I am investigating the path of a surfer who, starting at a point on the shore, is trying to paddle to a certain point directly in front of him (perpendicularly to the shore) while a current pushes perpendicularly on him. I have found the equation of the path he will take, which is in fact the same as is answered to the following question: Name that curve! It is a curve that looks like this and here is its equation:
$y(x) = x\sinh(\sinh^{-1}(y_0)+k\ln(x))$
Now I was wondering whether I could somehow take into account the waves periodically crashing on the surfer. Instead of going in a smooth path, he would follow a similar route except every couple meters get pushed backwards. Any idea how I could mathematically show this, would I incorporate some sine function to make it cyclical?
I would greatly appreciate any ideas and feedback.