Standing wave-organ pipe

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Homework Statement


An organ pipe is 2.5 m long and has a cross-sectional area of 30 cm^2. The pipe is open at one end and closed at the other. The equation of the displacement s of an air molecule from its equilibrium position is
s=(0.5 cm) sin (pi x) cos (900pi t) where x is the distance from the closed end in meters, and t is the time in seconds starting with a time when its displacements are maximum.
(i) What is the average speed of an air molecule at a displacement antinode?
(ii) What would be the wave speed of a sound wave that travelled down an organ pipe of infinite length containing air at the same temperature and pressure as in the pipe containing the stationary wave described above?
(iii) What is the lowest possible frequency of a stationary wave in the organ pipe described above?

(iv) Once you know all constant characteristics of a given gas, which of the following variable properties, by itself, determines the speed of sound waves in that gas: pressure, temperature, volume, density? Justify your answer.

Homework Equations


k=2 π/ λ
w= 2 π f
v=λf
2A sin kx is the maximum antinode where x = (2k+1) (λ/4)


The Attempt at a Solution


For i) I think at the displacement antinode, the speed of a particle will be 0 because it's the maximum displacement or it could be the greatest average vertical speed?
ii) Not sure for when it says infinite length? We can find λ and f and use v=fλ?
iii) f=v/L? Not sure when it says described above whether its referring to part i or ii?
c) Either pressure or density-not sure
 
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Answers and Replies

  • #2
haruspex
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at the displacement antinode, the speed of a particle will be 0 because it's the maximum displacement
An antinode is the minimum of whatever, so a displacement antinode is zero displacement.
We can find λ and f and use v=fλ?
Yes.
f=v/L?
What do standing waves look like in half open pipes? Are the ends nodes, antinodes, or one of each? What's the longest wavelength in a pipe length L consistent with that?
Either pressure or density-not sure
What formulae do you know for the speed of sound in a gas?
 
  • #3
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Isn't an antinode maximum displacement and node is zero displacement?

In half open pipes, there's a node at one end and an antinode at the open end.

v= sqrt of (γRT / M) or v = sqrt of (γ P/ρ)
 
  • #4
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Also if L = 2.5 and λ=2 then L = (5/4)λ so we know which harmonic this is and we can find the fundamental frequency by using fn = n f1
(λ = 2π/k = 2π/π=2)
 
  • #5
haruspex
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Isn't an antinode maximum displacement and node is zero displacement
Whoops, sorry - you are right.
In half open pipes, there's a node at one end and an antinode at the open end.
Yes, so what's the smallest number or fraction of a wavelength that might be present?
v= sqrt of (γRT / M) or v = sqrt of (γ P/ρ)
Ok, so ignore all the terms in there which relate to inherent properties of the gas. What's left?
 
  • #6
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Also if L = 2.5 and λ=2 then L = (5/4)λ so we know which harmonic this is and we can find the fundamental frequency by using fn = n f1
(λ = 2π/k = 2π/π=2)

So what will the average speed at the anitnode be? Will it just be zero? I know average speed is the total distance/time taken and I know that the x position of the antinode is given by 2A sin(kx) so solving that between bounds gives x = (2k+1)(λ/4)? But it says at the antinode which kind of means like an instantaneous speed?

Pressure, volume and temperature are inherent properties of gas ? So density?
 
  • #7
haruspex
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Also if L = 2.5 and λ=2
But why should the wavelength be 2m? (It isn't.)

So what will the average speed at the anitnode be? Will it just be zero?
Yes.
 
  • #8
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We know that λ=2π / k and k is given to us by the standing wave equation 2A sin(kx) cos (wt) so we can calculate λ

Edit: "The average speed of each particle is not the same at one cycle. The anti-node will be the fastest as it travels the farthest in one cycle" found from https://en.wikibooks.org/wiki/IB_Physics/Oscillations_and_Waves
 
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  • #9
haruspex
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We know that λ=2π / k and k is given to us by the standing wave equation 2A sin(kx) cos (wt) so we can calculate λ
Question iii isasking about the lowest possible frequency. The wave equation will be different.
 
  • #10
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But it says of the same stationary wave as described above? 5λ/4 is a different harmonic and we have to find the fundamental harmonic and we can do that using fn=nfo where we know fn and n?

5λ/4 is the fifth harmonic so fn=n fo so fo = fn / n = 450 / 5 = 90

I got f=450 using w= 2πf where w is 900 in the given standing wave equation.

