How Do Sound Waves Interfere in Dual Speaker Setups?

[chinnie15]

Homework Statement

Two speakers are emitting identical sound waves with a wavelength of 4.0 m. The speakers are 8.0 m apart, directed toward each other. At each of the noted points (a-e) in the above diagram (at 2,3,4,5, and 6 meters, respectively), is the interference constructive, destructive, or somewhere in between?

Homework Equations

Δr=|r₂-r₁|

The Attempt at a Solution

Ok, so, mathematically, I understand this with no problem. I know that a, c, and e are constructive and that b and d are destructive, based off the formula.

For example, for a:
r₁= 2m → .5λ and r₂= 6m → 1.5λ
Δr= |1.5λ-.5λ|= 1λ. It's a whole wavelength, so it's constructive.

I get that. What I don't understand, is how exactly the waves appear on the picture. The image we're given has the line without the wave included (attached if interested). They are equally spaced and have the same exact wavelength, so wouldn't that mean that the waves perfectly overlap? I am having the hardest time imagining how these two waves will look. When I draw them both out, they are both literally identical (they go one right on top of the other). What am I failing to see? Sorry if this is a stupid question, but I am not so great at seeing these sort of things sometimes.

Thank you for any help!

 

[Simon Bridge]

They are equally spaced and have the same exact wavelength, so wouldn't that mean that the waves perfectly overlap? I am having the hardest time imagining how these two waves will look. When I draw them both out, they are both literally identical (they go one right on top of the other).

The waves should not be identical - they should have the same phase at their respective sources and travel in opposite directions.

It may help you see if you sketch cosine waves instead of sine waves.
It may also help if you repeat the sketch for several different times so you see how the resulting superposition evolves over time. What kind of wave should you have?

 

[TSny]

Look here and scroll down to Two sine waves traveling in opposite directions create a standing wave and see if that helps.

 

[chinnie15]

Ohhh, ok. I think I get it now (points a, c, and e are constantly flipping between constructive interference up top or at the bottom, while points b and d are never constructive). Thank you so much!

 

[Nickie]

I can understand your confusion and it is completely normal to have difficulty visualizing wave interference. Let me try to explain it in a different way.

Think of each speaker as a source of waves. These waves travel towards each other and when they meet, they interfere with each other. At certain points, the crests and troughs of the two waves align, resulting in constructive interference, where the amplitude of the resulting wave is increased. This is what is happening at points a, c, and e in the diagram.

On the other hand, at points b and d, the crests of one wave align with the troughs of the other, resulting in destructive interference, where the amplitude of the resulting wave is decreased. This is why the lines are closer together at these points.

In between these points, the interference is somewhere in between constructive and destructive, resulting in a varying amplitude of the resulting wave.

I hope this helps to clarify the concept of wave interference. It is important to remember that the waves are not actually overlapping or combining, they are simply passing through each other and the resulting wave is a combination of the two. Keep practicing and you will become better at visualizing wave interference.