , wavelength,
the direction of motion rotates with angular speed,
; and
any particle must undergo a rotation below deep water
waves (Fig. 4.42). The radii of these water orbitals as they
are called, decreases exponentially below the surface.
4.9.2
Shallow water surface gravity waves
, and celerity (wave speed), c . Attempts to
investigate wave motion in a more rigorous manner
assume that the wave surface displacement may be approx-
imated by curves of various shapes, the simplest of which is
a harmonic motion used in linear (Airy) wave theory
(Cookie 14). Sinusoidal waves of small amplitude in deep
water cause motions that cannot reach the bottom. Small-
amplitude wave theory approach assumes the water is
inviscid and irrotational. The result shows that surface
gravity waves traveling over very deep water are dispersive
in the sense that their rate of forward motion is directly
dependant upon wavelength: wave height and water depth
play no role in determining wave speed (Fig. 4.42). An
important consequence of dispersion is that if a variation
of wavelength occurs among a population of deep water
waves, perhaps sourced as different wave trains, then the
longer waves travel through the shorter ones, tending to
amplify when in phase and canceling when out of phase.
This causes production of wave groups , with the group
speed, c g being 50 percent less than the individual wave
speeds, c (Cookie 15).
At any fixed point on or within the water column the
fluid speed caused by wave motion remains constant while
Deep water wave theory fails when water depth falls below
about 0.5
. This can occur even in the deepest oceans for
the tidal wave and for very long (10s to 100s km) wave-
length tsunamis (see below). Shallow water waves are
quite different in shape and dynamics from that predicted
by the simple linear theory of sinusoidal deep water waves.
As deep water waves pass into shallow water, defined as
h
/20, they suffer attenuation through bottom friction
and significant horizontal motions are induced in the
developing wave boundary layer (Figs 4.43 and 4.44). The
waves take on new forms, with more pointed crests and
flatter troughs. After a transitional period, when wave
speed becomes increasingly affected by water depth,
shallow-water gravity waves move with a velocity that is
proportional to the square root of the water depth, inde-
pendent of wavelength or period (Cookie 16). The disper-
sive effect thus vanishes and wave speed equals wave group
speed. The wave orbits are elliptical at all depths with
increasing ellipticity toward the bottom, culminating at
the bed as horizontal straight line flow representing to-
and-fro motion. Steepening waves may break in very shal-
low water or when intense wind shear flattens wave crests
(Section 6.6). In both cases air is entrained into the surface
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