Let's be specific and use a simpler example than what I actually need to solve. $$ \begin{split} x(t) &= t + A\sin(wt) \\ y(t) &= B \cos(wt) \end{split} $$ How would I obtain the derivative of x in y? My maths are pretty rusty :(
I'd like to be able to derive Gerstner wave normal equations by myself, and that would help. Any good book I could find the answer in (the theory) would also be welcome!
Hint: Use
$$\frac{dy}{dx}=\frac{dy}{dt}.\frac{dt}{dx}$$
If you want to find: $\frac{dx}{dy}$, Then use
$$\frac{dx}{dy}=\frac{dx}{dt}.\frac{dt}{dy}$$
We have that
$$t=\frac1w\arccos\frac yB$$
Also, $$\sin(wt)=\sqrt{1-\frac{y^2}{B^2}}$$ for $t\in[0,\frac{\pi}w]$.
Therefore, $$x=\frac1w\arccos\frac yB+A\sqrt{1-\frac{y^2}{B^2}}$$
Now you can differentiate in the usual way.
