a standing wave

Wiggle a jump rope at just the right rhythm and a striking thing happens: the rope settles into a pattern that does not travel along it at all. Instead it stays put, with certain points pinned motionless and others swinging up and down with maximum swing. That frozen-in-place, breathing pattern is a standing wave.

Mathematically a standing wave is a solution of the wave equation that separates into a fixed shape in space times an oscillation in time: u(x,t) = sin(n pi x / L) cos(omega t). The space factor sets the shape once and for all; the time factor just makes the whole shape grow and shrink, swelling up and flipping over, again and again. The points where the space factor is zero are the nodes — they never move; the points of largest swing are the antinodes. Crucially, a standing wave is not a fundamentally new thing: it is exactly two equal travelling waves, one going left and one going right, added together. Where they reinforce you get antinodes; where they always cancel you get nodes. The travelling parts move, but their sum appears to stand still.

Standing waves are the visible form of the normal modes, and they are everywhere music and resonance live: the vibrating string of a violin, the air column of a flute or organ pipe, the skin of a drum, the electromagnetic field in a laser cavity or microwave oven (the cold spots in a microwave are its nodes), even the electron orbitals of an atom. A standing wave is what you get when waves are trapped between boundaries and only certain frequencies fit cleanly — which is why a bounded system rings at discrete pitches rather than a continuous smear.