How Are Nodes and Antinodes Located in Standing Waves Between Two Speakers?

[Gear300]

Two speakers are driven in phase by a common oscillator at 800 Hz and face each other at a distance of 1.25 m. Locate the points along a line joining the two speakers where relative minima of sound pressure amplitude would be expected (Use v = 343 m/s).

This problem ended up screwing my mind until I found out that the waves are supposed to be, as the problem said, in phase. I was able to get the answer, which are .518 m, .303 m, .0891 m, .732 m, .947 m, and 1.16 m from either speaker. However, in the solutions book, the way they did it was by stating that the point halfway between the speakers (.625 m) is an antinode of pressure, and since the distance between adjacent nodes is .214 m and adjacent node-antinodes is .214/2 m, they used .625 - .214/2 = .518 m to find a pressure node. Then they added/subtracted .214 m to get the other values. The statement about the midpoint being a pressure antinode began to screw my mind. They didn't prove anything for it, they stated it as seemingly a postulation out of nowhere...how do they know the midpoint is an antinode?

 

[Gear300]

d-_-b

 

[LowlyPion]

Two speakers are driven in phase by a common oscillator at 800 Hz and face each other at a distance of 1.25 m. Locate the points along a line joining the two speakers where relative minima of sound pressure amplitude would be expected (Use v = 343 m/s).

This problem ended up screwing my mind until I found out that the waves are supposed to be, as the problem said, in phase. I was able to get the answer, which are .518 m, .303 m, .0891 m, .732 m, .947 m, and 1.16 m from either speaker. However, in the solutions book, the way they did it was by stating that the point halfway between the speakers (.625 m) is an antinode of pressure, and since the distance between adjacent nodes is .214 m and adjacent node-antinodes is .214/2 m, they used .625 - .214/2 = .518 m to find a pressure node. Then they added/subtracted .214 m to get the other values. The statement about the midpoint being a pressure antinode began to screw my mind. They didn't prove anything for it, they stated it as seemingly a postulation out of nowhere...how do they know the midpoint is an antinode?

The point that is equidistant means that the sound arrives from each speaker at the same time, if they are in phase. Shouldn't you expect a maximum at that point of twice the amplitude? The sound from one speaker 1/4 shy of the mid point added to the one from the other side that has been traveling 1/4 longer then is 1/2 out of phase isn't it?

 

[Gear300]

But how do we know it is not a node?

 

[LowlyPion]

But how do we know it is not a node?

It can't be a node. At the mid point there is by definition constructive interference because the sound is in phase and equidistant from the 2 sources. Destructive interference must be 180 degrees out of phase.

 

[Gear300]

O_O! Good point you make...but to make sure of things...this requires that if a wave immediate to one speaker is, say, compressional, then the wave immediate to the other speaker is also compressional, right?

 

[LowlyPion]

O_O! Good point you make...but to make sure of things...this requires that if a wave immediate to one speaker is, say, compressional, then the wave immediate to the other speaker is also compressional, right?

How would they be in phase otherwise?

 

[Gear300]

I see...Thanks for the clarification