14.2 The Principle of Superposition

Superposition is local in space and time. You do not add wave amplitudes globally once; you add displacements point-by-point while overlap exists. After overlap, each pulse continues with its original shape and speed in ideal linear media. This is the core contrast with colliding particles, which exchange momentum and do not pass through each other unchanged.

The word algebraic is essential. Opposite signs can cancel and same signs can reinforce. If two equal pulses with opposite displacements perfectly overlap, instantaneous displacement can be zero everywhere, but that does not mean energy disappears permanently. It is redistributed in the oscillatory field and pulse evolution.

Constructive overlap occurs when displacements share sign at a point, increasing magnitude. Destructive overlap occurs when signs oppose, reducing magnitude. Most exam sketches ask for an intermediate overlap shape, so the fastest workflow is to draw each pulse lightly and sum their heights at several reference points.

Linear superposition also underpins interference patterns later in this chapter. Two-slit bright and dark fringes are not a separate principle; they are repeated superposition outcomes across many observation points with systematically varying phase difference.

At a fixed end, boundary displacement must remain zero, so the reflected pulse inverts (180 degree phase shift). At a free end, boundary force condition differs and reflection can occur without inversion. These boundary phase rules become important in standing-wave formation in the next chapter.

A useful mental model is to enforce the boundary condition first, then infer reflected sign. If the endpoint cannot move, the reflected contribution must cancel incident displacement at that point. If the endpoint can move freely, cancellation is not required in the same way.