Gerstner waves

A few months ago I was interested in recreating the Gerstner algorithm to recreate ocean waves in without going into fluid simulation.

To undestand it better I decided to start with graphing it in Desmos, and so I did.
Gerstner waves are parametric functions, which means that the function's x and y depend both on a third parameter, which in our case is goning to be time.

The parameter t is time, k is a constant related to the wavelength and c is another constant that involves gravity. Finally a is another constant that regards the steepnes of the wave, or how sharp they are.

Here you can find two different Gerstner waves with different values of k, c and a.
The blue curve is the result of the addition of the previous two. You can see that the blue one get it's highest crest when the black and the orange are as aligned as they can be. This is a clear example of  the constructive and destructive interference properties of waves.

As a final exarcise I tried to find the tangent and normal vectors for a given point along the curve, which can be acomplished by getting the derivatives of those funcitons at a ceratin point:

 

 

Have a great rest of your day,