Disclaimer: I am not a "maths guy" and someone else might produce a better explanation.

Step 1: Figure out how the sine wave maps to your physical dimensions.

For instance, let's say we have a 2D sidescroller and you want the object to move like in that picture with +X = right and +Y = up. Then the direct equations are:

positionX = x + cX

positionY = amplitude * sin(x * 2 * pi / wavelength + phase) + cY

(cX, cY) is an arbitary start offset.

Now, as you can see, this expresses the positionY of the object as a function of x which can be either the x coordinate on the screen (in case you want to render it), or something else.

If you want to use time as the independent variable, you need to substitute:

x = speedX * time

positionX = speedX * time + cX

positionY = amplitude * sin(speedX * time * 2 * pi / wavelength + phase) + cY

The equations above are great if you can exactly position the object (rather than move it relative to some undefined position, as is more common in games). In that case, you to take the derivative of the above equations with respect to time to produce equations for velocity. Let's do that:

velocityX = speedX

velocityY = amplitude * cos(speedX * time * 2 * pi / wavelength + phase) * speedX * 2 * pi / wavelength

Then you can do integration with delta time in each frame, a basic (crude) way is:

positionX += velocityX * deltaTime

positionY += velocityY * deltaTime

In reality, you might not want to move the object exactly along the X-axis. In that case, you transform the velocity by a rotation matrix to make the object move in another direction.

Might also add that the complicated looking equation for velocityY can be simplified if you're satisfied with just tweaking values without needing to understand the units, i.e:

velocityY = A * cos(F * time + phase)

In this case, just find good looking values for A, F and phase.