| The fractal dimension of measured ocean wave profiles is found to be in the range of 1.5–1.8. This non-integer dimension indicates the fractal nature of the waves. Standard formulations to analyze waves are based on a differential approach. Since fractals are non-differentiable, this formulation fails for waves with fractal characteristics. Integral solutions for long waves that are valid for a nondifferentiable fractal surfaces are developed. Field observations show a positive correlation between the fractal dimension and the degree of nonlinearity of the waves, wave steepness, and breaking waves. Solutions are developed for a variety of linear cases. As waves propagate shoreward and become more nonlinear, the fractal dimension increases. The linear solutions are unable to reproduce the change in fractal dimension evident in the ocean data. However, the linear solutions do demonstrate a finite speed of propagation.; The correlation of the fractal dimension with the nonlinearity of the waves suggests using a nonlinear wave equation. We first confirm the nonlinear behavior of the waves using the finite difference method with continuous function as the initial condition. Next, we solve the system using a Runge-Kutta method to integrate the characteristics of the nonlinear wave equation. For small times, the finite difference and Runge-Kutta solutions are similar. At longer times, however, the Runge-Kutta solution shows the leading edge of the wave extending beyond the base of the wave corresponding to over-steepening and breaking.; A simple long wave solution on multi-step bottom is developed in order to calculate the reflection coefficient for a sloping beach. Multiple reflections and transmissions are allowed at each step, and the resulting reflection coefficient is calculated. The reflection coefficient is also calculated for model with thousands of small steps where the waves are reflected and transmitted once over each step. The effect of depth-limited breaking waves is also considered. |
