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For example, if we consider a rope that can move in a pipe such that it can have

  • both ends free to move (Case 1)
  • one end free and one end fixed (Case 2)
  • both ends fixed (Case 3).

Each of these cases is slightly different because the free or fixed end determines whether a node or anti-node will form when a standingwave is created in the rope. These are the main restrictions when we determine the wavelengths of potential standing waves. These restrictions are known as boundary conditions and must be met.

In the diagram below you can see the three different cases. It is possible to create standing waves with different frequencies andwavelengths as long as the end criteria are met.

The longer the wavelength the less the number of anti-nodes in the standing waves. We cannot have a standing wave with no anti-nodes becausethen there would be no oscillations. We use n to number the anti-nodes. If all of the tubes have a length L and we know the end constraints we can find the wavelength, λ , for a specific number of anti-nodes.

One node

Let's work out the longest wavelength we can have in each tube, i.e. the case for n = 1 .

Case 1 : In the first tube, both ends must be anti-nodes, so we must place onenode in the middle of the tube. We know the distance from one anti-node to another is 1 2 λ and we also know this distance is L. So we can equate the two and solve for the wavelength:

1 2 λ = L λ = 2 L

Case 2 : In the second tube, one end must be a node and the other must be ananti-node. Since we are looking at the case with one node, we are forced to have it at the end. We know the distance from onenode to another is 1 2 λ but we only have half this distance contained in the tube. So :

1 2 1 2 λ = L λ = 4 L

Case 3 : Here both ends are closed and so we must have two nodes so it isimpossible to construct a case with only one node.

Two nodes

Next we determine which wavelengths could be formed if we had two nodes. Remember that we are dividing the tubeup into smaller and smaller segments by having more nodes so we expect the wavelengths toget shorter.

Case 1 : Both ends are open and so they must be anti-nodes. We can have twonodes inside the tube only if we have one anti-node contained inside the tube and one on each end. This means we have 3 anti-nodes in thetube. The distance between any two anti-nodes is half a wavelength. This means there is half wavelength between the left sideand the middle and another half wavelength between the middle and the right side so there must be one wavelength inside the tube. The safestthing to do is work out how many half wavelengths there are and equate this to the length of the tube L and then solve for λ .

2 ( 1 2 λ ) = L λ = L

Case 2 : We want to have two nodes inside the tube. The left end must be anode and the right end must be an anti-node. We can have one node inside the tube as drawn above. Again we can count the number ofdistances between adjacent nodes or anti-nodes. If we start from the left end we have one half wavelength between the end and the nodeinside the tube. The distance from the node inside the tube to the right end which is an anti-node is half of the distance to anothernode. So it is half of half a wavelength. Together these add up to the length of the tube:

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 10 physical science. OpenStax CNX. Aug 29, 2011 Download for free at http://cnx.org/content/col11245/1.3
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