A new solution to the linearized shallow-water wave equations is introduced for the case of cnoidal waves climbing up a plane beach. The solution is used to calculate the maximum runup. It is shown that the maximum relative runup of cnoidal waves is significantly larger than the runup of monochromatic waves with the same wave height and wavelength far from the shore. It is also shown that the maximum relative runup of cnoidal waves is not a monotonically varying function of the normalized wavelength.
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Synolakis et al. (1988) studied this question.
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