Equation of sine wave around a circle

Consider a sine wave having $4$ cycles wrapped around a circle of radius 1 unit (its center needs not be the origin of a Cartesian coordinate system). Assume that the length of axis of the sine wave is as same as the circumference of the circle.

The circumference of circle is assumed to be mapped to $2\pi\ \mathrm{rad}$. Therefore, the sine wave represents the equation along the $x$-axis:

$$ y = \sin(4x) $$

To find the equation of the sine wave with the circumference of circle acting as the $x$-axis, one approach is to consider the sine wave along a rotated line like aligned $\frac\pi4\ \mathrm{rad}$ to $x$-axis. But it doesn’t suffice for the circular path. This is where the problem of finding the equation is stuck. A hint/help taking to a right answer would be appreciated.

For more clarity, here is a rough image. In the image, four lines are drawn to clearly distinguish between crests and troughs.

Solutions Collecting From Web of "Equation of sine wave around a circle"

Do it first for the circle centered at the origin in polar coordinates.

Then switch do Cartesian coordinates, then shift to the actual center of the circle.

it should be, in cartesian coordinates

x = (R + a · sin(n·θ)) · cos(θ) + xc

y = (R + a · sin(n·θ)) · sin(θ) + yc

where

R is circle’s radius

a is sinusoid amplitude

θ is the parameter (angle), from 0 to 2π

xc,yc is circle’s center point

n is number of sinusoids on circle

you can also get a pure cartesian equation (non-parametric) on x/y, but just for half circle, solving second for sin(θ) and replacing it on first one.