Árbók VFÍ/TFÍ - 01.06.2004, Síða 216

(hardening spring effect) when the amplitude to the shaking is large as is the case for
these conditions. The final plot shows the wave heights for /i = 1.50. This frequency is
beyond the jump frequency, that is, the wave heights are significantly lower than for the
smaller frequencies. Again, the wave heights in the middle of the tanks are small and the
waves are not broken.
Figure 4 shows the maximum and minimum wave heights for all frequencies for this par-
ticular case. Plot (a) in the figure shows the wave heights at the end wall and plot (c) the
wave heights in the middle of the tank. It is evident that the wave heights increase with
increasing shaking frequency until they abruptly decrease at the jump frequency, which
in this case is at about /J = 1.38. The behavior at the end of the tank and in the middle is
similar as expected. The two plots show clearly that the jump frequency is not uniquely
defined. Plots b) and d) show a enlarge of the segment around the jump frequency for
both wave gages. The arrows on the plots indicate in what direction the frequency was
changed for each branch of the response. Plot b) shows that the wave height remains
almost unchanged until it abruptly decreases when the frequency is increased in steps
from /J = 1.397 to p = 1.402 when the frequency is increased in steps from a low frequen-
cy to higher frequencies. However, when the frequency is decreased the wave heights do
not increase until the frequency is changed from /J = 1.367 to /3 = 1.362, which is consid-
erably lower than for the other branch. Therefore, the wave behavior takes on two dis-
tinctly different forms in the frequency range from p = 1.367 to /f = 1.397, even though the
conditions such as water depth, shaking frequency etc., are exactly same. One is a violent
wave breaking solution but the other relatively
smooth wave behavior. The only thing that creates
these very different behaviors is the prior conditions
of the sloshing in time.
Figure 5 shows a snapshot of the water surface from
the image capturing of the experiments, illuminated
with the laser sheet. The pictures in the figure show
the water surface profile just as the maximum wave
height is approaching the end wall for exactly the
same frequency, [i = 1.397. The upper plot in the fig-
ure is for the upper branch on Figure 4b) and the
lower plot is for the lower branch. It is evident from
the figure how different the wave behavior is for the
two branches. For the upper branch the wave break-
ing is easily observed but for the lower branch the
water surface is smooth.
The image capturing was used to capture pictures of
the water surface at a rate of 30 pictures every sec-
ond. They were processed by image processing soft-
ware to generate a sequence of water surface pro-
files for a complete three dimensional image of the
profile in the middle of the tank in time. Results from the image processing are shown in
Figure 6 for two different frequencies. Plots a) and b) in the figure are for /J = 1.397. The
figure shows about 1 /3 of the tank length from the end wall (the length along the tank is
normalized with L which is the length of the tank) and the vertical axis is as before, 1] /Iiq
which is the normalized deviation from quiescent water. The time axis is normalized by
the shaking table period, T. These plots show clearly how different the wave conditions
are for exactly the same conditions. Plot a) shows the upper branch. It shows that there is
one main wave front traveling back and forth with a step leading front indicating a wave
break. A second smaller wave trails the main wave front. The run-up along the walls of
the tank is quite large and the reflected wave is quite pronounced. Plot b) shows the
lower branch which shows completely different behavior. The wave behavior is more like
Figure 5. Snapshots of the water surface
from the video tape of the experiment illumi-
nated by the laser sheet for p = 1.397.
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Arbók VFl/TFl 2004