Is there constructive or destructive interference?

[thalassa]

Homework Statement

Two sound sources are coherent and in phase. Is there constructive or destructive interference at the midpoint between the speakers?

Homework Equations

Generally, there is constructive interference whenever the path-length difference is an integer times the wavelength.

The Attempt at a Solution

At the midpoint between the speakers, the path-length difference is zero, which corresponds to constructive interference.

However, can we apply the usual conditions for constructive and destructive interference to points between the speakers? Sound is a longitudinal wave. Consider the midpoint between the speakers and apply the superposition principle. At the midpoint, away from one speaker is toward the other. Thus, at a given time at this point, if the molecular motion due to one speaker is in one direction, the molecular motion due to the other will be in the opposite direction. Do we not, then, have destructive interference at the midpoint?

 

[Bystander]

coherent and in phase

motion due to one speaker is in one direction, the molecular motion due to the other will be in the opposite direction

"In phase" means "in phase."

 

[thalassa]

"In phase" means "in phase."

Yes, the sources are in phase. Is that a vote for constructive or destructive interference at the midpoint? According to the superposition principle, there will be no displacement at the midpoint.

 

[Bystander]

there is constructive interference whenever the path-length difference is an integer times the wavelength.

"In phase" means "in phase," and "yes, it's constructive interference" means "it's constructive interference."

 

[Bystander]

You have stated that the sources are in phase, that is, that the compressions and rarefactions of one source are matched by the compressions and rarefactions of the second source; you are interested in constructive or destructive interferences at the midpoint between the sources. The times required for travel of the compression (or rarefaction) fronts from the sources to that midpoint are equal.

 

[thalassa]

You have stated that the sources are in phase, that is, that the compressions and rarefactions of one source are matched by the compressions and rarefactions of the second source; you are interested in constructive or destructive interferences at the midpoint between the sources. The times required for travel of the compression (or rarefaction) fronts from the sources to that midpoint are equal.

Yes, the sources are in phase. That is, the diaphragms of each speaker move in a synchronized way, so that when the diaphragm of one speaker is moving outward, then so is that of the other speaker, etc.

The times required to travel to the midpoint will be the same, as you state. However, upon arriving at the midpoint, the molecular motion due to one speaker will be opposite in direction to that of the other. According to the superposition principle, then, the net molecular motion at the midpoint will be zero. This, to me, suggests destructive interference, not constructive.

 

[ehild]

Sound is a pressure wave. It is the pressure that changes periodically in space and time. In a high-pressure region, the molecules move inward, and they move outward from a place of low pressure. There are not the velocities of the molecules that add up.

http://www.physicsclassroom.com/class/sound/Lesson-1/Sound-is-a-Pressure-Wave

 

[thalassa]

Sound is a pressure wave. It is the pressure that changes periodically in space and time. In a high-pressure region, the molecules move inward, and they move outward from a place of low pressure. There are not the velocities of the molecules that add up.

http://www.physicsclassroom.com/class/sound/Lesson-1/Sound-is-a-Pressure-Wave

The superposition principle, on which wave interference is based, makes a definitive statement about the net displacement of the medium when two or more waves are present. Namely--The net displacement of the medium at a particular point and time is the sum of the displacements of the individual waves. Unless we are to abandon the superposition principle, we must start with describing the displacements of the molecules and from that deduce the pressure variations.

 

[HallsofIvy]

I'm not sure what your point is but the answer is very simple- if the waves are in phase and the distance to the two sources are equal then the maxima from the two sources will arrive at the same time so this is constructive interference as Bystander said.

 

[ehild]

The superposition principle, on which wave interference is based, makes a definitive statement about the net displacement of the medium when two or more waves are present. Namely--The net displacement of the medium at a particular point and time is the sum of the displacements of the individual waves. Unless we are to abandon the superposition principle, we must start with describing the displacements of the molecules and from that deduce the pressure variations.

Sound wave is considered a scalar wave - that of pressure. But you can consider it as the displacement of the molecules. Displacement is a vector quantity, and equal displacement means both equal magnitude and direction. If the waves are in phase the air molecules move in the same direction at both source.
The question is, how you define sources in phase. If you have two tuning forks as sources, you have to hit them from the same direction to get waves in phase. So one will move towards the centre and the other one away from it.

 

[thalassa]

I'm not sure what your point is but the answer is very simple- if the waves are in phase and the distance to the two sources are equal then the maxima from the two sources will arrive at the same time so this is constructive interference as Bystander said.

My point is that the answer is not simple. I agree that "maxima" will arrive at the midpoint at the same time, but suppose that the wave amplitude is 5 cm. In this case, when two maxima arrive simultaneously, one corresponds to a 5 cm displacement in one direction, while the other corresponds to a 5 cm displacement in the opposite direction. According to the superposition principle, then, there is a net displacement of zero at the midpoint.

