Standing Waves: Synchronization between a Tube & a Stick

[Const@ntine]

Homework Statement

A wooden stick, part of a musical instrument, which produces a musical sound when hit, oscillates by creating a transverse standing wave, with three antinodes and two nodes (3 "valleys", 2 "ground levels"). The lowest note has a frequency of f = 87.0 Hz, and is produced by a stick of L = 0.4 m

a) Find the speed of the transverse waves in that stick.

b) A vertical tube is hanging bellow the centre of the stick and boosts the intensity of the sound. If only the top part is open, how much of the tube's length must synchronize with the stick from (a) ?

Homework Equations

f = n*λ/4L, n =1,3,5,7,9...

The Attempt at a Solution

a) Okay, so the Standing Wave has the form of the Picture bellow:

So, 4*(λ/4) = L <=> λ = 0.4 m
v = λ*f
f = 87.0 Hz
___________________________
v = 34.8 m/s

b) Alright, I'm completely lost here. From what I'm getting, the tube follows the one end open/one end closed model. But doesn't the stick, that is inserted into the tube essentially "close" the open part as well? Even if that wasn't the case, all the formulas I know have to do with just a tube/stick/string, meaning that in this case where I have two kinds of material, I don't really know what to do.

Furthermore, I don't really get how I'm supposed to approach this. It's not like I can put in the frequency and the speed to find the length or anything. Also, as far as "synchonization" goes, my book has no exercises on the subject, and just a small paragraph that explains the phenomenon, no formulas or anything. What I know about synchronization comes from the osciallation part, which is widly different from the waves, and has no bearing at this exercise. I'm really at a loss here.

Any help is appreciated!

 

[scottdave]

From the description, it sounds something like a xylophone bar (the stick) with attachment points at the two nodes. If the stick is horizontal, and the hollow tube sits below it, to resonate with the frequency and make it sound louder.
I don't think the stick is inside or in any way blocking up the tube opening.

 

[Const@ntine]

From the description, it sounds something like a xylophone bar (the stick) with attachment points at the two nodes. If the stick is horizontal, and the hollow tube sits below it, to resonate with the frequency and make it sound louder.
I don't think the stick is inside or in any way blocking up the tube opening.

It says it's a "Mariba", so I assume the tube it's talking about is the thing underneath the stick:

I have no idea wha to do here though. I went to the chapter about tubes bellow sticks, but the theory doesn't fit. The problem says that the tube hangs bellow the centre, whereas in the chapter it says that the tube has to move to achieve different standing waves. But if the tube is in the centre/middle fo the stick, the standing wave does not fit the shape of the one in the problem (3 antinodes, 2 nodes).

 

[scottdave]

It says it's a "Mariba", so I assume the tube it's talking about is the thing underneath the stick:
The problem says that the tube hangs bellow the centre, whereas in the chapter it says that the tube has to move to achieve different standing waves. But if the tube is in the centre/middle fo the stick, the standing wave does not fit the shape of the one in the problem (3 antinodes, 2 nodes).

The tube does not have to move to have resonance within it. Take a look at this video, by Steve Mould.

Your vibrating marimba bar is the source, like the speaker at the end of his tube.

 

[Const@ntine]

The tube does not have to move to have resonance within it. Take a look at this video, by Steve Mould.

Your vibrating marimba bar is the source, like the speaker at the end of his tube.

I focused on that, and tackled it this way: The stick is justa source of sound, right? So, I just have sound being inserted into a tube with one end open, one end closed. Right away, I know that I will have to use this formula: f = nv/4L, n = 1,3,5,7,9,...

Now, here's where I get stuck. See, the book gives me the result "0.986 m", the question being "how much of the tube's length must resonate with the stick". To get that result, I need to put in n = 1, v =343 m/s & f = 87.0 Hz. I understand that we use the speed of sound, because the stick is essentially just a source, and what the wave that travels inside the tube does so through the air, not some other material. As for the frequency, it's the frequency the stick has, so if I want the two of them to resonate, that's what I should have in the tube as well, right? What perplexes me is the n. In my book, for n = 1, the wave in the closed tube has 1 node and 1 antinode. But the wave in the stick has 3 antinodes and 2 nodes. If they are to resonate, shouldn't the two waves have the same shape?

 

[scottdave]

The mode with which the tube resonates, will not be the same as the bar, but the wave, which has bounced off the bottom and come back to the top should reinforce the antinode to make it stronger.

 

[Const@ntine]

The mode with which the tube resonates, will not be the same as the bar, but the wave, which has bounced off the bottom and come back to the top should reinforce the antinode to make it stronger.

Okay, that makes sense. But why couldn't the tube have, dunno, 3 antinodes, or 5, for example? In each case, the "end point" of the wave at the open part is always an antinode. Why does n have to be 1, thus producing just one antinode at the open part, and a node at the bottom/closed part? The book doesn't specify anything, which is why I'm asking.