Also, I don't know why but I have a feeling the average speed won't be 0 at the displacement antinode? The word average is bugging me.

Also, I think density isn't an inherent property of gases?
 
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  • #11
haruspex
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But it says of the same stationary wave as described above?
No, it says
in the organ pipe described above
The word average is bugging me.
What's got me bothered is the word speed. Don't forget that air molecules move around even without sound waves passing through.
Also, I think density isn't an inherent property of gases?
Right, and neither are pressure, mass, number of moles, nor temperature. Is there an equation for velocity which only uses one non-inherent property?
 
  • #12
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But then again, they're all referring to the same standing wave in part b and c? How will we find a wave equation for an infinitely long tube?

These are the only 2 equations I know of for ideal gases:
v= sqrt of (γRT / M) or v = sqrt of (γ P/ρ)
 
  • #13
haruspex
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But then again, they're all referring to the same standing wave in part b and c? How will we find a wave equation for an infinitely long tube?
No, ii) is for a travellng wave, not a standing wave at all.
What does a generic equation for a travelling wave look like?

iii) is for a different standing wave from that originally described.
What does the lowest frequency note look like in a half open tube, in terms of nodes and antinodes?

These are the only 2 equations I know of for ideal gases:
One of those equations will do it. I'm not asking you to find a different equation.
Which of them only includes one non-inherent property?
 
  • #14
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ii) y =A sin(kx-wt)

iii)One node and one antinode

v= sqrt of (γRT / M)
 
  • #15
haruspex
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ii) y =A sin(kx-wt)
Yes. Anyway, you already answered this question correctly, so no need to go any further with that.
iii)One node and one antinode
Right, so what is that in terms of wavelengths?
v= sqrt of (γRT / M)
Right. So which is the only non-inherent property in there?
 
  • #16
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ii) so v is 900 m/s?

iii) λ/4

Temperature
 
  • #17
haruspex
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ii) so v is 900 m/s?

iii) λ/4

Temperature
Yes to all three.
So what frequency do you get for iii)?

Returning to i), do you understand what I wrote about air molecules in still air, with no sound waves? What average speed would they have? What average velocity would they have?
 
  • #18
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I know f = v / λ/4 but now I'm not so sure what λ is going to be? Is it 2 m as I said before?

No I didn't get that part.
 
  • #19
haruspex
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I know f = v / λ/4 but now I'm not so sure what λ is going to be? Is it 2 m as I said before?
No.
You know the length of the tube. How does that relate to the wavelength, according to your post #16?
No I didn't get that part.
Do you understand the difference between speed and velocity?
What would be the average velocity of a molecule of still air?
 
  • #20
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L = λ/4 and I can solve for this yes but we're not considering it as infinite as in part ii? That's what I was confused about.

I do know the difference:speed magnitude only and velocity magnitude and direction.
av. Velocity=displacement/time. It would just be zero if it's still? Unless there's some direction change?
 
  • #21
haruspex
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L = λ/4 and I can solve for this yes but we're not considering it as infinite as in part ii?
I agree it could be worded more clearly, but in iii) it says stationary wave, so we must be discussing a pipe of finite length.
av. Velocity=displacement/time. It would just be zero if it's still?
Yes. How will it be different within a stationary sound wave?
 
  • #22
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Well at the nodes the particles are not moving at all but at the antinodes the particles are moving about a mean position but since they're coming back to the same position, their displacement would be 0 I presume but they are still covering some distance which I is why I was wondering why the average speed is 0. And then they mentioned at a particular point which is the antinode and that makes it instantaneous and in this case it should be 0.
 
  • #23
haruspex
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Well at the nodes the particles are not moving at all but at the antinodes the particles are moving about a mean position but since they're coming back to the same position, their displacement would be 0 I presume but they are still covering some distance which I is why I was wondering why the average speed is 0. And then they mentioned at a particular point which is the antinode and that makes it instantaneous and in this case it should be 0.
You are right to conclude that the average velocity will be zero regardless of the sound wave and whether the molecules are at nodes, antinodes, or wherever.
But as I noted, the question as you stated it asks for average speed. What is the average speed of a molecule of air, under everyday conditions, in still air, no sound waves?
 
  • #24
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It's around 500m/s I think
 
  • #25
haruspex
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It's around 500m/s I think
Yes, something like that. So how do you think you should answer the question? I am not sure whether they really meant speed or intended to ask for velocity.
 

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