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Espectro de um Sinal

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(amplitudes) obtidas a partir do Espectro e com um conjunto de fases
obtidas aleatoriamente, reconstitui-se um novo sinal, com as mesmas componentes de
ondas. Pore´m com o novo conjunto de fases.
4. A figura 9 mostra esquematicamente va´rias realizac¸o˜es de um mesmo conjunto de com-
ponentes de ondas, variando-se suas fases.
5. A figura 10 mostra esquematicamente va´rias ondas se propagando em diferentes direc¸o˜es
com diferentes per´ıodos e diferentes amplitudes e sua superposic¸a˜o.
Texto Preliminar, SH Sphaier 10
5.3 Irregular Waves
Now that the background of regular waves has been discussed, attention shifts to a more
realistic image of the sea surface. Often the sea surface looks very confused; its image
changes continuously with time without repeating itself. Both the wave length between
two successive crests or troughs and the vertical distance between a crest and a trough
vary continuously.
5.3.1 Wave Superposition
It was stated in the …rst lines of this chapter that it is possible to represent the irregular
sea surface using a linear superposition of wave components. This will be demonstrated
here. In fact, a superposition of only two components with identical directions but di¤erent
speeds are needed to show this; see …gure 5.22.
Figure 5.22: Superposition of Two Uni-Directional Harmonic Waves
A superposition of three components yields a more realistic record, even though these still
come from one direction; see …gure 5.23. A wave train such as this can easily be generated
in a wave ‡ume, for example. Since all of the energy travels in the same direction, the
wave crests will be (theoretically) in…nitely long; everything of interest can be observed in
a single x-z plane.
Note that for a realistic record of uni-directional irregular waves, a superposition of at
least 15 or 20 components is required in practice if one is only interested in the mean value
of an output. More components are handy if additional information such as statistical
distributions are needed.
If a third dimension - direction - is added to this, then the sea (as seen from above) becomes
even more realistic as is shown in …gures 5.24 and 5.25.
One sees from these …gures that the length of the wave crests is now limited. This is a sign
of the fact that wave energy is simultaneously propagating in several directions.
5.3.2 Wave Measurements
Most waves are recorded at a single …xed location. The simplest instrumentation can be
used to simply record the water surface elevation as a function of time at that location.
Since only a scalar, single point measurement is being made, the resulting record will yield
no information about the direction of wave propagation.
Figura 6: Superposic¸a˜o de duas ondas monocroma´ticas5-40 CHAPTER 5. OCEAN SURFACE WAVES
Figure 5.32 gives a graphical interpretation of the meaning of a wave spectrum and how
it relates to the waves. The irregular wave history, ³(t) in the time domain at the lower
left hand part of the …gure can be expressed via Fourier series analysis as the sum of a
large number of regular wave components, each with its own frequency, amplitude and
phase in the frequency domain. These phases will appear to be rather random, by the way.
The value 12³
a(!)=¢! - associated with each wave component on the !-axis - is plotted
vertically in the middle; this is the wave energy spectrum, S³ (!). This spectrum, S³ (!),
can be described nicely in a formula; the phases cannot and are usually thrown away.
Figure 5.32: Wave Record Analysis
Spectrum Axis Transformation
When wave spectra are given as a function of frequency in Hertz (f = 1=T) instead of
! (in radians/second), they have to be transformed. The spectral value for the waves,
S³(!), based on !, is not equal to the spectral value, S³ (f), based on f. Because of
the requirement that an equal amount of energy must be contained in the corresponding
frequency intervals ¢! and ¢f, it follows that:
jS³(!) ¢ d! = S³(f) ¢ dfj or: S³(!) = S³(f )d!
The relation between the frequencies is:
! = 2¼ ¢ f or: d!
= 2¼ (5.113)
Figura 7: Espectro de um sinal
Texto Preliminar, SH Sphaier 11
When obtaining the wave spectrum S³(!) from the irregular wave history, the phase an-
gles "n have been thrown away. New "n-values must be selected from a set of uniformly
distributed random numbers in the range 0 · "n < 2¼: While "n is needed to generate the
time record - and they may not all be set equal to zero!- the exact (randomly selected) "n
do not in‡uence the record’s statistics. Looked at another way: By choosing a new random
set of "n values, one can generate a new, statistically identical but in detail di¤erent time
This procedure is illustrated by extending …gure 5.32 as shown below in …gure 5.38. Note
that the original (on the left in the …gure) and the newly obtained wave history (on the
right hand part of the …gure) di¤er because di¤erent phase angles have been used. However,
they contain an equal amount of energy and are statistically identical.
Figure 5.38: Wave Record Analysis and Regeneration
Directional spreading can be introduced as well by breaking each frequency component
up into a number of directional components. One should not underestimate the (extra)
computational e¤ort however.
5.5 Wave Prediction and Climatology
In 1805, the British Admiral Sir Francis Beaufort devised an observation scale for measuring
winds at sea. His scale measures winds by observing their e¤ects on sailing ships and waves.
Beaufort’s scale was later adapted for use on land and is still used today by many weather
stations. A de…nition of this Beaufort wind force scale is given in …gure 5.39.
The pictures in …gure 5.40 give a visual impression of the sea states in relation to Beaufort’s
scale. Storm warnings are usually issued for winds stronger than Beaufort force 6.
Figura 8: Gerac¸a˜o de um sinal
Figura 9: Va´rias realizac¸o˜es de um sinal
Texto Preliminar, SH Sphaier 12
Figure 5.1: A Sum of Many Simple Sine Waves Makes an Irregular Sea
location along the ‡ume; it looks similar in many ways to the other …gure, but time t has
replaced x on the horizontal axis.
Notice that the origin of the coordinate system is at the still water level with the positive
z-axis directed upward; most relevant values of z will be negative. The still water level
is the average water level or the level of the water if no waves were present. The x-axis is
positive in the direction of wave propagation. The water depth, h, (a positive value) is
measured between the sea bed (z = ¡h) and the still water level.
The highest point of the wave is called its crest and the lowest point on its surface is the
trough. If the wave is described by a sine wave, then its amplitude ³a is the distance
from the still water level to the crest, or to the trough for that matter. The subscript a
denotes amplitude, here. The wave height H is measured vertically from wave trough level
to the wave crest level. Obviously:
jH = 2³aj for a sinusoidal wave (5.1)
The horizontal distance (measured in the direction of wave propagation) between any two
successive wave crests is the wave length, ¸. The distance along the time axis is the wave
period, T: The ratio of wave height to wave length is often referred to as the dimensionless
Figura 10: Superposic¸a˜o com diferentes direc¸o˜es de propagac¸a˜o