Waves As Disturbances That Superpose

A wave is a traveling disturbance in some physical quantity. In a water wave that quantity is the surface displacement from equilibrium. In a sound wave it is the air pressure and density relative to their equilibrium values. The disturbance moves, and energy moves with it, but the material does not flow along with the wave in the same way a thrown object moves through space.

In a typical mechanical wave, each bit of the medium oscillates about an equilibrium position. For a stretched string, a small segment moves mainly up and down while the wave pattern moves along the string. For sound in air, a small parcel of air oscillates back and forth while regions of compression and rarefaction move through the room. This separation matters because it stops you from treating waves as lumps of matter that collide.

A wave model becomes predictively useful when the medium has a linear response. Linear response means the restoring effects inside the medium are proportional to the size of the disturbance. Double the displacement and the restoring force per unit mass doubles, so the resulting acceleration doubles too. In that regime, the wave equation is linear and disturbances can pass through each other without permanently changing shape.

Nonlinear response is the opposite regime. The restoring effects are not proportional to disturbance size, so large disturbances change the local wave speed or generate harmonics. The consequence for this lesson is simple. Linear waves superpose cleanly, nonlinear waves do not.

Superposition is point by point

The superposition principle says that when two or more waves overlap, the net disturbance at a given position and time equals the algebraic sum of the individual disturbances at that same position and time. The word same does the work here. You add what each wave contributes at the identical location, at the identical instant.

To state it with symbols, let $y(x,t)$ be the transverse displacement of a string from equilibrium at position $x$ in metres and time $t$ in seconds. If two waves would separately produce $y_1(x,t)$ and $y_2(x,t)$, and the medium stays in the linear regime, then the combined disturbance is

$$y(x,t)=y_1(x,t)+y_2(x,t).$$

This equation describes overlap, not collision. The waves do not bounce off each other in the linear model. They pass through, and once they no longer overlap, each continues as if the other had not been there.

Limiting case check. If wave 2 is absent, then $y_2(x,t)=0$ everywhere, so $y(x,t)=y_1(x,t)$. The rule reduces to the single wave you started with.