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Objects in low Earth orbit move on a wavy or sinusoidal path around Earth's surface only because that is how the path appears on a flat surface like a map.  An orbit is a complete circle around the Earth, and a spacecraft rises above and drops below the equator as it goes around to make the circle.  If you unwrap the globe to make it a flat map, the three-dimensional circle flattens out too and appears as a wavy line.  If the orbit path were drawn on a spherical globe, the line would form a complete circle around the globe.  You can see an example of a spacecraft creating a sinusoidal path over Earth’s surface via NASA’s live tracking of the International Space Station in the National Air and Space Museum’s Moving Beyond Earth exhibition or on NASA’s website.