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Solitary Wave on a Canonical Beach

Figure 1: Definition sketch for canonical bathymetry, i.e., sloping beach connected to a constant-depth region.
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In this set of experiments, the 31.73 m-long, 60.96 cm-deep, and 39.97 cm-wide California Institute of Technology, Pasadena, California wave tank was used with water at varying depths (Fig. ). The tank is described by Hammack (1972), Goring (1978), and Synolakis (1986, 1987). The bottom of the tank consisted of painted stainless steel plates. Carriage rails run along the whole length of the tank, permitting the arbitrary movement of instrument carriages. A ramp was installed at one end of the tank to model the bathymetry of the canonical problem of a constant-depth region adjoining a sloping beach. The ramp had a slope of 1:19.85. The ramp was sealed to the tank side walls. The toe of the ramp was distant 14.95 m from the rest position of the piston generator used to generate waves.

A total of more than 40 experiments with solitary waves running up the sloping beach were performed, with wave depths ranging from 6.25 cm to 38.32 cm. Solitary waves are uniquely defined by their maximum height $\widetilde{H}$ to depth $\widetilde{d}$ ratio and the depth, i.e., $\widetilde{H}/\widetilde{d}$ and $\widetilde{d}$ are sufficient to specify the wave. $\widetilde{H}/\widetilde{d}$ ranged from $0.021$ to $0.626$. Breaking occurs when $\widetilde{H}/\widetilde{d}>0.045$, for this particular beach. This is the same set of experiments used to validate the maximum runup analytical predictions presented in section A2.1.1.

Initial location, $X_{s}$ in the analytical consideration ([*]), changes with different wave heights. The reason $X_{s}$ distance varies is that solitary waves of different heights have different effective ``wavelengths.'' A measure of the ``wavelength'' of a solitary wave is the distance between the point $x_{f}$ on the front and the tail $x_{t}$ where the local height is 1% of the maximum, i.e., $\eta (x_{f},t=0)=\eta(x_{t},t=0)=(\widetilde{H}/\widetilde{d})/100$. The distance $X_{s}$ is at an offshore location where only 5% of the solitary wave is already over the beach, so that scaling can work. Therefore, in the laboratory experiments initial wave heights are identified at a point $X_{s}=X_{0}+(L/2)$ where $L/2=(1/\gamma) arccosh\sqrt{20}$ with $\gamma =\sqrt{3(\widetilde{H}/\widetilde{d})/4}.$ In the laboratory, even solitary waves can dissipate. If the wave height is measured far offshore and used as an initial condition for non-dissipative numerical models, the comparisons will be less meaningful, as the solitary wave will slightly change as it propagates in the laboratory. By keeping the same relative offshore distance for defining the initial condition, meaningful comparisons are assured.

While only 10 wave gages were used in each experimental run, the generation was extremely repeatable. As experiments were repeated for each wave height, the wave gages were moved to different locations, and the same $\widetilde{H}/\widetilde{d}$ wave was generated again until a sufficient number of data points existed to resolve the entire wave profile. In Synolakis (1987) two different comparisons are presented: one is the amplitude variation at specific $x$-locations, and the second is the amplitude variation at specific $t$-times, the latter resembling the image that a photograph from the side with a large depth of field and angular range would show.

Figure 2: Time evolution of $H$ = 0.0185 initial wave over a sloping beach with $\cot \beta = 19.85$ from $t = 25$ to 65 with 10 increments. Constant depth-segment starts at $X_0 = 19.85$. While markers show experimental results of Synolakis (1986, 1987), solid lines show nonlinear analytical solution of Synolakis (1986, 1987) Experimental data is provided from $t = 30$ to 70 with 10 increments.
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Figure 3: Time evolution of $H = 0.3$ initial wave over a sloping beach with $\cot \beta = 19.85$ from $t = 10$ to 30 with 5 increments. Constant-depth segment starts at $X_0 = 19.85$. Markers show a different realization of experimental results of Synolakis (1987). Experimental data is provided from $t = 15$ to 30 with 5 increments.
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References:

Synolakis, C.E. (1986): The Runup of Long Waves. Ph.D. Thesis, California Institute of Technology, Pasadena, California, 91125, 228 pp.

Synolakis, C.E. (1987): The runup of solitary waves. J. Fluid Mech., 185, 523-545.