With a function such as
$$y = \frac{1}{8}\sin(\pi x)$$
How can the function be modified so that the wavelength increases by 2 each period?
[Math] Sine Waves with Increasing Wavelength
geometrylinear algebratrigonometry
Related Solutions
Start with a "triangle" function of period $2 \pi/n$, minimum value $0$ and maximum value $\pi/n$: a suitable choice is $T_n(t) = \frac{1}{n} \arccos(\cos(n t))$. A regular $n$-gon of inradius $r_0$ is given, in polar coordinates, by $r = r_0/\cos(T_n(\theta - \theta_0))$. Its $y$ coordinate is then $$y = \dfrac{r_0 \sin(\theta)}{\cos(T_n(\theta - \theta_0))}$$
Here is a similar animation using a pentagon ($n=5$):
Hint:
You can use an equation in polar coordinates of the form: $$ r=R+a\sin(n\theta) $$ or $$ r=R+a\cos(n\theta) $$ that represent an oscillation around a circle of radius R with $n$ oscillations of amplitude $a$.
See here for an exemple.
Best Answer
If $$ f(x)=-1+\sqrt{1+4x} $$ then $$ f\left(n^2+n\right)=2n $$ Therefore, the "period" of $$ \frac18\sin(\pi f(x)) $$ increases by $2$ since $\left(n^2+n\right)-\left(n^2-n\right)=2n$: