Wave run-up is the height to which waves run up the slope of a revetment, bank or dike, regardless of whether the waves are breaking or not. Conversely, wave run-down is the height to which waves recede. These heights are always measured vertically (and not along the slope). The wave run-up height, denoted by R u {\displaystyle R_{u}} , R u p {\displaystyle {R_{up}}} , or z {\displaystyle z} , is a very important parameter in coastal engineering as, together with the design highest still water level, it determines the required crest height of a dike or revetment.
History
The first scientific measurements of wave run-up were carried out by the Lorentz Committee in preparation for the works to close off the Zuiderzee. The Committee measured the wave height and wave run-up at various locations in 1920, but established that state of the art methods for measuring waves in the field during storms were inadequate. As a result, scale tests were also undertaken, but these also proved to be of very limited efficacy due to the fact that only regular waves (idealised, periodic waves with constant amplitude and a fixed time period between successive wave crests, following a sinusoidal pattern) could be modelled at the time. The methods and technology available to the committee at the time did not permit model testing of the more realistic and complex irregular waves (consisting of varying heights, periods and directions), which provide a more accurate representation of the actual conditions faced by coastal structures and shorelines. It was found, however, that the depth in front of the dike is very important for wave run-up and that, at least for the range of observations in the committee's measurements, the slope ratio does not play a major role. Nearly all dikes in the Netherlands at that time had a slope of 1:3. Current knowledge indicates that during storms and on gentle coastal slopes, the significant wave height is approximately half the water depth. This relationship appears to be accurate, and the observation is more pronounced for slopes around 1:3. This research was continued during the Zuiderzee Works, and eventually led to the (old) Delft formula for wave run-up:
R u = 8 H tan α {\displaystyle Ru=8H\tan \alpha }
in which:
R u {\displaystyle Ru} is the run-up,
H {\displaystyle H} is the (regular) wave height at the toe and
α {\displaystyle \alpha } is the slope of the construction under consideration. This formula proved to be generally applicable for smooth slopes and relatively steep (storm) waves. Subsequently, it was discovered that longer (swell) waves resulted in higher run-up. To account for this, the wave period was incorporated into the formula using the Iribarren number, ξ {\displaystyle \xi } , leading to the development of Hunt's Formula:
R u H = ξ = tan α ( H / L 0 ) 1 2 {\displaystyle {\frac {Ru}{H}}=\xi ={\frac {\tan \alpha }{(H/L_{0})^{\frac {1}{2}}}}}
This formula was also valid for regular waves. The Old Delft Formula and Hunt's Formula are identical for waves with a steepness of 1/64, or about 2%. For higher values of ξ {\displaystyle \xi } , Hunt's formula has a limit value:
ξ > 2 . 5 {\displaystyle \xi >2{.}5} → {\displaystyle \rightarrow } R u / H = 2 . 5 {\displaystyle Ru/H=2{.}5} .
van der Meer, TAW and continuing development of formulae In 1988, van der Meer provided formulae for wave run-up on rubble mound breakwaters, based on tests with rock-armoured straight slopes. He also introduced a notional permeability factor P {\displaystyle P} for the structure. This factor also accounts for the effect of the pore volume. Defining R u p {\displaystyle R_{up}} as the run-up level of exceedance probability p {\displaystyle p} , the formula, valid for 0.1 ≤ P ≤ 0.6 {\displaystyle 0.1\leq P\leq 0.6} and head-on waves, is:
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