Strings are not described very accurately in popular science, because much of the physics of strings was only understood long after the mathematical theory was somewhat advanced, and an accurate classical analog for the string wasn't available until relatively recently.
The classical analog people often use is a vibrating band of energy, but this is mostly wrong, because strings don't interact by bumping into each other, like rubber bands do. They can only interact in a strange way determined by the consistency of having infinitely many different particles, all conspiring to make a consistent theory. This conspiracy makes it that when strings merge, they do a complicated thing, and the only correct classical analog to this complicated thing is a black hole merger. This wasn't understood very well until at least the late 1990s, so pop-sci has not caught up.
Ordinary astrophysical black holes are big, and complicated, and black, and gooey. They are electrically resistive, their oscillations decay quickly, they are very irreversible in the statistical sense. The string describes the case that the black hole is extended and charged so much that it is in the non-viscous extremal limit, so that strings are not gooey and lossy like ordinary black holes, but shiny and reversible. The one dimensional shiny black holes are the strings, in the limit that these black holes are weakly interacting, so that the lightest ones are light, and consequently weakly charged (because extremality relates charge and mass).
When black holes collide, the oscillations do not combine in a simple way, they combine the way black holes combine, in a strange acausal way that only makes sense teleologically. So you can't say "this pattern made this pattern", at least not in a causal way precisely, you need to describe the whole thing at once. The ripples running along the string doesn't add to the ripple running along another string when they combine, but if the two combine at high energies to make a long string with many ripples running in a statistical way, so that it has a classical interpretation, they combine to an object which is gooey because it has a lot of junk running around on it, and this is no different than a viscous liquid. When the strings cool down again by shooting out other cold strings, the oscillations die down. The laws of combination are not like superposing waves on a pond, but they are described over the entire space-time world-sheet.
Strings are more elementary than black holes, they are small and simple. You shouldn't be intimidated by the above to thinking that the laws of string collisions require you to understand the classical collisions of black holes! This is no more true than saying that describing a collision in the Born approximation in quantum mechanics requires you to solve the much more difficult classical problem of particles scattering around in that potential. The quantum behavior is simpler than the classical behavior in many cases.
The laws of string collisions, for those strings which are lightest at the lowest energies, that is for those distance scales which reproduce our experience, are extremely simple: they just reproduce the laws of Feynman diagrams, so that the particles combine into other particles the same way they do in the standard model. On the string world sheet, these laws of particle combination are the laws of algebraic products of operators, they describe how to expand a product of two operators in an infinite series of a third operator. This process takes a limit where the points of merger of the incoming particles are smooshed close together by a conformal transformation, so that each of their oscillations is no longer distinct, but merged into a combined oscillation a long long time ago (the collision limit is not really a short-distance limit, but a long-time limit). The combined oscillation just looks like an infinite series of particles in the theory, an infinite series of operators on the world sheet which create the oscillation corresponding to sending in one of these particles from infinity.
This is just like a Taylor expansion, except for fluctuating quantities, and the operator product laws are the laws of string merger and vibration-adding. You can't make nothing, because you always have a world sheet, and the addition law is strange, not by the laws of superposition (like water waves, or oscillations on rubber bands) but by the rules of operator product expansion.
