Superposition and Reflection

When multiple waves occupy the same place, the medium does not choose one wave and ignore the others. The visible disturbance is the superposition of all individual waves:

That means wave amplitudes add point by point, and two waves can amplify or cancel one another locally.

Boundaries as Constraints

Now consider waves near a boundary. An incident wave is the wave arriving at the boundary. A reflected wave is the part sent back into the original medium. A transmitted wave is the part that continues into the next medium, if the next medium can respond.

Although the particular physics that cause reflection and transmission (i.e. medium-dependent forces) may differ, it is convenient to reason about the so-called boundary conditions. These are local demands placed on the total wave. For example, the displacement must be zero at a fixed end of string. The reflected wave therefore comes back inverted so the incident and reflected displacements cancel exactly at this fixed end.

Air Boundaries

Like strings, air-filled tubes (flutes, PVC pipes, pipe organs etc.) support similar wave phenomena. A full description of what individual air molecules are actually doing is rather complex, but there are two macroscopic quantities that are convenient to describe:

• Pressure tells us how compressed (more dense) or rarefied (less dense) the air is.
• Displacement tells us how the air, when treated as a fluid, is moving back and forth.

The two descriptions are related, but they do not peak in the same place. Pressure is about crowding: if neighboring regions of air move toward each other, the air between them compresses; if they move apart, it rarefies. Displacement is about how regions of air have shifted from equilibrium. In a traveling sound pulse, the crowding shows up slightly ahead of the largest displacement.

When starting, pick whichever is more intuitive; in the long run both descriptions give equivalent results. For example, a closed pipe wall stops the air motion. This means a closed end is a displacement node and a pressure antinode. An open pipe end does almost the opposite: the pressure variation near the opening stays close to the outside atmospheric pressure, so the open end is a pressure node and a displacement antinode.

It is tempting to think an open end should simply let the wave leave. Some energy really does radiate outward as sound. But an open end is still a sharp change from a narrow guided air column to a large outside space.

From Reflections to Standing Waves

When a source keeps driving the same system, the incident and reflected waves overlap again and again. Most driving frequencies result in a mess of partial reinforcement and cancellation. Special frequencies make the phase line up so the total pattern repeats in place. Those are the standing waves and harmonics on the next page.