Two-Dimensional Wave Equations and Wave Characteristics

2.10 Summary

Experiments conducted in wave tanks (Wiegel, 1950; Eagleson, 1956; LeMehaute et al., 1968) give some indication of the accuracy of small-amplitude wave theory in predicting the transformation of monochromatic two-dimensional waves as they travel into intermediate and shallow water depths, and of the accuracy in predicting particle kinematics given the wave height and period and the water depth. A summary follows:

• 1. For most typical bottom slopes the dispersion equation is satisfactory for predicting the wave celerity and length up to the breaker zone.

• 2. For increasing beach slopes and wave steepnesses, the wave height predictions given by Eq. (2.42) will be lower than the real wave heights. This discrepancy increases as the relative depth decreases. As an example, on a 1:10 slope, for a relative depth of 0.1 and a deep water wave steepness of 0.02, the experimental wave height exceeded the calculated wave height by 15%.

• 3. For waves on a relatively flat slope and having a relative depth greater than about 0.1, the small-amplitude theory is satisfactory for predicting horizontal water particle velocities. At lesser relative depths the small-amplitude theory still predicts reasonably good values for horizontal velocity near the bottom, but results are poorer (up to 50% errors on the low side) near the surface.

Limitations of the small-amplitude theory in shallow water and for high waves in deep water suggest a need to consider nonlinear or finite-amplitude wave theories for some engineering applications. The next chapter presents an overview of selected aspects of the more useful finite-amplitude wave theories, as well as their application and the improved understanding of wave characteristics that they provide.