 

[thalassa]

Sound wave is considered a scalar wave - that of pressure. But you can consider it as the displacement of the molecules. Displacement is a vector quantity, and equal displacement means both equal magnitude and direction. If the waves are in phase the air molecules move in the same direction at both source.
The question is, how you define sources in phase. If you have two tuning forks as sources, you have to hit them from the same direction to get waves in phase. So one will move towards the centre and the other one away from it.

Coherent, in-phase sources is standards physics terminology. For sound speakers, it means that the diaphragms oscillate with the same frequency and those oscillations are temporally in phase.

 

[thalassa]

The following video, I think, is evidence that supports my contention that there is, in fact, destructive interference at the midpoint, not constructive interference:

http://video.mit.edu/watch/sound-wave-interference-8396/

 

[HallsofIvy]

My point is that the answer is not simple. I agree that "maxima" will arrive at the midpoint at the same time, but suppose that the wave amplitude is 5 cm. In this case, when two maxima arrive simultaneously, one corresponds to a 5 cm displacement in one direction, while the other corresponds to a 5 cm displacement in the opposite direction. According to the superposition principle, then, there is a net displacement of zero at the midpoint.

No, that's not two "maxima". That's one maximum and one minimum. That is not "in phase" that is "180 degrees out of phase"! "In phase", means that maxima "leave the source" at the same time and so will reach the midpoint at the same time and minima reach the midpoint at the same time. At the midpoint you will have a wave up to 10 cm and down to -10 cm.

 

[berkeman]

My point is that the answer is not simple. I agree that "maxima" will arrive at the midpoint at the same time, but suppose that the wave amplitude is 5 cm. In this case, when two maxima arrive simultaneously, one corresponds to a 5 cm displacement in one direction, while the other corresponds to a 5 cm displacement in the opposite direction. According to the superposition principle, then, there is a net displacement of zero at the midpoint.

The amplitude of the sound wave is not measured in cm. It is measured in units of pressure. That is what you are missing. The longitudinal sound wave is alternating high and low pressure areas, that move away from the source. At the midpoint between the facing, in-phase speakers, the high pressure points add to higher pressure as they pass the midpoint, and the low pressure areas make an even lower pressure area as they pass the midpoint.

Does that help clear up your confusion?

 

[berkeman]

See for example: https://en.wikipedia.org/wiki/Sound_pressure

.

 

[thalassa]

The amplitude of the sound wave is not measured in cm. It is measured in units of pressure. That is what you are missing. The longitudinal sound wave is alternating high and low pressure areas, that move away from the source. At the midpoint between the facing, in-phase speakers, the high pressure points add to higher pressure as they pass the midpoint, and the low pressure areas make an even lower pressure area as they pass the midpoint.

Does that help clear up your confusion?

Respectfully, not at all. Why can't I choose to measure the amplitude of a sound wave in cm rather than rather than pressure units? The answer is that I can, and the superposition principle is framed in terms of displacements, not pressures. There is no superposition principle for pressure, so we must always begin by describing the displacements, then deduce the pressure variations.

 

[berkeman]

Respectfully, not at all. Why can't I choose to measure the amplitude of a sound wave in cm rather than rather than pressure units? The answer is that I can, and the superposition principle is framed in terms of displacements, not pressures. There is no superposition principle for pressure, so we must always begin by describing the displacements, then deduce the pressure variations.

Sorry, that is just plain wrong and non-physical. Tell you what, if you are convinced, go ahead and put down as your answer on this problem that the sound waves cancel out at the mid-point between the speakers. You will find out that it is the wrong answer. And it's because the sound wave intensity is measured in units of pressure, not displacement of anything.

Just because the speaker cone moves to create the alternating pressure wave, does not mean that the wave propagates with the same displacement of air molecules...

 

[Dale]

Closed pending moderation.

For what it's worth, why ask a question for which you believe you already know the answer and about which you are unwilling to consider the alternative.

 

[berkeman]

OP's question has been answered as best as we can. Thread will remain closed.

 

[SammyS]

Coherent, in-phase sources is standards physics terminology. For sound speakers, it means that the diaphragms oscillate with the same frequency and those oscillations are temporally in phase.

Yes "in-phase" is standard physics terminology.

If these sources are speakers, that are facing each other, then the issue of them being in-phase has everything to do with the type of wave under consideration.

Assuming, as you have stated in post #7, so that when the diaphragm of one speaker is moving outward, then so is that of the other speaker, then we have the following results.

Considering the wave to be sound as fluctuations in air pressure:

Both speakers produce high pressure simultaneously and low pressure simultaneously, thus they are said to be in phase with each other.
At the midway point there is constructive interference. The fluctuation in pressure is enhanced over what it would be due to either speaker individually.​

Considering the wave as a longitudinal wave involving the displacement of the medium (air) or its velocity:

While one speaker pushes air to the right, the other speaker simultaneously pushes air to the left. In this sense, the speakers are out of phase. Moreover they're 180° out of phase.
At the midway point, the air is stationary, as you have stated. We have a node and destructive interference, just as we expect with sources out of phase.​

There is no contradiction here. All of this has everything to due with the details of what you are considering the wave phenomena to consist of, as my good colleague ehild has pointed out.