Wave Runup
Definition
Wave runup is the maximum vertical extent of wave uprush on a beach or structure above the still water level (SWL) (Sorensen, 1997). This definition is depicted in the figure below.
Two different wave runup values were investigated in this study: mean and two percent wave runup. Mean runup is simply the average runup (R) of all waves observed. The two percent wave runup (R_{2%}) is the runup that only two percent of the wave runup values observed will reach or exceed.
Wave Runup Prediction Formulas
Five different formulas for predicting wave runup were investigated in this study. The following table provides the wave runup equations used in this study with some comments on the different formulas.
General equations and variables defined for wave runup formulas.
Method 
Source 
Slopes 
Waves 
Setting 
Equations 
Comments 
Hunt 
Hunt, 1959; Battjes, 1974 
Smooth, impermeable, continuous 
Regular 
Laboratory 

0.1 < x < 0.3 
CERC 
CERC Shore Protection Manual, 1984 
Smooth, impermeable, continuous 
Regular 
Laboratory 
Chart solution, see figure below 

Mase 
Mase, 1989 
Smooth, impermeable, continuous, gentle: q = 1.9°  11.3° 
Irregular 
Laboratory 


Nielsen and Hanslow 
Nielsen and Hanslow, 1991 
Natural sand beaches with q = 1.5°  10.8° and Mean grain size = 0.18  0.8 mm 
Irregular 
Field 
for tan b_{F} > 0.10
for tan b_{F} < 0.10 

Ahrens and Seelig 
Ahrens and Seelig, 1996 
Sand and gravel beaches 
Irregular 
Laboratory and Field 

d in mm, w_{sr} in cm/s
w_{sr} for fresh water and 0.15 < d_{sr} < 0.85 mm

CERC wave runup chart solution, reproduced from Sorensen (1997).
Each of the wave runup formulas requires input wave data. For this study deep water wave data from three different sources was used. NOAA's Nation Buoy Data Center (NDBC) has a wave buoy (station 45006) in the western half of Lake Superior that records wave data for nearly 75% of the year (see figure below). USACE wave Information Study (WIS) data stations, which are locations at which a computer model was used to hindcast deep water wave data, are another source of deep water wave data. The final method was to use wind data hindcast deep water wave data from measurements of fetch, time of duration, and wind speed.
Wave runup was measured in the field to determine the appropriate wave runup equations to be used for the study sites.
Video cameras were used to record 60 minute segments of wave runup at most of the study sites. For calm conditions a line of stakes was set up at a known interval (see figure below). The slope of the beach was measured using an inclinometer. Wave runup videos were manually digitized to record the peak wave runup of each wave. Wave runup was calculated knowing the hypotenuse length of runup (WR_{L}) and the angle of the beach (b):
R_{i} = (WR_{L})_{i }sin b
where _{i} denotes an individual wave
Example of field wave runup recording setup (calm conditions). Camera is on tripod in back; orange stakes mark 0.3 m intervals
Determination of Appropriate Wave Runup Equation
The wave runup measured in the field is compared to the R and R_{2%} formulas discussed above. The most suitable equation was chosen as the equation yielding the lowest errors overall. Mean wave runup values were calculated for the wave runup data, and two percent wave runup values were determined from cumulative frequency distributions of wave runup.