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Definition/Summary
Why is a sound wave reflected at the open end of a pipe? Why doesn't it just "escape"?
Because when a pulse expands, conservation of momentum combined with constancy of speed requires a reaction pulse of "opposite mass" to be created in the "backward" direction.
So a high density pulse emerging from the open end of a pipe produces a low density reaction pulse back into the pipe, and vice versa, and similarly a wave emerging produces a reaction wave 180º out of phase.
Equations
Extended explanation
Sound is a pressure difference:
Sound in air is a traveling longitudinal pressure difference (unlike, for example, sound in a violin string, which is a transverse displacement).
A pulse of sound is a moving region of either higher or lower pressure than average.
A moving high-pressure pulse has more mass than the surrounding air, and so carries momentum in the direction in which it is moving, but a moving low-pressure pulse has less mass, and so carries momentum in the opposite direction.
Effectively, a high-pressure pulse has positive mass, and therefore "forward" momentum, but a low-pressure pulse has negative mass, and therefore "backward" momentum.
Reflection of a single pulse:
First consider a single pulse (not a wave).
At a closed end of a pipe, a pulse of high pressure is reflected as a pulse of high pressure: there is no conservation of momentum, and the pulse "bounces" exactly as a ball bounces off a wall.
At an open end, a pulse of high pressure is reflected as a pulse of low pressure (and vice versa) … a 180º "change in phase".
This is because when a pulse leaves the pipe, it spreads out spherically.
Before spreading out, all the energy of the pulse was associated with momentum in the "forward" direction.
But after spreading out, some of the energy is associated with "sideways" momentum.
However, total "forward" momentum must be conserved.
In an ordinary "explosion" of something moving forwards, in which mass is conserved, there is a similar spreading of velocities of the parts, but momentum is conserved by some parts going forward faster after the explosion than before.
Since sound travels at a constant speed (the speed of sound in that medium at that temperature), it is impossible for the loss of total "forward" momentum to be made up by a change in speed, and the only alternative is for negative "effective mass" to be created.
This is achieved by creation of a low pressure pulse (also traveling at the speed of sound) in the backward direction: compared with the average air density, this constitutes a negative mass, whose "backward" velocity gives it a positive "forward" momentum.
In other words, when a "positive-mass" pulse spreads out at the open end of a pipe, conservation of momentum combined with constancy of speed produces a "negative-mass" pulse back down the pipe (and vice versa).
Reflection of a wave:
The principle is the same for a wave, which is basically a regular series of pulses, and for which momentum is conserved by continuous production of a reflection wave, 180º out of phase.
For a lot of detail, including a rather good animation, see this Australian site: http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html#time
(Another way of looking at it is that the open end of the pipe is at high pressure, and the air beyond it is at ordinary pressure, so it behaves like a boundary between two fluids of different pressure, and so a reflection is to be expected.
)
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
Why is a sound wave reflected at the open end of a pipe? Why doesn't it just "escape"?
Because when a pulse expands, conservation of momentum combined with constancy of speed requires a reaction pulse of "opposite mass" to be created in the "backward" direction.
So a high density pulse emerging from the open end of a pipe produces a low density reaction pulse back into the pipe, and vice versa, and similarly a wave emerging produces a reaction wave 180º out of phase.
Equations
Extended explanation
Sound is a pressure difference:
Sound in air is a traveling longitudinal pressure difference (unlike, for example, sound in a violin string, which is a transverse displacement).
A pulse of sound is a moving region of either higher or lower pressure than average.
A moving high-pressure pulse has more mass than the surrounding air, and so carries momentum in the direction in which it is moving, but a moving low-pressure pulse has less mass, and so carries momentum in the opposite direction.
Effectively, a high-pressure pulse has positive mass, and therefore "forward" momentum, but a low-pressure pulse has negative mass, and therefore "backward" momentum.
Reflection of a single pulse:
First consider a single pulse (not a wave).
At a closed end of a pipe, a pulse of high pressure is reflected as a pulse of high pressure: there is no conservation of momentum, and the pulse "bounces" exactly as a ball bounces off a wall.
At an open end, a pulse of high pressure is reflected as a pulse of low pressure (and vice versa) … a 180º "change in phase".
This is because when a pulse leaves the pipe, it spreads out spherically.
Before spreading out, all the energy of the pulse was associated with momentum in the "forward" direction.
But after spreading out, some of the energy is associated with "sideways" momentum.
However, total "forward" momentum must be conserved.
In an ordinary "explosion" of something moving forwards, in which mass is conserved, there is a similar spreading of velocities of the parts, but momentum is conserved by some parts going forward faster after the explosion than before.
Since sound travels at a constant speed (the speed of sound in that medium at that temperature), it is impossible for the loss of total "forward" momentum to be made up by a change in speed, and the only alternative is for negative "effective mass" to be created.
This is achieved by creation of a low pressure pulse (also traveling at the speed of sound) in the backward direction: compared with the average air density, this constitutes a negative mass, whose "backward" velocity gives it a positive "forward" momentum.
In other words, when a "positive-mass" pulse spreads out at the open end of a pipe, conservation of momentum combined with constancy of speed produces a "negative-mass" pulse back down the pipe (and vice versa).
Reflection of a wave:
The principle is the same for a wave, which is basically a regular series of pulses, and for which momentum is conserved by continuous production of a reflection wave, 180º out of phase.
For a lot of detail, including a rather good animation, see this Australian site: http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html#time
(Another way of looking at it is that the open end of the pipe is at high pressure, and the air beyond it is at ordinary pressure, so it behaves like a boundary between two fluids of different pressure, and so a reflection is to be expected.
)* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!