Hold a steel water bottle under a running tap and listen. A hollow note comes from the bottle, and it climbs steadily as the bottle fills. Just before the water reaches the top, the note is high and thin. The tap and the stream of water have not changed. Only the air left in the bottle has changed.
Standing waves in air columns
A guitar string can vibrate in a standing wave, as the chapter Standing Waves on a String showed. A column of air inside a pipe or a tube can also carry standing sound waves. Sound travels along the pipe, reflects at its ends and interferes with itself. The chapter Interference and Reflection of Sound describes those reflections.
When the air column is driven at the right frequency, its standing wave grows large. This is resonance, exactly as on a string. Resonance in an air column is the physical basis of every wind instrument, from a huge pipe organ to a simple flute. The resonant frequencies are set by the length of the air column. Crucially, they are also set by what happens at its two ends.
An air column resonates when it is driven at one of its resonant frequencies. Its sound reflects at both ends and builds a large standing wave. The resonant frequencies depend on the length of the column and on whether each end is closed or open.
A sound wave is longitudinal. So in a pipe the air moves to and fro along the pipe, not across it. Every layer of air oscillates about its own place. A node is a layer that never moves, and an antinode is a layer that swings most. The words are the same as on a string. Only the direction of the motion has changed.
A pipe open at both ends sounds its lowest note. The air inside is drawn as columns of particles. Which columns will swing the most?
The figure above shows the fundamental standing wave in a pipe open at both ends, drawn with the air particles. When played, every column oscillates about its own position. The columns at the open ends, marked A, move most. The column at the centre, marked N, does not move at all. No column travels along the pipe, so the wave stands in place.
Watch the air around the centre column. The columns on its two sides move towards it together, and then away from it together. So the air at the centre is squeezed and then stretched in turn. The centre does not move, yet its pressure changes the most. The next topic explains why.
The dashed curves drawn inside a pipe are a graph. Their width at each point is the amplitude of the air there. The air itself does not bulge towards the walls. It moves only along the pipe, as the particles show.
Boundary conditions for sound waves in pipes
For a longitudinal sound wave inside a pipe, the boundaries are the ends of the pipe. The behaviour of the air molecules at these ends decides where nodes and antinodes can form. These rules are the boundary conditions of the air column.
The closed end
A closed end is a solid barrier. The air molecules there are not free to move back and forth. Therefore a closed end must always be a displacement node.
The same end is a pressure antinode. When the air nearby moves towards the wall, it piles up against the wall, and the pressure rises. When the air moves away, the pressure falls. So the pressure swings most at the wall.
This matches what the chapter Propagation of Sound Waves found. The pressure variation is \(\Delta P = -B\,\partial s/\partial x\), where \(s\) is the displacement of the air and \(B\) is the bulk modulus. Where \(s\) is always zero, its slope \(\partial s/\partial x\) is largest. So a displacement node is a pressure antinode. Displacement and pressure are \(90°\) apart, in a pipe as in a travelling wave.
At a closed end of an air column, there is always a displacement
The open end
An open end is exposed to the atmosphere, where the pressure stays nearly constant. This means that an open end must be a pressure node. To keep the pressure constant there, the air molecules must be free to move with the largest amplitude. Therefore an open end must always be a displacement antinode.
Think of the column of air that reaches the mouth of the pipe. Beyond the mouth lies the whole atmosphere. Air pushed out of the pipe spreads away at once, and air drawn in comes from all around. Nothing crowds the air at the mouth, so its pressure cannot build up. Its motion, on the other hand, is not held back at all.
Press play on Fig 1.1 and watch the two painted columns. The column next to the wall hardly moves, and the displacement graph stays pinned to zero there. The column at the open end swings widely, and the pressure graph stays pinned to zero there. Each end fixes one quantity at zero, and the other quantity then swings most.
A closed end of a pipe is a displacement node and a pressure antinode. An open end of a pipe is a displacement antinode and a pressure node.
At the open end of a sounding pipe, the variation of pressure is
A tiny microphone responds to changes of pressure. A pipe closed at one end sounds its lowest note. Where should the microphone be placed for the loudest signal?
Which statements about the ends of a sounding pipe are true?
Why must the open end of a pipe be a displacement antinode?
The open end meets the atmosphere, whose pressure stays nearly constant, so the pressure variation there must be zero: a pressure node. The pressure variation is \(\Delta P = -B\,\partial s/\partial x\), so the slope of the displacement is zero at the open end. Where the displacement has zero slope it is at its largest, so the air swings most there: a displacement antinode.
Now return to the singing bottle. The water surface acts as a closed end, because the air cannot move into the water. The mouth of the bottle is an open end. As the water rises, the air column between them gets shorter. A shorter column holds a standing wave of shorter wavelength. The speed of sound is unchanged, so the frequency rises, and so does the note.
In 1866 the German physicist August Kundt laid a glass tube on its side and sprinkled a fine powder along its floor. He closed one end with a stopper. A metal rod carrying a light disc reached into the other end. Stroking the rod made it ring, and the disc drove the air. At resonance the powder danced where the air swung most and was swept into small heaps where the air stayed still. The heaps marked the displacement nodes, half a wavelength apart. By measuring their spacing, Kundt found the speed of sound in gases and in the rod itself.
- Treating both ends of a pipe like the tied ends of a string, as nodes.Only a closed end is a displacement node. An open end is a displacement antinode, because the air there is free to move.
- Thinking the pressure is largest where the air moves most.Pressure follows the slope of the displacement, not its size. It swings most at a displacement node, such as a closed end.
- Reading the curves drawn inside a pipe as the shape of the air.They are a graph of the amplitude. The air moves only along the pipe.