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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 $ $rad$. 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 $ \pi/4 $ $rad$ 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.

Update: From comments, it appears that the question is not clear. So, for more clarity, a rough image is uploaded at http://imgur.com/l0cY9. In the image, four lines are drawn to clearly distinguish between crests and troughs. Look at it and read the above question once again. Thanks.

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Let $P(x,y)$ be a point on your desired wave, and let $M$ be its midpoint. Write the desired function of the distance $|PM|$. –  barto Oct 30 '12 at 17:11
Do you mean something like the curve given by the polar equation $r=1+a \sin \theta$? –  Matthew Conroy Oct 30 '12 at 18:30
@Matthew: assuming we're interpreting the quedtion correctly, wouldn't it be $\sin 4\theta$? –  Javier Badia Oct 30 '12 at 18:41
@NeerajTuteja: Try a smaller $a$, such as $a=0.25$: wolframalpha.com/input/… –  Blue Oct 31 '12 at 7:24
I don't see the distinction you're making; polar and Cartesian coordinates are simply different ways of expressing the same graph. For example, you can substitute $r = \sqrt{x^2+y^2}$ and $\sin 4\theta=4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta=(4x^3y-4xy^3)/r^4$ into the polar equation to get $4axy(x^2-y^2)=(x^2+y^2)^2(\sqrt{x^2+y^2}-1)$, which is the same graph in Cartesian form. –  Rahul Nov 1 '12 at 20:53
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up vote 0 down vote accepted

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.

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Apology for counter-commenting. Irrespective of the location of center, your solution is not pointing to a right answer, does it? –  Neeraj T Oct 30 '12 at 17:50
@NeerajTuteja Your comment is not telling me where your are stuck, is it? Maybe you are not familiar with polar coordinates, maybe you do not really know what you mean by "wrapping around the circle", maybe you have trouble switching to Cartesian coordinates, how should I know when you just tell me my answer must be wrong? –  Phira Oct 30 '12 at 22:00
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