Resonance and air columns open at one end and closed at the other.

[Spookie71]

Homework Statement

The question I'm working on is

What is the third longest wavelength that resonates within an air column 33cm in length, open at one end and closed at the other.

I actually have the answer to this one as it is a sample question in the textbook however I'm not clear on 3 resonances and the diagrams drawn for them and the eventual equation to solve.

First the diagram:http://http://i215.photobucket.com/albums/cc10/Spookie71/image0-12.jpg

Why is this considered 3 resonances, I believe I see the first two with each completed with a node. Then there is the start of another which goes for about a 1/4 of a wavelength and ends up as an antinode at the end of the column.

There is the matter of the equation I use to solve the equation:

l = \frac{(2n - 1)\lambda}{4}

The n = the number of resonances but what does the -1 / 4 stand for, the 1/4 wavelength extending from the node to the antinode at the end of the column?

Can you please explain for me?

Scott

 

[Anadyne]

A resonance is not a part of the diagram. The entire diagram is of one particular resonance.

In order to draw a resonating wave, all you have to do for this situation (open and free ends) is make sure that the node is at the closed end and the antinode is at the free end. The first resonance refers to the situation where there's half a full loop. (Top one in this picture: http://www.umanitoba.ca/faculties/arts/linguistics/russell/138/sec4/sh2-15.gif )

The second resonance refers to the situation where there's a two nodes and two antinodes (one loop and a half). This is the second way of making sure that there's a node at one end and an antinode at the third end.

Here's an analogy:
You're a physics teacher and your boss tells you to make a line of students. He INSISTS that the first person be male and the last person be female. The simplest way of doing this is to make a line using one male student and one female student. This is your first solution.
If he comes back and says "Don't be lazy! Find a different way!" then you can put one male, another student, and one female in a line. This would be your second solution.

If you think of the first person being male and the last being female that's like having the ends open and closed. If those weren't the case then the waves would just go crazy and not resonate (not keep the node positions and whatnot). The first solution would be the first resonance and the second solution would be the second solution, etc etc.

As for lengths, I would never use a formula. I'd much rather draw it and figure out the ratio between the length and the wavelength.

 

[Spookie71]

Anadyne

Thank you for your quick response. You actually helped clear things up for me quite a bit. Where you said.

The second resonance refers to the situation where there's a two nodes and two antinodes (one loop and a half)

I took a look at my diagrams and sure enough that's how it goes for example three nodes and three antinodes would refer to a third resonance.

I'm sure quite obvious to you, but this simple hint cleared things up for me. My text never explained that. However it's probably so obvious lol that It was assumed.

Do you no where I could do some more reading about Resonances on the web in terms of open and closed air columns.

again Thanks
Scott