Geoscience Reference
In-Depth Information
One of the main results of the analysis of non-linear run-up problems consists
in the proof that run-up characteristics depend linearly on the wave amplitude far
from the coast. This fact provides for the possibility of applying linear theory in
calculating the run-up. A rigorous substantiation of such possibility can be found in
[Pelinovsky (1982)] and [Kaistrenko et al. (1991)].
Thus, for instance, in the case of the run-up of a monochromatic wave on a plane
slope, resolution of the linear problem results in it being possible to construct the fol-
lowing simple approximation for the maximum run-up value:
⎧
⎨
2
,
L
<
0
.
05
λ
;
2
L
λ
1
/
2
R
=
ξ
0
(5.51)
⎩
2
π
,
L
>
0
.
05
λ
.
If the length of the slope is insignificant as compared with the length of the inci-
dent wave, then the run-up process will proceed like in the case of a vertical wall,
i.e. the run-up height will turn out to be twice the amplitude of the incident wave.
An increase in the slope length
L
(a decrease of the angle
β
) will lead to a certain
enhancement of the run-up height.
Similar calculations were performed, also, for the run-up of a solitary wave. The
maximum run-up in this case, also, is described by a formula identical to (5.51), but
with a somewhat different numerical coefficient. It is interesting that oscillations of
the shoreline on steep slopes repeat the form of the initial wave. Shoreline oscilla-
tions on a gentle slope are related to the form of the incident wave in a more complex
manner. Thus, for instance, when a solitary wave (of positive sign) is incident upon
a slope, shoreline oscillations turn out to alternate in sign.
The relation between the wave height far from the coast and its run-up height on
a plane slope being linear is confirmed by results of laboratory and numerical ex-
periments. Figure 5.13 presents such a relation, obtained for the run-up of solitary
(a)
(b)
Fig. 5.13 Normalized maximum run-up of solitary waves on plane slope (a—1:1, b—1:19.85)
versus normalized height of incident wave. Squares—nonbreaking data, rhombs—breaking data.
Full and empty symbols correspond to laboratory and numerical experiments, respectively. Solid
line—run-up height on vertical wall, calculated by formula (5.46) (Adapted from [Titov, Synolakis
(1995)])
