Please Read Everything Fully and Carefully Before Responding!!
I'm trying to formulate parametric equations for a sine wave where the wavelength grows by a constant on alternate sides, in this case, 1.3, and where the amplitude decreases in an inverse manner. (See my image. It MUST be noted that the image probably does NOT look like the actual, properly graphed wave will!)
To give you an idea of the sort of equations i'm looking for:
$$y(t)=(function)$$
$$x(t)=sin(t)(function)^{-1}$$
NOTE. I'm using the term 'wavelength' in a way that is, to my understanding not in conformity with the standard definition. But, the above image should clarify what I'm talking about satisfactorily.
NOTE. A note to everyone: I'd rather deal with this problem in the form that is given by my example equations, unless there is some issue with doing so…
Best Answer
There's a lovely function $y = f(x)$ which is simply turned into a parametric equation by defining it as $\langle t, f(t) \rangle$.
Therefore, $$y = \cos\left(\pi\left(\frac{\ln\Big(\big(x+\frac1{r+1}\big)(r^2-1)+1\Big)}{\ln r}\right)\right)$$
For your case, $r=1.3$.
This function matches the above graph, where it starts at $(0,-1)$. If you prefer a different starting location, please drop a comment and I'll be back with an improved answer!
$$y = \cos\left(\pi\left(\frac{\ln\Big(\big(x+\frac{r^{(n)}-1}{r^-1}\big)(r^2-1)+1\Big)}{\ln r}\right)\right)$$ See the tiny $n$ in the exponent? Mess with that, and you have phase shift. You can also drop a negative in front of the cosine, or change it to sine, or etc. Just to preempt any questions.
Alrighty!
$$\langle\frac{r^{2t}-r^{n}}{r^{2}-1},r^{\left(\frac{r^{2t}-r^{n}}{r^{2}-1}\right)}\cos\left(2\pi t\right)\rangle$$