Standing Waves

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

Standing Waves

The term standing wave is often applied to a resonant mode of an extended vibrating object. The resonance is created by constructive interference of two waves which travel in opposite directions in the medium, but the visual effect is that of an entire system moving in simple harmonic motion. The sketches illustrate the fundamental and second harmonic standing waves for a stretched string.

Displacement and Pressure

The standing waves associated with resonance in air columns have been discussed mainly in terms of the displacement of air in the columns. They can also be visualized in terms of the pressure variations in the column. A node for displacement is always an antinode for pressure and vice versa, as illustrated below. When the air is constrained to a node, the air motion will be alternately squeezing toward that point and expanding away from it, causing the pressure variation to be at a maximum. This view of resonant modes in terms of pressure waves makes it easier to see why the mouthpiece end of a wind instrument is a node for the resonances. For example, the clarinet is acoustically a closed-end cylindrical air column because the mouthpiece end acts as a pressure antinode. An oboe is induced to produce its upper register by opening a hole near the mouthpiece, releasing pressure to make that point a pressure node and therefore a displacement antinode.

Pressure and Displacement, Air Column

The standing waves associated with resonance in air columns have been discussed mainly in terms of the displacement of air in the columns. They can also be visualized in terms of the pressure variations in the column. A node for displacement is always an antinode for pressure and vice versa, as illustrated below. When the air is constrained to a node, the air motion will be alternately squeezing toward that point and expanding away from it, causing the pressure variation to be at a maximum. This view of resonant modes in terms of pressure waves makes it easier to see why the mouthpiece end of a wind instrument is a node for the resonances. For example, the clarinet is acoustically a closed-end cylindrical air column because the mouthpiece end acts as a pressure antinode.

One way to demonstrate standing waves in an air column is by stroking a metal rod to set up the longitudinal standing wave in the rod. If a disc is placed on the end of the rod, it can set up standing waves in the air column. The motion of the air at the antinodes is sufficient to move cork dust to produce a pattern in the dust. The illustration at left is part of a Kundt's tube designed to produce the standing waves. Kundt tube video