Waves along fractal coastlines: From fractal arithmetic to wave equations

TL;DR

This paper develops an intrinsic wave equation model for wave propagation along fractal coastlines, specifically using fractal arithmetic on Koch-type curves, without replacing the fractal with a continuum.

Contribution

It introduces the first intrinsic wave equation formulation directly on a fractal curve, avoiding continuum approximation.

Findings

Successfully formulated and solved a wave equation on a fractal coastline

Demonstrated wave propagation intrinsic to the fractal structure

Pioneered a new approach to modeling waves on fractal geometries

Abstract

Beginning with addition and multiplication which are intrinsic to a Koch-type curve, I formulate and solve a wave equation that describes wave propagation along a fractal coastline. As opposed to the examples known from the literature I do not replace the fractal by the continuum in which it is embedded. This seems to be the first example of a truly intrinsic description of wave propagation along a fractal curve.