Parametric Equations For A Sine Wave With Changing Angle Relative To Growth / Decrease Factor

Question

I've been working to create parametric equations for a sine wave that corresponds to the expressions beneath (Please Read My Description After The Expressions, or my post may not make sense!):

If, $ $$0≤t≤ 360$

then:

$ $$x(t) = exp\left(t\frac{ln(1.4)}{360(1.2)}\right)sin(t)$

$ $$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360}\right)$

If, $ $$360≤t≤ 360(2)$

then:

$ $$x(t) = exp\left(t\frac{ln(1.4)}{360}\right)sin(t)$

$ $$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360(\frac{1}{1.2})}\right)$

If, $ $$360(2)≤t≤ 360(3)$

then:

$ $$x(t) = exp\left(t\frac{ln(1.4)}{360(\frac{1}{1.2})}\right)sin(t)$

$ $$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360(\frac{1}{1.2})^2}\right)$

If, $ $$360(3)≤t≤ 360(4)$

then:

$ $$x(t) = exp\left(t\frac{ln(1.4)}{360(\frac{1}{1.2})^2}\right)sin(t)$

$ $$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360(\frac{1}{1.2})^3}\right)$

And so on....

What I want to do can be viewed as combining the above sine waves at the given values. I can't seem to find a way of doing this myself. But, I've reached the conclusion that there are two general forms the answer could take:

1. A sine wave where the angle shifts seamlessly, reaching exactly my values for the angle at the input values given, and never changing other than at multiples of 360 i.e., full revolutions.

2. A sine wave where my angle values are reached but only at the given inputs, that is, one where they do not remain constant in-between full revolutions; a wave where they gradually decrease to my values.

Ideally, I would like to find both solutions, or even another way of combining these waves which I haven't thought of but which still corresponds to the above.

Sorry for a super long post, but I really would be very interested in the answer as I can't seem to discover it myself nor does it appear to be in any of my books or online!

Thank you guys so much for your time and talent.

Please Post Your Answers In Parametric Form!

Answer 1

These are all of the form: $$ x(t) = \exp(\alpha(t) t)\sin(t), y(t) = \exp(\beta(t) t)\sin(t) $$ where $\alpha(t)$ and $\beta(t)$ are constant within each subinterval, but jump discontinuously at the subinterval boundaries. This time dependence can be modeled using step functions. For example, we can define a function that is 1 between $t=0$ and $t=1$ and 0 outside that interval: $$ \theta(t) = \begin{cases} 1, & 0\le t \lt 1 \\ 0 & \text{otherwise} \end{cases} $$ Then if you have a piecewise constant function that changes values at $t = 0, 1, 2, 3$ and is 0 if $t < 0$ or $t >= 4$ say, you could write: $$ f(t) = f_0 \theta(t) + f_1 \theta(t-1) + f_2 \theta(t-2) + f_3 \theta(t-3) +\ ... $$ where $f_0$ is the constant value in the $0\le t \lt 1$ subinterval and similarly for the others. You can change the scale of t in order to accommodate your subintervals, so since you want your subinternvals to be e.g $0\le t \lt 360$, you could write the $\alpha(t)$ function above: $$ \alpha(t) = \alpha_0 \theta(t/360) + \alpha_1 \theta(t/360 - 1) + \alpha_2 \theta(t/360-2) + \alpha_3 \theta(t/360-3) +\ ... $$ where $$ \alpha_0 = \frac{\ln(1.4)}{1.2\cdot 360}\\ \alpha_n = ({1.2})^n \alpha_0 $$ and similarly for $\beta(t)$.

Answer 2

Maybe something like this,

$ $$x(t) = exp\left(t\frac{ln(1.4)}{360\left(\frac{1}{1.2}\right)^\frac{t}{360}}\right)sin(t)$

but I'm really unsure though